From 54da0617a511aab4529a12bee360d8df696172bc Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 10 Oct 2025 15:54:25 -0700
Subject: [PATCH 001/134] Reorganize signed arithmetic on rational numbers

---
 src/elementary-number-theory.lagda.md         |   2 +
 ...closed-intervals-rational-numbers.lagda.md |   2 +-
 .../absolute-value-rational-numbers.lagda.md  | 255 ++++++++----------
 ...tion-nonnegative-rational-numbers.lagda.md |  10 +-
 ...ddition-positive-rational-numbers.lagda.md |  14 +-
 .../addition-rational-numbers.lagda.md        |   5 +-
 ...roperty-positive-rational-numbers.lagda.md |   4 +-
 ...imedean-property-rational-numbers.lagda.md |   8 +-
 .../difference-rational-numbers.lagda.md      |   2 +-
 .../distance-rational-numbers.lagda.md        |   2 +-
 ...ive-and-negative-rational-numbers.lagda.md | 106 ++++++++
 .../inequality-rational-numbers.lagda.md      |  11 +-
 ...imum-nonnegative-rational-numbers.lagda.md |   4 +-
 ...inima-and-maxima-rational-numbers.lagda.md |   8 +-
 ...closed-intervals-rational-numbers.lagda.md |   1 +
 ...closed-intervals-rational-numbers.lagda.md |   6 +-
 ...ication-negative-rational-numbers.lagda.md | 159 +++++++++++
 ...tion-nonnegative-rational-numbers.lagda.md |  48 +++-
 ...tion-nonpositive-rational-numbers.lagda.md |  14 +-
 ...ive-and-negative-rational-numbers.lagda.md |   9 +-
 ...ication-positive-rational-numbers.lagda.md |  12 +-
 ...icative-group-of-rational-numbers.lagda.md |   4 +-
 ...closed-intervals-rational-numbers.lagda.md |   6 +-
 .../negative-rational-numbers.lagda.md        | 227 ++++------------
 .../nonnegative-rational-numbers.lagda.md     |  98 +++----
 .../nonpositive-rational-numbers.lagda.md     |  55 ++--
 ...ive-and-negative-rational-numbers.lagda.md | 195 ++++++++++----
 .../positive-rational-numbers.lagda.md        | 104 +++----
 .../rational-numbers.lagda.md                 |  21 +-
 .../squares-rational-numbers.lagda.md         |  19 +-
 ...lity-nonnegative-rational-numbers.lagda.md |  10 -
 ...quality-positive-rational-numbers.lagda.md |   5 +-
 ...trict-inequality-rational-numbers.lagda.md |   7 +-
 ...at-bounded-distance-metric-spaces.lagda.md |   8 +-
 .../located-metric-spaces.lagda.md            |   4 +-
 .../metric-space-of-rational-numbers.lagda.md |   8 +-
 ...onal-sequences-approximating-zero.lagda.md |   1 +
 src/order-theory.lagda.md                     |   1 +
 .../order-preserving-maps-posets.lagda.md     |   6 +-
 ...rder-preserving-maps-total-orders.lagda.md | 193 +++++++++++++
 ...ition-lower-dedekind-real-numbers.lagda.md |  16 +-
 ...ition-upper-dedekind-real-numbers.lagda.md |   4 +-
 ...thmetically-located-dedekind-cuts.lagda.md |   6 +-
 .../multiplication-real-numbers.lagda.md      |   1 +
 ...lower-upper-dedekind-real-numbers.lagda.md |  16 +-
 .../negation-real-numbers.lagda.md            |   2 +-
 .../nonnegative-real-numbers.lagda.md         |   5 +-
 .../positive-real-numbers.lagda.md            |  16 +-
 .../strict-inequality-real-numbers.lagda.md   |   2 +-
 49 files changed, 1052 insertions(+), 670 deletions(-)
 create mode 100644 src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
 create mode 100644 src/order-theory/order-preserving-maps-total-orders.lagda.md

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 662f2615dc5..d6d8fc4df2d 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -93,6 +93,7 @@ open import elementary-number-theory.group-of-integers public
 open import elementary-number-theory.half-integers public
 open import elementary-number-theory.hardy-ramanujan-number public
 open import elementary-number-theory.inclusion-natural-numbers-conatural-numbers public
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers public
 open import elementary-number-theory.inequality-arithmetic-geometric-means-integers public
 open import elementary-number-theory.inequality-arithmetic-geometric-means-rational-numbers public
 open import elementary-number-theory.inequality-conatural-numbers public
@@ -137,6 +138,7 @@ open import elementary-number-theory.multiplication-integers public
 open import elementary-number-theory.multiplication-interior-closed-intervals-rational-numbers public
 open import elementary-number-theory.multiplication-lists-of-natural-numbers public
 open import elementary-number-theory.multiplication-natural-numbers public
+open import elementary-number-theory.multiplication-negative-rational-numbers public
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers public
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers public
 open import elementary-number-theory.multiplication-positive-and-negative-integers public
diff --git a/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md
index 21651d39f03..c0568e3e299 100644
--- a/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md
@@ -69,7 +69,7 @@ abstract
               ( r≤0))
             ( transitive-leq-ℚ (neg-ℚ r) (neg-ℚ p) (rational-abs-ℚ p)
               ( neg-leq-abs-ℚ p)
-              ( neg-leq-ℚ _ _ p≤r))))
+              ( neg-leq-ℚ p≤r))))
       ( λ 0≤r →
         transitive-leq-ℚ
           ( rational-abs-ℚ r)
diff --git a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
index 93b5e7586ee..fa1c1a6995c 100644
--- a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
@@ -12,12 +12,15 @@ module elementary-number-theory.absolute-value-rational-numbers where
 open import elementary-number-theory.addition-nonnegative-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -48,29 +51,27 @@ its negation.
 rational-abs-ℚ : ℚ → ℚ
 rational-abs-ℚ q = max-ℚ q (neg-ℚ q)
 
-opaque
-  unfolding neg-ℚ
-
+abstract
   is-nonnegative-rational-abs-ℚ : (q : ℚ) → is-nonnegative-ℚ (rational-abs-ℚ q)
   is-nonnegative-rational-abs-ℚ q =
     rec-coproduct
-      ( λ 0≤q →
+      ( λ is-nonpos-q →
         inv-tr
           ( is-nonnegative-ℚ)
-          ( right-leq-left-max-ℚ
-            ( q)
-            ( neg-ℚ q)
-            ( transitive-leq-ℚ (neg-ℚ q) zero-ℚ q 0≤q (neg-leq-ℚ zero-ℚ q 0≤q)))
-          ( is-nonnegative-leq-zero-ℚ q 0≤q))
-      ( λ q≤0 →
+          ( left-leq-right-max-ℚ _ _
+            ( leq-nonpositive-nonnegative-ℚ
+              ( q , is-nonpos-q)
+              ( neg-ℚ⁰⁻ (q , is-nonpos-q))))
+          ( is-nonnegative-neg-is-nonpositive-ℚ is-nonpos-q))
+      ( λ is-nonneg-q →
         inv-tr
           ( is-nonnegative-ℚ)
-          ( left-leq-right-max-ℚ
-            ( q)
-            ( neg-ℚ q)
-            ( transitive-leq-ℚ q zero-ℚ (neg-ℚ q) (neg-leq-ℚ q zero-ℚ q≤0) q≤0))
-          ( is-nonnegative-leq-zero-ℚ (neg-ℚ q) (neg-leq-ℚ q zero-ℚ q≤0)))
-      ( linear-leq-ℚ zero-ℚ q)
+          ( right-leq-left-max-ℚ _ _
+            ( leq-nonpositive-nonnegative-ℚ
+              ( neg-ℚ⁰⁺ (q , is-nonneg-q))
+              ( q , is-nonneg-q)))
+          ( is-nonneg-q))
+      ( is-nonpositive-or-nonnegative-ℚ q)
 
 abs-ℚ : ℚ → ℚ⁰⁺
 pr1 (abs-ℚ q) = rational-abs-ℚ q
@@ -92,41 +93,43 @@ abstract
   le-abs-le-le-neg-ℚ p q = le-max-le-both-ℚ q p (neg-ℚ p)
 ```
 
-### The absolute value of a nonnegative rational number is the number itself
+### The absolute value of the negation of `q` is the absolute value of `q`
 
 ```agda
-opaque
-  unfolding neg-ℚ
+abstract
+  abs-neg-ℚ : (q : ℚ) → abs-ℚ (neg-ℚ q) ＝ abs-ℚ q
+  abs-neg-ℚ q =
+    eq-ℚ⁰⁺
+      ( ap (max-ℚ (neg-ℚ q)) (neg-neg-ℚ q) ∙ commutative-max-ℚ _ _)
+
+  rational-abs-neg-ℚ : (q : ℚ) → rational-abs-ℚ (neg-ℚ q) ＝ rational-abs-ℚ q
+  rational-abs-neg-ℚ q = ap rational-ℚ⁰⁺ (abs-neg-ℚ q)
+```
+
+### The absolute value of a nonnegative rational number is the number itself
 
+```agda
+abstract
   abs-rational-ℚ⁰⁺ : (q : ℚ⁰⁺) → abs-ℚ (rational-ℚ⁰⁺ q) ＝ q
-  abs-rational-ℚ⁰⁺ (q , nonneg-q) =
+  abs-rational-ℚ⁰⁺ q =
     eq-ℚ⁰⁺
-      ( right-leq-left-max-ℚ
-        ( q)
-        ( neg-ℚ q)
-        ( transitive-leq-ℚ
-          ( neg-ℚ q)
-          ( zero-ℚ)
-          ( q)
-          ( leq-zero-is-nonnegative-ℚ q nonneg-q)
-          ( neg-leq-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ q nonneg-q))))
+      ( right-leq-left-max-ℚ _ _ (leq-nonpositive-nonnegative-ℚ (neg-ℚ⁰⁺ q) q))
 
   rational-abs-zero-leq-ℚ : (q : ℚ) → leq-ℚ zero-ℚ q → rational-abs-ℚ q ＝ q
   rational-abs-zero-leq-ℚ q 0≤q =
-    ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (q , is-nonnegative-leq-zero-ℚ q 0≤q))
+    ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (q , is-nonnegative-leq-zero-ℚ 0≤q))
 
   rational-abs-leq-zero-ℚ :
     (q : ℚ) → leq-ℚ q zero-ℚ → rational-abs-ℚ q ＝ neg-ℚ q
   rational-abs-leq-zero-ℚ q q≤0 =
-    left-leq-right-max-ℚ
-      ( q)
-      ( neg-ℚ q)
-      ( transitive-leq-ℚ
-        ( q)
-        ( zero-ℚ)
-        ( neg-ℚ q)
-        ( neg-leq-ℚ q zero-ℚ q≤0)
-        ( q≤0))
+    equational-reasoning
+      rational-abs-ℚ q
+      ＝ rational-abs-ℚ (neg-ℚ q)
+        by inv (rational-abs-neg-ℚ q)
+      ＝ neg-ℚ q
+        by
+          ap rational-ℚ⁰⁺
+            ( abs-rational-ℚ⁰⁺ (neg-ℚ⁰⁻ (q , is-nonpositive-leq-zero-ℚ q≤0)))
 ```
 
 ### The absolute value of a positive rational number is the number itself
@@ -139,16 +142,6 @@ abstract
     ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ q))
 ```
 
-### The absolute value of the negation of `q` is the absolute value of `q`
-
-```agda
-abstract
-  abs-neg-ℚ : (q : ℚ) → abs-ℚ (neg-ℚ q) ＝ abs-ℚ q
-  abs-neg-ℚ q =
-    eq-ℚ⁰⁺
-      ( ap (max-ℚ (neg-ℚ q)) (neg-neg-ℚ q) ∙ commutative-max-ℚ _ _)
-```
-
 ### `q` is less than or equal to `abs-ℚ q`
 
 ```agda
@@ -163,36 +156,31 @@ abstract
 ### The absolute value of `q` is zero iff `q` is zero
 
 ```agda
-opaque
-  unfolding neg-ℚ
-
-  eq-zero-eq-abs-zero-ℚ : (q : ℚ) → abs-ℚ q ＝ zero-ℚ⁰⁺ → q ＝ zero-ℚ
-  eq-zero-eq-abs-zero-ℚ q abs=0 =
-    rec-coproduct
-      ( λ 0≤q →
-        antisymmetric-leq-ℚ
-          ( q)
-          ( zero-ℚ)
-          ( tr (leq-ℚ q) (ap rational-ℚ⁰⁺ abs=0) (leq-abs-ℚ q)) 0≤q)
-      ( λ q≤0 →
-        antisymmetric-leq-ℚ
-          ( q)
-          ( zero-ℚ)
-          ( q≤0)
-          ( tr
-            ( leq-ℚ zero-ℚ)
-            ( neg-neg-ℚ q)
-            ( neg-leq-ℚ
-              ( neg-ℚ q)
-              ( zero-ℚ)
-              ( tr
-                ( leq-ℚ (neg-ℚ q))
-                ( ap rational-ℚ⁰⁺ abs=0)
-                ( neg-leq-abs-ℚ q)))))
-      ( linear-leq-ℚ zero-ℚ q)
+abstract
+  eq-zero-eq-abs-zero-ℚ : {q : ℚ} → abs-ℚ q ＝ zero-ℚ⁰⁺ → q ＝ zero-ℚ
+  eq-zero-eq-abs-zero-ℚ {q} abs=0 =
+    antisymmetric-leq-ℚ
+      ( q)
+      ( zero-ℚ)
+      ( transitive-leq-ℚ
+        ( q)
+        ( rational-abs-ℚ q)
+        ( zero-ℚ)
+        ( leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ abs=0))
+        ( leq-abs-ℚ q))
+      ( binary-tr
+        ( leq-ℚ)
+        ( ap (neg-ℚ ∘ rational-ℚ⁰⁺) abs=0 ∙ neg-zero-ℚ)
+        ( neg-neg-ℚ q)
+        ( neg-leq-ℚ (neg-leq-abs-ℚ q)))
 
   abs-zero-ℚ : abs-ℚ zero-ℚ ＝ zero-ℚ⁰⁺
-  abs-zero-ℚ = eq-ℚ⁰⁺ (left-leq-right-max-ℚ _ _ (refl-leq-ℚ zero-ℚ))
+  abs-zero-ℚ =
+    eq-ℚ⁰⁺
+      ( equational-reasoning
+        max-ℚ zero-ℚ (neg-ℚ zero-ℚ)
+        ＝ max-ℚ zero-ℚ zero-ℚ by ap-max-ℚ refl neg-zero-ℚ
+        ＝ zero-ℚ by idempotent-max-ℚ zero-ℚ)
 ```
 
 ### The triangle inequality
@@ -236,78 +224,53 @@ abstract
 ### `|ab| = |a||b|`
 
 ```agda
-opaque
-  unfolding neg-ℚ
-
-  abs-left-mul-nonnegative-ℚ :
-    (q : ℚ) (p : ℚ⁰⁺) → abs-ℚ (rational-ℚ⁰⁺ p *ℚ q) ＝ p *ℚ⁰⁺ abs-ℚ q
-  abs-left-mul-nonnegative-ℚ q p⁰⁺@(p , nonneg-p) with linear-leq-ℚ zero-ℚ q
-  ... | inl 0≤q =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ p *ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-rational-ℚ⁰⁺
-                ( p⁰⁺ *ℚ⁰⁺ (q , is-nonnegative-leq-zero-ℚ q 0≤q)))
-        ＝ p *ℚ rational-abs-ℚ q
-          by ap (p *ℚ_) (inv (rational-abs-zero-leq-ℚ q 0≤q)))
-  ... | inr q≤0 =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ rational-abs-ℚ (neg-ℚ (p *ℚ q))
-          by ap rational-ℚ⁰⁺ (inv (abs-neg-ℚ (p *ℚ q)))
-        ＝ rational-abs-ℚ (p *ℚ neg-ℚ q)
-          by ap rational-abs-ℚ (inv (right-negative-law-mul-ℚ p q))
-        ＝ p *ℚ neg-ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-rational-ℚ⁰⁺
-                ( p⁰⁺ *ℚ⁰⁺
-                  ( neg-ℚ q ,
-                    is-nonnegative-leq-zero-ℚ
-                      ( neg-ℚ q)
-                      ( neg-leq-ℚ q zero-ℚ q≤0))))
-        ＝ p *ℚ rational-abs-ℚ q
-          by ap (p *ℚ_) (inv (rational-abs-leq-zero-ℚ q q≤0)))
+abstract
+  rational-abs-left-mul-nonnegative-ℚ :
+    (q : ℚ) (p : ℚ⁰⁺) →
+    rational-abs-ℚ (rational-ℚ⁰⁺ p *ℚ q) ＝ rational-ℚ⁰⁺ p *ℚ rational-abs-ℚ q
+  rational-abs-left-mul-nonnegative-ℚ q p⁰⁺@(p , _) =
+    equational-reasoning
+      rational-abs-ℚ (p *ℚ q)
+      ＝ max-ℚ (p *ℚ q) (p *ℚ neg-ℚ q)
+        by ap-max-ℚ refl (inv (right-negative-law-mul-ℚ p q))
+      ＝ p *ℚ rational-abs-ℚ q
+        by inv (left-distributive-mul-max-ℚ⁰⁺ p⁰⁺ _ _)
 
   abs-mul-ℚ : (p q : ℚ) → abs-ℚ (p *ℚ q) ＝ abs-ℚ p *ℚ⁰⁺ abs-ℚ q
-  abs-mul-ℚ p q with linear-leq-ℚ zero-ℚ p
-  ... | inl 0≤p =
+  abs-mul-ℚ p q =
     eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ p *ℚ rational-abs-ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-left-mul-nonnegative-ℚ
-                ( q)
-                ( p , is-nonnegative-leq-zero-ℚ p 0≤p))
-        ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
-          by ap (_*ℚ rational-abs-ℚ q) (inv (rational-abs-zero-leq-ℚ p 0≤p)))
-  ... | inr p≤0 =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ rational-abs-ℚ (neg-ℚ (p *ℚ q))
-          by ap rational-ℚ⁰⁺ (inv (abs-neg-ℚ (p *ℚ q)))
-        ＝ rational-abs-ℚ (neg-ℚ p *ℚ q)
-          by ap rational-abs-ℚ (inv (left-negative-law-mul-ℚ p q))
-        ＝ neg-ℚ p *ℚ rational-abs-ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-left-mul-nonnegative-ℚ
-                ( q)
-                ( neg-ℚ p ,
-                  is-nonnegative-leq-zero-ℚ (neg-ℚ p) (neg-leq-ℚ p zero-ℚ p≤0)))
-        ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
-          by ap (_*ℚ rational-abs-ℚ q) (inv (rational-abs-leq-zero-ℚ p p≤0)))
+      ( rec-coproduct
+        ( λ is-nonpos-p →
+          equational-reasoning
+            rational-abs-ℚ (p *ℚ q)
+            ＝ rational-abs-ℚ (neg-ℚ (p *ℚ q))
+              by inv (rational-abs-neg-ℚ _)
+            ＝ rational-abs-ℚ (neg-ℚ p *ℚ q)
+              by ap rational-abs-ℚ (inv (left-negative-law-mul-ℚ p q))
+            ＝ neg-ℚ p *ℚ rational-abs-ℚ q
+              by
+                rational-abs-left-mul-nonnegative-ℚ
+                  ( q)
+                  ( neg-ℚ⁰⁻ (p , is-nonpos-p))
+            ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
+              by
+                ap-mul-ℚ
+                  ( inv
+                    ( rational-abs-leq-zero-ℚ
+                      ( p)
+                      ( leq-zero-is-nonpositive-ℚ is-nonpos-p)))
+                  ( refl))
+        ( λ is-nonneg-p →
+          equational-reasoning
+            rational-abs-ℚ (p *ℚ q)
+            ＝ p *ℚ rational-abs-ℚ q
+              by rational-abs-left-mul-nonnegative-ℚ q (p , is-nonneg-p)
+            ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
+              by
+                ap-mul-ℚ
+                  ( inv (ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (p , is-nonneg-p))))
+                  ( refl))
+        ( is-nonpositive-or-nonnegative-ℚ p))
 
 abstract
   rational-abs-mul-ℚ :
diff --git a/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md
index cf3e298f8bb..fa24223764b 100644
--- a/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md
@@ -35,12 +35,12 @@ nonnegative.
 
 ```agda
 opaque
-  unfolding add-ℚ
+  unfolding add-ℚ is-nonnegative-ℚ
 
   is-nonnegative-add-ℚ :
-    (p q : ℚ) → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
+    {p q : ℚ} → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
     is-nonnegative-ℚ (p +ℚ q)
-  is-nonnegative-add-ℚ p q nonneg-p nonneg-q =
+  is-nonnegative-add-ℚ {p} {q} nonneg-p nonneg-q =
     is-nonnegative-rational-fraction-ℤ
       ( is-nonnegative-add-fraction-ℤ
         { fraction-ℚ p}
@@ -50,7 +50,7 @@ opaque
 
 add-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
 add-ℚ⁰⁺ (p , nonneg-p) (q , nonneg-q) =
-  ( p +ℚ q , is-nonnegative-add-ℚ p q nonneg-p nonneg-q)
+  ( p +ℚ q , is-nonnegative-add-ℚ nonneg-p nonneg-q)
 
 infixl 35 _+ℚ⁰⁺_
 _+ℚ⁰⁺_ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
@@ -73,7 +73,7 @@ abstract
         ( q)
         ( zero-ℚ)
         ( p)
-        ( leq-zero-is-nonnegative-ℚ p nonneg-p))
+        ( leq-zero-is-nonnegative-ℚ nonneg-p))
 
   is-inflationary-map-right-add-rational-ℚ⁰⁺ :
     (p : ℚ⁰⁺) → is-inflationary-map-Poset ℚ-Poset (_+ℚ rational-ℚ⁰⁺ p)
diff --git a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
index 5b5cc12dc1b..85668704afe 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -57,7 +57,7 @@ positive.
 
 ```agda
 opaque
-  unfolding add-ℚ
+  unfolding add-ℚ is-positive-ℚ
 
   is-positive-add-ℚ :
     {x y : ℚ} → is-positive-ℚ x → is-positive-ℚ y → is-positive-ℚ (x +ℚ y)
@@ -147,9 +147,7 @@ module _
         ( rational-ℚ⁺ x)
         ( zero-ℚ)
         ( rational-ℚ⁺ y)
-        ( le-zero-is-positive-ℚ
-          ( rational-ℚ⁺ y)
-          ( is-positive-rational-ℚ⁺ y)))
+        ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ y)))
 
   le-right-add-ℚ⁺ : le-ℚ⁺ y (x +ℚ⁺ y)
   le-right-add-ℚ⁺ =
@@ -160,9 +158,7 @@ module _
         ( rational-ℚ⁺ y)
         ( zero-ℚ)
         ( rational-ℚ⁺ x)
-        ( le-zero-is-positive-ℚ
-          ( rational-ℚ⁺ x)
-          ( is-positive-rational-ℚ⁺ x)))
+        ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x)))
 ```
 
 ### The positive difference of strictly inequal positive rational numbers
@@ -268,9 +264,7 @@ abstract
         ( x)
         ( zero-ℚ)
         ( rational-ℚ⁺ d)
-        ( le-zero-is-positive-ℚ
-          ( rational-ℚ⁺ d)
-          ( is-positive-rational-ℚ⁺ d)))
+        ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ d)))
 
   le-right-add-rational-ℚ⁺ : (x : ℚ) (d : ℚ⁺) → le-ℚ x (x +ℚ (rational-ℚ⁺ d))
   le-right-add-rational-ℚ⁺ x d =
diff --git a/src/elementary-number-theory/addition-rational-numbers.lagda.md b/src/elementary-number-theory/addition-rational-numbers.lagda.md
index 8d887c25a69..0603a977979 100644
--- a/src/elementary-number-theory/addition-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-rational-numbers.lagda.md
@@ -154,8 +154,7 @@ abstract
 
 ```agda
 opaque
-  unfolding add-ℚ
-  unfolding neg-ℚ
+  unfolding add-ℚ neg-ℚ
 
   left-inverse-law-add-ℚ : (x : ℚ) → (neg-ℚ x) +ℚ x ＝ zero-ℚ
   left-inverse-law-add-ℚ x =
@@ -199,7 +198,7 @@ opaque
   distributive-neg-add-ℚ :
     (x y : ℚ) → neg-ℚ (x +ℚ y) ＝ neg-ℚ x +ℚ neg-ℚ y
   distributive-neg-add-ℚ (x , dxp) (y , dyp) =
-    ( inv (preserves-neg-rational-fraction-ℤ (x +fraction-ℤ y))) ∙
+    ( inv (neg-rational-fraction-ℤ (x +fraction-ℤ y))) ∙
     ( eq-ℚ-sim-fraction-ℤ
       ( neg-fraction-ℤ (x +fraction-ℤ y))
       ( add-fraction-ℤ (neg-fraction-ℤ x) (neg-fraction-ℤ y))
diff --git a/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md b/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md
index 3affd486cfb..458395e0a84 100644
--- a/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md
@@ -13,7 +13,6 @@ open import elementary-number-theory.archimedean-property-rational-numbers
 open import elementary-number-theory.integers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -22,7 +21,6 @@ open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
-open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
@@ -57,7 +55,7 @@ abstract
         asymmetric-le-ℚ
           ( zero-ℚ)
           ( y)
-          ( le-zero-is-positive-ℚ y pos-y)
+          ( le-zero-is-positive-ℚ pos-y)
           ( tr
             ( le-ℚ y)
             ( equational-reasoning
diff --git a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
index 59ab9240074..f2fdcf6a303 100644
--- a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
@@ -15,7 +15,7 @@ open import elementary-number-theory.integers
 open import elementary-number-theory.multiplication-integer-fractions
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.natural-numbers
-open import elementary-number-theory.positive-rational-numbers
+-- open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
@@ -42,7 +42,9 @@ of `ℚ` is that for any two
 an `n : ℕ` such that `y` is less than `n` as a rational number times `x`.
 
 ```agda
-abstract
+opaque
+  unfolding is-positive-ℚ
+
   bound-archimedean-property-ℚ :
     (x y : ℚ) →
     is-positive-ℚ x →
@@ -90,5 +92,5 @@ abstract
   exists-greater-natural-ℚ q =
     map-tot-exists
       ( λ _ → tr (le-ℚ q) (right-unit-law-mul-ℚ _))
-      ( archimedean-property-ℚ one-ℚ q _)
+      ( archimedean-property-ℚ one-ℚ q (is-positive-rational-ℚ⁺ one-ℚ⁺))
 ```
diff --git a/src/elementary-number-theory/difference-rational-numbers.lagda.md b/src/elementary-number-theory/difference-rational-numbers.lagda.md
index 45b52b7ba80..21f068f27b5 100644
--- a/src/elementary-number-theory/difference-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/difference-rational-numbers.lagda.md
@@ -183,7 +183,7 @@ abstract
     equational-reasoning
       rational-ℤ x -ℚ rational-ℤ y
       ＝ rational-ℤ x +ℚ rational-ℤ (neg-ℤ y)
-        by ap (rational-ℤ x +ℚ_) (inv (preserves-neg-rational-ℤ y))
+        by ap (rational-ℤ x +ℚ_) (inv (neg-rational-ℤ y))
       ＝ rational-ℤ (x -ℤ y)
         by add-rational-ℤ x (neg-ℤ y)
 ```
diff --git a/src/elementary-number-theory/distance-rational-numbers.lagda.md b/src/elementary-number-theory/distance-rational-numbers.lagda.md
index 8576576ca69..066898a463e 100644
--- a/src/elementary-number-theory/distance-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-rational-numbers.lagda.md
@@ -141,7 +141,7 @@ abstract
       ( equational-reasoning
         p *ℚ rational-dist-ℚ q r
         ＝ rational-abs-ℚ (p *ℚ (q -ℚ r))
-          by ap rational-ℚ⁰⁺ (inv (abs-left-mul-nonnegative-ℚ _ p⁰⁺))
+          by inv (rational-abs-left-mul-nonnegative-ℚ _ p⁰⁺)
         ＝ rational-abs-ℚ (p *ℚ q +ℚ p *ℚ (neg-ℚ r))
           by ap rational-abs-ℚ (left-distributive-mul-add-ℚ p q (neg-ℚ r))
         ＝ rational-abs-ℚ (p *ℚ q -ℚ p *ℚ r)
diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..3c35c0ff087
--- /dev/null
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -0,0 +1,106 @@
+# Inequalities between positive, negative, nonnegative, and nonpositive rational numbers
+
+```agda
+module elementary-number-theory.inequalities-positive-and-negative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.dependent-pair-types
+```
+
+</details>
+
+## Idea
+
+This page describes
+[inequalities](elementary-number-theory.inequalities-rational-numbers.md) and
+[strict inequalities](elementary-number-theory.strict-inequality-rational-numbers.md)
+between [positive](elementary-number-theory.positive-rational-numbers.md),
+[negative](elementary-number-theory.negative-rational-numbers.md),
+[nonnegative](elementary-number-theory.nonnegative-rational-numbers.md), and
+[nonpositive](elementary-number-theory.nonpositive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md).
+
+## Properties
+
+### Inequalities between rational numbers with different signs
+
+Some inequalities can be deduced directly from the sign of a rational number:
+for example, every negative rational number is less than every nonnegative
+rational number.
+
+#### Any negative rational number is less than any nonnegative rational number
+
+```agda
+abstract
+  le-negative-nonnegative-ℚ :
+    (p : ℚ⁻) (q : ℚ⁰⁺) → le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁰⁺ q)
+  le-negative-nonnegative-ℚ (p , neg-p) (q , nonneg-q) =
+    concatenate-le-leq-ℚ p zero-ℚ q
+      ( le-zero-is-negative-ℚ neg-p)
+      ( leq-zero-is-nonnegative-ℚ nonneg-q)
+
+  leq-negative-nonnegative-ℚ :
+    (p : ℚ⁻) (q : ℚ⁰⁺) → leq-ℚ (rational-ℚ⁻ p) (rational-ℚ⁰⁺ q)
+  leq-negative-nonnegative-ℚ p q = leq-le-ℚ (le-negative-nonnegative-ℚ p q)
+```
+
+#### A nonpositive rational number is less than a positive rational number
+
+```agda
+abstract
+  le-nonpositive-positive-ℚ :
+    (p : ℚ⁰⁻) (q : ℚ⁺) → le-ℚ (rational-ℚ⁰⁻ p) (rational-ℚ⁺ q)
+  le-nonpositive-positive-ℚ (p , nonpos-p) (q , pos-q) =
+    concatenate-leq-le-ℚ p zero-ℚ q
+      ( leq-zero-is-nonpositive-ℚ nonpos-p)
+      ( le-zero-is-positive-ℚ pos-q)
+```
+
+#### A nonpositive rational number is less than or equal to a nonnegative rational number
+
+```agda
+abstract
+  leq-nonpositive-nonnegative-ℚ :
+    (p : ℚ⁰⁻) (q : ℚ⁰⁺) → leq-ℚ (rational-ℚ⁰⁻ p) (rational-ℚ⁰⁺ q)
+  leq-nonpositive-nonnegative-ℚ (p , nonpos-p) (q , nonneg-q) =
+    transitive-leq-ℚ p zero-ℚ q
+      ( leq-zero-is-nonnegative-ℚ nonneg-q)
+      ( leq-zero-is-nonpositive-ℚ nonpos-p)
+```
+
+### Inequalities showing the sign of a rational number
+
+#### If `p < q` and `p` is nonnegative, then `q` is positive
+
+```agda
+abstract
+  is-positive-le-ℚ⁰⁺ :
+    (p : ℚ⁰⁺) (q : ℚ) → le-ℚ (rational-ℚ⁰⁺ p) q → is-positive-ℚ q
+  is-positive-le-ℚ⁰⁺ (p , nonneg-p) q p<q =
+    is-positive-le-zero-ℚ
+      ( concatenate-leq-le-ℚ _ _ _ (leq-zero-is-nonnegative-ℚ nonneg-p) p<q)
+```
+
+#### If `p < q` and `q` is nonpositive, then `p` is negative
+
+```agda
+abstract
+  is-negative-le-ℚ⁰⁻ :
+    (q : ℚ⁰⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁰⁻ q) → is-negative-ℚ p
+  is-negative-le-ℚ⁰⁻ (q , nonpos-q) p p<q =
+    is-negative-le-zero-ℚ
+      ( concatenate-le-leq-ℚ p q zero-ℚ
+        ( p<q)
+        ( leq-zero-is-nonpositive-ℚ nonpos-q))
+```
diff --git a/src/elementary-number-theory/inequality-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
index 39552d30824..14d1a9398e4 100644
--- a/src/elementary-number-theory/inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
@@ -340,9 +340,7 @@ module _
   where
 
   opaque
-    unfolding add-ℚ
-    unfolding leq-ℚ-Prop
-    unfolding neg-ℚ
+    unfolding add-ℚ leq-ℚ-Prop neg-ℚ
 
     iff-translate-diff-leq-zero-ℚ : leq-ℚ zero-ℚ (y -ℚ x) ↔ leq-ℚ x y
     iff-translate-diff-leq-zero-ℚ =
@@ -445,11 +443,10 @@ abstract
 
 ```agda
 opaque
-  unfolding leq-ℚ-Prop
-  unfolding neg-ℚ
+  unfolding leq-ℚ-Prop neg-ℚ
 
-  neg-leq-ℚ : (x y : ℚ) → leq-ℚ x y → leq-ℚ (neg-ℚ y) (neg-ℚ x)
-  neg-leq-ℚ x y = neg-leq-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
+  neg-leq-ℚ : {x y : ℚ} → leq-ℚ x y → leq-ℚ (neg-ℚ y) (neg-ℚ x)
+  neg-leq-ℚ {x} {y} = neg-leq-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
 ```
 
 ### Transposing additions on inequalities of rational numbers
diff --git a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
index 1bd6c1e4d2f..f7b3619d5e2 100644
--- a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
@@ -36,13 +36,13 @@ abstract
   is-nonnegative-max-ℚ⁰⁺ :
     (p q : ℚ⁰⁺) → is-nonnegative-ℚ (max-ℚ (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q))
   is-nonnegative-max-ℚ⁰⁺ (p , is-nonneg-p) (q , is-nonneg-q) =
-    is-nonnegative-leq-zero-ℚ _
+    is-nonnegative-leq-zero-ℚ
       ( transitive-leq-ℚ
         ( zero-ℚ)
         ( p)
         ( max-ℚ p q)
         ( leq-left-max-ℚ p q)
-        ( leq-zero-is-nonnegative-ℚ p is-nonneg-p))
+        ( leq-zero-is-nonnegative-ℚ is-nonneg-p))
 
 max-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
 max-ℚ⁰⁺ p⁰⁺@(p , _) q⁰⁺@(q , _) = (max-ℚ p q , is-nonnegative-max-ℚ⁰⁺ p⁰⁺ q⁰⁺)
diff --git a/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md b/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md
index 9c62ce02864..87e2c01b901 100644
--- a/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md
@@ -41,10 +41,10 @@ module _
       rec-coproduct
         ( λ p≤q →
           ( ap neg-ℚ (left-leq-right-min-ℚ p q p≤q)) ∙
-          ( inv (right-leq-left-max-ℚ _ _ (neg-leq-ℚ _ _ p≤q))))
+          ( inv (right-leq-left-max-ℚ _ _ (neg-leq-ℚ p≤q))))
         ( λ q≤p →
           ( ap neg-ℚ (right-leq-left-min-ℚ p q q≤p)) ∙
-          ( inv (left-leq-right-max-ℚ _ _ (neg-leq-ℚ _ _ q≤p))))
+          ( inv (left-leq-right-max-ℚ _ _ (neg-leq-ℚ q≤p))))
         ( linear-leq-ℚ p q)
 ```
 
@@ -61,9 +61,9 @@ module _
       rec-coproduct
         ( λ p≤q →
           ( ap neg-ℚ (left-leq-right-max-ℚ _ _ p≤q)) ∙
-          ( inv (right-leq-left-min-ℚ _ _ (neg-leq-ℚ _ _ p≤q))))
+          ( inv (right-leq-left-min-ℚ _ _ (neg-leq-ℚ p≤q))))
         ( λ q≤p →
           ( ap neg-ℚ (right-leq-left-max-ℚ _ _ q≤p)) ∙
-          ( inv (left-leq-right-min-ℚ _ _ (neg-leq-ℚ _ _ q≤p))))
+          ( inv (left-leq-right-min-ℚ _ _ (neg-leq-ℚ q≤p))))
         ( linear-leq-ℚ p q)
 ```
diff --git a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
index 761616c9dcb..a13fd9dd5b3 100644
--- a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
@@ -23,6 +23,7 @@ open import elementary-number-theory.maximum-nonnegative-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minima-and-maxima-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
+open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
diff --git a/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
index b0d806d19f3..b05bb91cc05 100644
--- a/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
@@ -9,10 +9,12 @@ module elementary-number-theory.multiplication-interior-closed-intervals-rationa
 ```agda
 open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.decidable-total-order-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.interior-closed-intervals-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
+open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
@@ -333,7 +335,7 @@ abstract
               inv-tr
                 ( is-nonnegative-ℚ)
                 ( min'=a'c')
-                ( is-nonnegative-mul-ℚ a' c' is-nonneg-a' is-nonneg-c')
+                ( is-nonnegative-mul-ℚ is-nonneg-a' is-nonneg-c')
           in
             rec-coproduct
               ( λ is-neg-a →
@@ -567,7 +569,7 @@ abstract
           ( neg-ℚ)
           ( right-negative-law-lower-bound-mul-closed-interval-ℚ [a,b] [c,d])) ∙
         ( neg-neg-ℚ _))
-      ( neg-le-ℚ _ _
+      ( neg-le-ℚ
         ( le-lower-bound-mul-interior-closed-interval-ℚ
           ( [a,b])
           ( neg-closed-interval-ℚ [c,d])
diff --git a/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..164dcab84e8
--- /dev/null
+++ b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
@@ -0,0 +1,159 @@
+# Multiplication by negative rational numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module elementary-number-theory.multiplication-negative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.identity-types
+open import foundation.transport-along-identifications
+```
+
+</details>
+
+## Idea
+
+The product of two
+[negative rational numbers](elementary-number-theory.negative-rational-numbers.md)
+is their [product](elementary-number-theory.multiplication-rational-numbers.md)
+as [rational numbers](elementary-number-theory.rational-numbers.md), and is
+[positive](elementary-number-theory.positive-rational-numbers.md).
+
+## Properties
+
+### The product of two negative rational numbers is positive
+
+```agda
+opaque
+  unfolding is-negative-ℚ
+
+  is-positive-mul-negative-ℚ :
+    {x y : ℚ} → is-negative-ℚ x → is-negative-ℚ y → is-positive-ℚ (x *ℚ y)
+  is-positive-mul-negative-ℚ {x} {y} P Q =
+    tr
+      ( is-positive-ℚ)
+      ( negative-law-mul-ℚ x y)
+      ( is-positive-mul-ℚ P Q)
+```
+
+### Multiplication by a negative rational number reverses inequality
+
+```agda
+module _
+  (p : ℚ⁻)
+  (q r : ℚ)
+  (H : leq-ℚ q r)
+  where
+
+  abstract
+    reverses-leq-right-mul-ℚ⁻ : leq-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
+    reverses-leq-right-mul-ℚ⁻ =
+      binary-tr
+        ( leq-ℚ)
+        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
+        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
+        ( preserves-leq-right-mul-ℚ⁺
+          ( neg-ℚ⁻ p)
+          ( neg-ℚ r)
+          ( neg-ℚ q)
+          ( neg-leq-ℚ H))
+```
+
+### Multiplication by a negative rational number reverses strict inequality
+
+```agda
+module _
+  (p : ℚ⁻)
+  (q r : ℚ)
+  (H : le-ℚ q r)
+  where
+
+  abstract
+    reverses-le-right-mul-ℚ⁻ : le-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
+    reverses-le-right-mul-ℚ⁻ =
+      binary-tr
+        ( le-ℚ)
+        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
+        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
+        ( preserves-le-right-mul-ℚ⁺
+          ( neg-ℚ⁻ p)
+          ( neg-ℚ r)
+          ( neg-ℚ q)
+          ( neg-le-ℚ H))
+
+    reverses-le-left-mul-ℚ⁻ : le-ℚ (rational-ℚ⁻ p *ℚ r) (rational-ℚ⁻ p *ℚ q)
+    reverses-le-left-mul-ℚ⁻ =
+      binary-tr
+        ( le-ℚ)
+        ( commutative-mul-ℚ _ _)
+        ( commutative-mul-ℚ _ _)
+        ( reverses-le-right-mul-ℚ⁻)
+```
+
+### The negative rational numbers are invertible elements of the multiplicative monoid of rational numbers
+
+```agda
+opaque
+  inv-ℚ⁻ : ℚ⁻ → ℚ⁻
+  inv-ℚ⁻ q = neg-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q))
+
+  left-inverse-law-mul-ℚ⁻ :
+    (q : ℚ⁻) → rational-ℚ⁻ (inv-ℚ⁻ q) *ℚ rational-ℚ⁻ q ＝ one-ℚ
+  left-inverse-law-mul-ℚ⁻ q =
+    inv (swap-neg-mul-ℚ _ _) ∙
+    ap rational-ℚ⁺ (left-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
+
+  right-inverse-law-mul-ℚ⁻ :
+    (q : ℚ⁻) → rational-ℚ⁻ q *ℚ rational-ℚ⁻ (inv-ℚ⁻ q) ＝ one-ℚ
+  right-inverse-law-mul-ℚ⁻ q =
+    swap-neg-mul-ℚ _ _ ∙
+    ap rational-ℚ⁺ (right-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
+```
+
+### Inversion reverses inequality on negative rational numbers
+
+```agda
+opaque
+  unfolding inv-ℚ⁻
+
+  reverses-leq-inv-ℚ⁻ :
+    (p q : ℚ⁻) → leq-ℚ⁻ p q → leq-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
+  reverses-leq-inv-ℚ⁻ p q p≤q =
+    neg-leq-ℚ
+      ( inv-leq-ℚ⁺
+        ( neg-ℚ⁻ q)
+        ( neg-ℚ⁻ p)
+        ( neg-leq-ℚ p≤q))
+```
+
+### Inversion reverses strict inequality on negative rational numbers
+
+```agda
+opaque
+  unfolding inv-ℚ⁻
+
+  reverses-le-inv-ℚ⁻ :
+    (p q : ℚ⁻) → le-ℚ⁻ p q → le-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
+  reverses-le-inv-ℚ⁻ p q p<q =
+    neg-le-ℚ
+      ( inv-le-ℚ⁺
+        ( neg-ℚ⁻ q)
+        ( neg-ℚ⁻ p)
+        ( neg-le-ℚ p<q))
+```
diff --git a/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md
index 203f33961f2..e32b11124b7 100644
--- a/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of nonnegative rational numbers
+# Multiplication by nonnegative rational numbers
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
@@ -9,9 +9,12 @@ module elementary-number-theory.multiplication-nonnegative-rational-numbers wher
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.decidable-total-order-rational-numbers
 open import elementary-number-theory.inequality-integers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.maximum-rational-numbers
+open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-integer-fractions
 open import elementary-number-theory.multiplication-integers
 open import elementary-number-theory.multiplication-positive-and-negative-integers
@@ -23,6 +26,8 @@ open import elementary-number-theory.rational-numbers
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.identity-types
+
+open import order-theory.order-preserving-maps-total-orders
 ```
 
 </details>
@@ -41,12 +46,12 @@ itself nonnegative.
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-nonnegative-ℚ mul-ℚ
 
   is-nonnegative-mul-ℚ :
-    (p q : ℚ) → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
+    {p q : ℚ} → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
     is-nonnegative-ℚ (p *ℚ q)
-  is-nonnegative-mul-ℚ p q nonneg-p nonneg-q =
+  is-nonnegative-mul-ℚ {p} {q} nonneg-p nonneg-q =
     is-nonnegative-rational-fraction-ℤ
       ( is-nonnegative-mul-nonnegative-fraction-ℤ
         { fraction-ℚ p}
@@ -56,7 +61,7 @@ opaque
 
 mul-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
 mul-ℚ⁰⁺ (p , nonneg-p) (q , nonneg-q) =
-  ( p *ℚ q , is-nonnegative-mul-ℚ p q nonneg-p nonneg-q)
+  ( p *ℚ q , is-nonnegative-mul-ℚ nonneg-p nonneg-q)
 
 infixl 35 _*ℚ⁰⁺_
 _*ℚ⁰⁺_ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
@@ -77,8 +82,7 @@ abstract
 
 ```agda
 opaque
-  unfolding leq-ℚ-Prop
-  unfolding mul-ℚ
+  unfolding is-nonnegative-ℚ leq-ℚ-Prop mul-ℚ
 
   preserves-leq-right-mul-ℚ⁰⁺ :
     (p : ℚ⁰⁺) (q r : ℚ) → leq-ℚ q r →
@@ -123,3 +127,33 @@ abstract
       ( preserves-leq-right-mul-ℚ⁰⁺ s _ _ p≤q)
       ( preserves-leq-left-mul-ℚ⁰⁺ p _ _ r≤s)
 ```
+
+### Multiplication by a nonnegative rational number distributes over the minimum operation
+
+```agda
+abstract
+  left-distributive-mul-min-ℚ⁰⁺ :
+    (p : ℚ⁰⁺) (q r : ℚ) →
+    rational-ℚ⁰⁺ p *ℚ min-ℚ q r ＝
+    min-ℚ (rational-ℚ⁰⁺ p *ℚ q) (rational-ℚ⁰⁺ p *ℚ r)
+  left-distributive-mul-min-ℚ⁰⁺ p⁰⁺@(p , _) =
+    distributive-map-hom-min-Total-Order
+      ( ℚ-Total-Order)
+      ( ℚ-Total-Order)
+      ( p *ℚ_ , preserves-leq-left-mul-ℚ⁰⁺ p⁰⁺)
+```
+
+### Multiplication by a nonnegative rational number distributes over the maximum operation
+
+```agda
+abstract
+  left-distributive-mul-max-ℚ⁰⁺ :
+    (p : ℚ⁰⁺) (q r : ℚ) →
+    rational-ℚ⁰⁺ p *ℚ max-ℚ q r ＝
+    max-ℚ (rational-ℚ⁰⁺ p *ℚ q) (rational-ℚ⁰⁺ p *ℚ r)
+  left-distributive-mul-max-ℚ⁰⁺ p⁰⁺@(p , _) =
+    distributive-map-hom-max-Total-Order
+      ( ℚ-Total-Order)
+      ( ℚ-Total-Order)
+      ( p *ℚ_ , preserves-leq-left-mul-ℚ⁰⁺ p⁰⁺)
+```
diff --git a/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md
index 42b601b5891..f5bb0a14227 100644
--- a/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of nonpositive rational numbers
+# Multiplication by nonpositive rational numbers
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
@@ -36,7 +36,9 @@ as [rational numbers](elementary-number-theory.rational-numbers.md), and is
 ## Definition
 
 ```agda
-abstract
+opaque
+  unfolding is-nonpositive-ℚ
+
   is-nonnegative-mul-nonpositive-ℚ :
     {x y : ℚ} → is-nonpositive-ℚ x → is-nonpositive-ℚ y →
     is-nonnegative-ℚ (x *ℚ y)
@@ -44,7 +46,7 @@ abstract
     tr
       ( is-nonnegative-ℚ)
       ( negative-law-mul-ℚ x y)
-      ( is-nonnegative-mul-ℚ _ _ nonpos-x nonpos-y)
+      ( is-nonnegative-mul-ℚ nonpos-x nonpos-y)
 
 mul-nonpositive-ℚ : ℚ⁰⁻ → ℚ⁰⁻ → ℚ⁰⁺
 mul-nonpositive-ℚ (p , nonpos-p) (q , nonpos-q) =
@@ -56,7 +58,9 @@ mul-nonpositive-ℚ (p , nonpos-p) (q , nonpos-q) =
 ### Multiplication by a nonpositive rational number reverses inequality
 
 ```agda
-abstract
+opaque
+  unfolding is-nonpositive-ℚ
+
   reverses-leq-right-mul-ℚ⁰⁻ :
     (p : ℚ⁰⁻) (q r : ℚ) → leq-ℚ q r →
     leq-ℚ (r *ℚ rational-ℚ⁰⁻ p) (q *ℚ rational-ℚ⁰⁻ p)
@@ -65,6 +69,6 @@ abstract
       ( leq-ℚ)
       ( ap neg-ℚ (right-negative-law-mul-ℚ r p) ∙ neg-neg-ℚ _)
       ( ap neg-ℚ (right-negative-law-mul-ℚ q p) ∙ neg-neg-ℚ _)
-      ( neg-leq-ℚ _ _
+      ( neg-leq-ℚ
         ( preserves-leq-right-mul-ℚ⁰⁺ (neg-ℚ p , nonpos-p) q r q≤r))
 ```
diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index bf55c8c8e74..63a7876bb54 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of positive, negative, and nonnegative rational numbers
+# Multiplication by positive, negative, and nonnegative rational numbers
 
 ```agda
 module elementary-number-theory.multiplication-positive-and-negative-rational-numbers where
@@ -25,8 +25,9 @@ open import foundation.transport-along-identifications
 ## Idea
 
 When we have information about the sign of the factors of a
-[rational product](elementary-number-theory.multiplication-rational-numbers.md),
-we can deduce the sign of their product too.
+[rational](elementary-number-theory.rational-numbers.md)
+[product](elementary-number-theory.multiplication-rational-numbers.md), we can
+deduce the sign of their product too.
 
 ## Lemmas
 
@@ -34,7 +35,7 @@ we can deduce the sign of their product too.
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ mul-ℚ
 
   is-negative-mul-positive-negative-ℚ :
     {x y : ℚ} → is-positive-ℚ x → is-negative-ℚ y → is-negative-ℚ (x *ℚ y)
diff --git a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
index fd4ad86379d..1c43c7546b9 100644
--- a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of positive rational numbers
+# Multiplication by positive rational numbers
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
@@ -59,7 +59,7 @@ itself positive.
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ mul-ℚ
 
   is-positive-mul-ℚ :
     {x y : ℚ} → is-positive-ℚ x → is-positive-ℚ y → is-positive-ℚ (x *ℚ y)
@@ -147,7 +147,7 @@ module _
   where
 
   opaque
-    unfolding mul-ℚ
+    unfolding is-positive-ℚ mul-ℚ
 
     inv-is-positive-ℚ : ℚ
     pr1 inv-is-positive-ℚ = inv-is-positive-fraction-ℤ (fraction-ℚ x) P
@@ -215,8 +215,7 @@ module _
 
 ```agda
 opaque
-  unfolding le-ℚ-Prop
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ le-ℚ-Prop mul-ℚ
 
   preserves-le-left-mul-ℚ⁺ :
     (p : ℚ⁺) (q r : ℚ) →
@@ -256,8 +255,7 @@ opaque
 
 ```agda
 opaque
-  unfolding leq-ℚ-Prop
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ leq-ℚ-Prop mul-ℚ
 
   preserves-leq-left-mul-ℚ⁺ :
     (p : ℚ⁺) (q r : ℚ) → leq-ℚ q r →
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index 37fa4dc41d4..4db36a011c9 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -138,7 +138,9 @@ module _
       ( H)
       ( inv K)
 
-  abstract
+  opaque
+    unfolding is-negative-ℚ
+
     is-invertible-element-ring-is-nonzero-ℚ :
       is-nonzero-ℚ x → is-invertible-element-Ring ring-ℚ x
     is-invertible-element-ring-is-nonzero-ℚ H =
diff --git a/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md
index 27b21b5e0b1..36d0cbef5f4 100644
--- a/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md
@@ -32,7 +32,7 @@ of a
 ```agda
 neg-closed-interval-ℚ : closed-interval-ℚ → closed-interval-ℚ
 neg-closed-interval-ℚ ((a , b) , a≤b) =
-  ((neg-ℚ b , neg-ℚ a) , neg-leq-ℚ a b a≤b)
+  ((neg-ℚ b , neg-ℚ a) , neg-leq-ℚ a≤b)
 ```
 
 ## Properties
@@ -48,7 +48,7 @@ abstract
       ( neg-closed-interval-ℚ [a,b])
       ( neg-closed-interval-ℚ [c,d])
   is-interior-neg-closed-interval-ℚ ((a , b) , _) ((c , d) , _) (a<c , d<b) =
-    ( neg-le-ℚ d b d<b , neg-le-ℚ a c a<c)
+    ( neg-le-ℚ d<b , neg-le-ℚ a<c)
 ```
 
 ### Negation preserves nontriviality
@@ -58,5 +58,5 @@ abstract
   is-nontrivial-neg-closed-interval-ℚ :
     ([a,b] : closed-interval-ℚ) → is-proper-closed-interval-ℚ [a,b] →
     is-proper-closed-interval-ℚ (neg-closed-interval-ℚ [a,b])
-  is-nontrivial-neg-closed-interval-ℚ ((a , b) , _) a<b = neg-le-ℚ a b a<b
+  is-nontrivial-neg-closed-interval-ℚ ((a , b) , _) a<b = neg-le-ℚ a<b
 ```
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 0e179958514..be110dba249 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -13,9 +13,6 @@ open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integer-fractions
 open import elementary-number-theory.integers
-open import elementary-number-theory.multiplication-positive-rational-numbers
-open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.negative-integer-fractions
 open import elementary-number-theory.negative-integers
 open import elementary-number-theory.nonzero-rational-numbers
@@ -60,11 +57,12 @@ module _
   (q : ℚ)
   where
 
-  is-negative-ℚ : UU lzero
-  is-negative-ℚ = is-positive-ℚ (neg-ℚ q)
+  opaque
+    is-negative-ℚ : UU lzero
+    is-negative-ℚ = is-positive-ℚ (neg-ℚ q)
 
-  is-prop-is-negative-ℚ : is-prop is-negative-ℚ
-  is-prop-is-negative-ℚ = is-prop-is-positive-ℚ (neg-ℚ q)
+    is-prop-is-negative-ℚ : is-prop is-negative-ℚ
+    is-prop-is-negative-ℚ = is-prop-is-positive-ℚ (neg-ℚ q)
 
   is-negative-prop-ℚ : Prop lzero
   pr1 is-negative-prop-ℚ = is-negative-ℚ
@@ -100,16 +98,14 @@ module _
   is-negative-rational-ℚ⁻ = pr2 x
 
   opaque
-    unfolding neg-ℚ
+    unfolding is-negative-ℚ is-positive-ℚ neg-ℚ
 
     is-negative-fraction-ℚ⁻ : is-negative-fraction-ℤ fraction-ℚ⁻
     is-negative-fraction-ℚ⁻ =
       tr
-        ( is-negative-fraction-ℤ)
-        ( neg-neg-fraction-ℤ fraction-ℚ⁻)
-        ( is-negative-neg-positive-fraction-ℤ
-          ( neg-fraction-ℤ fraction-ℚ⁻)
-          ( is-negative-rational-ℚ⁻))
+        ( is-negative-ℤ)
+        ( neg-neg-ℤ _)
+        ( is-negative-neg-is-positive-ℤ is-negative-rational-ℚ⁻)
 
   is-negative-numerator-ℚ⁻ : is-negative-ℤ numerator-ℚ⁻
   is-negative-numerator-ℚ⁻ = is-negative-fraction-ℚ⁻
@@ -117,12 +113,6 @@ module _
   is-positive-denominator-ℚ⁻ : is-positive-ℤ denominator-ℚ⁻
   is-positive-denominator-ℚ⁻ = is-positive-denominator-ℚ rational-ℚ⁻
 
-  neg-ℚ⁻ : ℚ⁺
-  neg-ℚ⁻ = (neg-ℚ rational-ℚ⁻) , is-negative-rational-ℚ⁻
-
-neg-ℚ⁺ : ℚ⁺ → ℚ⁻
-neg-ℚ⁺ (q , q-is-pos) = neg-ℚ q , inv-tr is-positive-ℚ (neg-neg-ℚ q) q-is-pos
-
 abstract
   eq-ℚ⁻ : {x y : ℚ⁻} → rational-ℚ⁻ x ＝ rational-ℚ⁻ y → x ＝ y
   eq-ℚ⁻ {x} {y} = eq-type-subtype is-negative-prop-ℚ
@@ -141,7 +131,7 @@ is-set-ℚ⁻ = is-set-type-subtype is-negative-prop-ℚ is-set-ℚ
 
 ```agda
 opaque
-  unfolding neg-ℚ
+  unfolding is-negative-ℚ is-positive-ℚ neg-ℚ
 
   is-negative-rational-ℤ :
     (x : ℤ) → is-negative-ℤ x → is-negative-ℚ (rational-ℤ x)
@@ -156,41 +146,49 @@ negative-rational-negative-ℤ (x , x-is-neg) =
 
 ```agda
 opaque
-  unfolding neg-ℚ
-  unfolding rational-fraction-ℤ
+  unfolding is-negative-ℚ
 
   is-negative-rational-fraction-ℤ :
-    {x : fraction-ℤ} (P : is-negative-fraction-ℤ x) →
+    {x : fraction-ℤ} → is-negative-fraction-ℤ x →
     is-negative-ℚ (rational-fraction-ℤ x)
-  is-negative-rational-fraction-ℤ P =
-    is-positive-neg-is-negative-ℤ (is-negative-reduce-fraction-ℤ P)
+  is-negative-rational-fraction-ℤ {x} P =
+    tr
+      ( is-positive-ℚ)
+      ( neg-rational-fraction-ℤ _)
+      ( is-positive-rational-fraction-ℤ
+        ( is-positive-neg-negative-fraction-ℤ x P))
 ```
 
 ### A rational number `x` is negative if and only if `x < 0`
 
 ```agda
-module _
-  (x : ℚ)
-  where
+opaque
+  unfolding is-negative-ℚ
 
-  opaque
-    unfolding neg-ℚ
+  le-zero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → le-ℚ x zero-ℚ
+  le-zero-is-negative-ℚ {x} is-neg-x =
+    binary-tr
+      ( le-ℚ)
+      ( neg-neg-ℚ x)
+      ( neg-zero-ℚ)
+      ( neg-le-ℚ (le-zero-is-positive-ℚ is-neg-x))
+
+  is-negative-le-zero-ℚ : {x : ℚ} → le-ℚ x zero-ℚ → is-negative-ℚ x
+  is-negative-le-zero-ℚ {x} x<0 =
+    is-positive-le-zero-ℚ
+      ( tr
+        ( λ y → le-ℚ y (neg-ℚ x))
+        ( neg-zero-ℚ)
+        ( neg-le-ℚ x<0))
+
+is-negative-iff-le-zero-ℚ : (x : ℚ) → is-negative-ℚ x ↔ le-ℚ x zero-ℚ
+is-negative-iff-le-zero-ℚ x =
+  ( le-zero-is-negative-ℚ ,
+    is-negative-le-zero-ℚ)
 
-    le-zero-is-negative-ℚ : is-negative-ℚ x → le-ℚ x zero-ℚ
-    le-zero-is-negative-ℚ x-is-neg =
-      tr
-        ( λ y → le-ℚ y zero-ℚ)
-        ( neg-neg-ℚ x)
-        ( neg-le-ℚ zero-ℚ (neg-ℚ x) (le-zero-is-positive-ℚ (neg-ℚ x) x-is-neg))
-
-    is-negative-le-zero-ℚ : le-ℚ x zero-ℚ → is-negative-ℚ x
-    is-negative-le-zero-ℚ x-is-neg =
-      is-positive-le-zero-ℚ (neg-ℚ x) (neg-le-ℚ x zero-ℚ x-is-neg)
-
-    is-negative-iff-le-zero-ℚ : is-negative-ℚ x ↔ le-ℚ x zero-ℚ
-    is-negative-iff-le-zero-ℚ =
-      le-zero-is-negative-ℚ ,
-      is-negative-le-zero-ℚ
+abstract
+  leq-zero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → leq-ℚ x zero-ℚ
+  leq-zero-is-negative-ℚ is-neg-x = leq-le-ℚ (le-zero-is-negative-ℚ is-neg-x)
 ```
 
 ### The difference of a rational number with a greater rational number is negative
@@ -200,7 +198,9 @@ module _
   (x y : ℚ) (H : le-ℚ x y)
   where
 
-  abstract
+  opaque
+    unfolding is-negative-ℚ
+
     is-negative-diff-le-ℚ : is-negative-ℚ (x -ℚ y)
     is-negative-diff-le-ℚ =
       inv-tr
@@ -215,7 +215,9 @@ module _
 ### Negative rational numbers are nonzero
 
 ```agda
-abstract
+opaque
+  unfolding is-negative-ℚ
+
   is-nonzero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → is-nonzero-ℚ x
   is-nonzero-is-negative-ℚ {x} H =
     tr
@@ -227,98 +229,6 @@ nonzero-ℚ⁻ : ℚ⁻ → nonzero-ℚ
 nonzero-ℚ⁻ (x , N) = (x , is-nonzero-is-negative-ℚ N)
 ```
 
-### The product of two negative rational numbers is positive
-
-```agda
-opaque
-  unfolding mul-ℚ
-  unfolding rational-fraction-ℤ
-
-  is-positive-mul-negative-ℚ :
-    {x y : ℚ} → is-negative-ℚ x → is-negative-ℚ y → is-positive-ℚ (x *ℚ y)
-  is-positive-mul-negative-ℚ {x} {y} P Q =
-    is-positive-reduce-fraction-ℤ
-      ( is-positive-mul-negative-fraction-ℤ
-        { fraction-ℚ x}
-        { fraction-ℚ y}
-        ( is-negative-fraction-ℚ⁻ (x , P))
-        ( is-negative-fraction-ℚ⁻ (y , Q)))
-```
-
-### Multiplication by a negative rational number reverses inequality
-
-```agda
-module _
-  (p : ℚ⁻)
-  (q r : ℚ)
-  (H : leq-ℚ q r)
-  where
-
-  abstract
-    reverses-leq-right-mul-ℚ⁻ : leq-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
-    reverses-leq-right-mul-ℚ⁻ =
-      binary-tr
-        ( leq-ℚ)
-        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
-        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
-        ( preserves-leq-right-mul-ℚ⁺
-          ( neg-ℚ⁻ p)
-          ( neg-ℚ r)
-          ( neg-ℚ q)
-          ( neg-leq-ℚ q r H))
-```
-
-### Multiplication by a negative rational number reverses strict inequality
-
-```agda
-module _
-  (p : ℚ⁻)
-  (q r : ℚ)
-  (H : le-ℚ q r)
-  where
-
-  abstract
-    reverses-le-right-mul-ℚ⁻ : le-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
-    reverses-le-right-mul-ℚ⁻ =
-      binary-tr
-        ( le-ℚ)
-        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
-        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
-        ( preserves-le-right-mul-ℚ⁺
-          ( neg-ℚ⁻ p)
-          ( neg-ℚ r)
-          ( neg-ℚ q)
-          ( neg-le-ℚ q r H))
-
-    reverses-le-left-mul-ℚ⁻ : le-ℚ (rational-ℚ⁻ p *ℚ r) (rational-ℚ⁻ p *ℚ q)
-    reverses-le-left-mul-ℚ⁻ =
-      binary-tr
-        ( le-ℚ)
-        ( commutative-mul-ℚ _ _)
-        ( commutative-mul-ℚ _ _)
-        ( reverses-le-right-mul-ℚ⁻)
-```
-
-### The negative rational numbers are invertible elements of the multiplicative monoid of rational numbers
-
-```agda
-opaque
-  inv-ℚ⁻ : ℚ⁻ → ℚ⁻
-  inv-ℚ⁻ q = neg-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q))
-
-  left-inverse-law-mul-ℚ⁻ :
-    (q : ℚ⁻) → rational-ℚ⁻ (inv-ℚ⁻ q) *ℚ rational-ℚ⁻ q ＝ one-ℚ
-  left-inverse-law-mul-ℚ⁻ q =
-    inv (swap-neg-mul-ℚ _ _) ∙
-    ap rational-ℚ⁺ (left-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
-
-  right-inverse-law-mul-ℚ⁻ :
-    (q : ℚ⁻) → rational-ℚ⁻ q *ℚ rational-ℚ⁻ (inv-ℚ⁻ q) ＝ one-ℚ
-  right-inverse-law-mul-ℚ⁻ q =
-    swap-neg-mul-ℚ _ _ ∙
-    ap rational-ℚ⁺ (right-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
-```
-
 ### Inequality on negative rational numbers
 
 ```agda
@@ -329,42 +239,6 @@ le-ℚ⁻ : Relation lzero ℚ⁻
 le-ℚ⁻ p q = le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁻ q)
 ```
 
-### Inversion reverses inequality on negative rational numbers
-
-```agda
-opaque
-  unfolding inv-ℚ⁻
-
-  reverses-leq-inv-ℚ⁻ :
-    (p q : ℚ⁻) → leq-ℚ⁻ p q → leq-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
-  reverses-leq-inv-ℚ⁻ p q p≤q =
-    neg-leq-ℚ
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ p)))
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q)))
-      ( inv-leq-ℚ⁺
-        ( neg-ℚ⁻ q)
-        ( neg-ℚ⁻ p)
-        ( neg-leq-ℚ _ _ p≤q))
-```
-
-### Inversion reverses strict inequality on negative rational numbers
-
-```agda
-opaque
-  unfolding inv-ℚ⁻
-
-  reverses-le-inv-ℚ⁻ :
-    (p q : ℚ⁻) → le-ℚ⁻ p q → le-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
-  reverses-le-inv-ℚ⁻ p q p<q =
-    neg-le-ℚ
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ p)))
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q)))
-      ( inv-le-ℚ⁺
-        ( neg-ℚ⁻ q)
-        ( neg-ℚ⁻ p)
-        ( neg-le-ℚ _ _ p<q))
-```
-
 ### If `p ≤ q` for negative `q`, then `p` is negative
 
 ```agda
@@ -373,8 +247,7 @@ abstract
     (q : ℚ⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
   is-negative-leq-ℚ⁻ (q , neg-q) p p≤q =
     is-negative-le-zero-ℚ
-      ( p)
-      ( concatenate-leq-le-ℚ p q zero-ℚ p≤q (le-zero-is-negative-ℚ q neg-q))
+      ( concatenate-leq-le-ℚ p q zero-ℚ p≤q (le-zero-is-negative-ℚ neg-q))
 
   is-negative-le-ℚ⁻ :
     (q : ℚ⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
diff --git a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
index c7cd39a7481..621a2787146 100644
--- a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
@@ -60,11 +60,12 @@ module _
   (q : ℚ)
   where
 
-  is-nonnegative-ℚ : UU lzero
-  is-nonnegative-ℚ = is-nonnegative-fraction-ℤ (fraction-ℚ q)
+  opaque
+    is-nonnegative-ℚ : UU lzero
+    is-nonnegative-ℚ = is-nonnegative-fraction-ℤ (fraction-ℚ q)
 
-  is-prop-is-nonnegative-ℚ : is-prop is-nonnegative-ℚ
-  is-prop-is-nonnegative-ℚ = is-prop-is-nonnegative-fraction-ℤ (fraction-ℚ q)
+    is-prop-is-nonnegative-ℚ : is-prop is-nonnegative-ℚ
+    is-prop-is-nonnegative-ℚ = is-prop-is-nonnegative-fraction-ℤ (fraction-ℚ q)
 
   is-nonnegative-prop-ℚ : Prop lzero
   pr1 is-nonnegative-prop-ℚ = is-nonnegative-ℚ
@@ -99,20 +100,17 @@ module _
   is-nonnegative-rational-ℚ⁰⁺ : is-nonnegative-ℚ rational-ℚ⁰⁺
   is-nonnegative-rational-ℚ⁰⁺ = pr2 x
 
-  is-nonnegative-fraction-ℚ⁰⁺ : is-nonnegative-fraction-ℤ fraction-ℚ⁰⁺
-  is-nonnegative-fraction-ℚ⁰⁺ = pr2 x
+  opaque
+    unfolding is-nonnegative-ℚ
+
+    is-nonnegative-fraction-ℚ⁰⁺ : is-nonnegative-fraction-ℤ fraction-ℚ⁰⁺
+    is-nonnegative-fraction-ℚ⁰⁺ = pr2 x
 
   is-nonnegative-numerator-ℚ⁰⁺ : is-nonnegative-ℤ numerator-ℚ⁰⁺
   is-nonnegative-numerator-ℚ⁰⁺ = is-nonnegative-fraction-ℚ⁰⁺
 
   is-positive-denominator-ℚ⁰⁺ : is-positive-ℤ denominator-ℚ⁰⁺
   is-positive-denominator-ℚ⁰⁺ = is-positive-denominator-ℚ rational-ℚ⁰⁺
-
-zero-ℚ⁰⁺ : ℚ⁰⁺
-zero-ℚ⁰⁺ = zero-ℚ , _
-
-one-ℚ⁰⁺ : ℚ⁰⁺
-one-ℚ⁰⁺ = one-ℚ , _
 ```
 
 ## Properties
@@ -132,42 +130,49 @@ is-set-ℚ⁰⁺ : is-set ℚ⁰⁺
 is-set-ℚ⁰⁺ = is-set-type-subtype is-nonnegative-prop-ℚ is-set-ℚ
 ```
 
-### All positive rational numbers are nonnegative
-
-```agda
-abstract
-  is-nonnegative-is-positive-ℚ : (q : ℚ) → is-positive-ℚ q → is-nonnegative-ℚ q
-  is-nonnegative-is-positive-ℚ _ = is-nonnegative-is-positive-ℤ
-
-nonnegative-ℚ⁺ : ℚ⁺ → ℚ⁰⁺
-nonnegative-ℚ⁺ (q , H) = (q , is-nonnegative-is-positive-ℚ q H)
-```
-
 ### The rational image of a nonnegative integer is nonnegative
 
 ```agda
-abstract
+opaque
+  unfolding is-nonnegative-ℚ
+
   is-nonnegative-rational-ℤ :
-    (x : ℤ) → is-nonnegative-ℤ x → is-nonnegative-ℚ (rational-ℤ x)
-  is-nonnegative-rational-ℤ _ H = H
+    {x : ℤ} → is-nonnegative-ℤ x → is-nonnegative-ℚ (rational-ℤ x)
+  is-nonnegative-rational-ℤ H = H
 
 nonnegative-rational-nonnegative-ℤ : nonnegative-ℤ → ℚ⁰⁺
 nonnegative-rational-nonnegative-ℤ (x , x-is-neg) =
-  ( rational-ℤ x , is-nonnegative-rational-ℤ x x-is-neg)
+  ( rational-ℤ x , is-nonnegative-rational-ℤ x-is-neg)
 ```
 
 ### The images of natural numbers in the rationals are nonnegative
 
 ```agda
-nonnegative-rational-ℕ : ℕ → ℚ⁰⁺
-nonnegative-rational-ℕ n = (rational-ℕ n , is-nonnegative-int-ℕ n)
+module _
+  (n : ℕ)
+  where
+
+  opaque
+    unfolding is-nonnegative-ℚ
+
+    is-nonnegative-rational-ℕ : is-nonnegative-ℚ (rational-ℕ n)
+    is-nonnegative-rational-ℕ = is-nonnegative-int-ℕ n
+
+  nonnegative-rational-ℕ : ℚ⁰⁺
+  nonnegative-rational-ℕ = (rational-ℕ n , is-nonnegative-rational-ℕ)
+
+zero-ℚ⁰⁺ : ℚ⁰⁺
+zero-ℚ⁰⁺ = nonnegative-rational-ℕ zero-ℕ
+
+one-ℚ⁰⁺ : ℚ⁰⁺
+one-ℚ⁰⁺ = nonnegative-rational-ℕ 1
 ```
 
 ### The rational image of a nonnegative integer fraction is nonnegative
 
 ```agda
 opaque
-  unfolding rational-fraction-ℤ
+  unfolding is-nonnegative-ℚ rational-fraction-ℤ
 
   is-nonnegative-rational-fraction-ℤ :
     {x : fraction-ℤ} (P : is-nonnegative-fraction-ℤ x) →
@@ -182,30 +187,26 @@ opaque
 ### A rational number `x` is nonnegative if and only if `0 ≤ x`
 
 ```agda
-module _
-  (x : ℚ)
-  where
-
-  opaque
-    unfolding leq-ℚ-Prop
+opaque
+  unfolding is-nonnegative-ℚ leq-ℚ-Prop
 
-    leq-zero-is-nonnegative-ℚ : is-nonnegative-ℚ x → leq-ℚ zero-ℚ x
-    leq-zero-is-nonnegative-ℚ =
-      is-nonnegative-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
+  leq-zero-is-nonnegative-ℚ : {x : ℚ} → is-nonnegative-ℚ x → leq-ℚ zero-ℚ x
+  leq-zero-is-nonnegative-ℚ {x} =
+    is-nonnegative-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
 
-    is-nonnegative-leq-zero-ℚ : leq-ℚ zero-ℚ x → is-nonnegative-ℚ x
-    is-nonnegative-leq-zero-ℚ =
-      is-nonnegative-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
+  is-nonnegative-leq-zero-ℚ : {x : ℚ} → leq-ℚ zero-ℚ x → is-nonnegative-ℚ x
+  is-nonnegative-leq-zero-ℚ {x} =
+    is-nonnegative-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
 
-    is-nonnegative-iff-leq-zero-ℚ : is-nonnegative-ℚ x ↔ leq-ℚ zero-ℚ x
-    is-nonnegative-iff-leq-zero-ℚ =
-      ( leq-zero-is-nonnegative-ℚ ,
-        is-nonnegative-leq-zero-ℚ)
+is-nonnegative-iff-leq-zero-ℚ : (x : ℚ) → is-nonnegative-ℚ x ↔ leq-ℚ zero-ℚ x
+is-nonnegative-iff-leq-zero-ℚ x =
+  ( leq-zero-is-nonnegative-ℚ ,
+    is-nonnegative-leq-zero-ℚ)
 
 abstract
   leq-zero-rational-ℚ⁰⁺ : (p : ℚ⁰⁺) → leq-ℚ zero-ℚ (rational-ℚ⁰⁺ p)
   leq-zero-rational-ℚ⁰⁺ (p , is-nonneg-p) =
-    leq-zero-is-nonnegative-ℚ p is-nonneg-p
+    leq-zero-is-nonnegative-ℚ is-nonneg-p
 ```
 
 ### The successor of a nonnegative rational number is positive
@@ -216,12 +217,11 @@ abstract
     (q : ℚ) → is-nonnegative-ℚ q → is-positive-ℚ (succ-ℚ q)
   is-positive-succ-is-nonnegative-ℚ q H =
     is-positive-le-zero-ℚ
-      ( succ-ℚ q)
       ( concatenate-leq-le-ℚ
         ( zero-ℚ)
         ( q)
         ( succ-ℚ q)
-        ( leq-zero-is-nonnegative-ℚ q H)
+        ( leq-zero-is-nonnegative-ℚ H)
         ( le-left-add-rational-ℚ⁺ q one-ℚ⁺))
 
 positive-succ-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁺
diff --git a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
index 1f46950c690..b0506e21798 100644
--- a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
@@ -12,14 +12,12 @@ module elementary-number-theory.nonpositive-rational-numbers where
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integer-fractions
 open import elementary-number-theory.integers
-open import elementary-number-theory.multiplication-nonnegative-rational-numbers
-open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
-open import elementary-number-theory.nonpositive-integers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
+open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
@@ -48,10 +46,10 @@ module _
   (q : ℚ)
   where
 
-  is-nonpositive-ℚ : UU lzero
-  is-nonpositive-ℚ = is-nonnegative-ℚ (neg-ℚ q)
+  opaque
+    is-nonpositive-ℚ : UU lzero
+    is-nonpositive-ℚ = is-nonnegative-ℚ (neg-ℚ q)
 
-  abstract
     is-prop-is-nonpositive-ℚ : is-prop is-nonpositive-ℚ
     is-prop-is-nonpositive-ℚ = is-prop-is-nonnegative-ℚ (neg-ℚ q)
 
@@ -98,26 +96,27 @@ abstract
 ### A rational number is nonpositive if and only if it is less than or equal to zero
 
 ```agda
-module _
-  (q : ℚ)
-  where
-
-  opaque
-    unfolding neg-ℚ
-
-    leq-zero-is-nonpositive-ℚ : is-nonpositive-ℚ q → leq-ℚ q zero-ℚ
-    leq-zero-is-nonpositive-ℚ is-nonpos-q =
-      tr
-        ( λ p → leq-ℚ p zero-ℚ)
-        ( neg-neg-ℚ q)
-        ( neg-leq-ℚ _ _ (leq-zero-is-nonnegative-ℚ (neg-ℚ q) is-nonpos-q))
-
-    is-nonpositive-leq-zero-ℚ : leq-ℚ q zero-ℚ → is-nonpositive-ℚ q
-    is-nonpositive-leq-zero-ℚ q≤0 =
-      is-nonnegative-leq-zero-ℚ (neg-ℚ q) (neg-leq-ℚ _ _ q≤0)
-
-    is-nonpositive-iff-leq-zero-ℚ : is-nonpositive-ℚ q ↔ leq-ℚ q zero-ℚ
-    is-nonpositive-iff-leq-zero-ℚ =
-      ( leq-zero-is-nonpositive-ℚ ,
-        is-nonpositive-leq-zero-ℚ)
+opaque
+  unfolding is-nonpositive-ℚ
+
+  leq-zero-is-nonpositive-ℚ : {q : ℚ} → is-nonpositive-ℚ q → leq-ℚ q zero-ℚ
+  leq-zero-is-nonpositive-ℚ {q} is-nonpos-q =
+    binary-tr
+      ( leq-ℚ)
+      ( neg-neg-ℚ q)
+      ( neg-zero-ℚ)
+      ( neg-leq-ℚ (leq-zero-is-nonnegative-ℚ is-nonpos-q))
+
+  is-nonpositive-leq-zero-ℚ : {q : ℚ} → leq-ℚ q zero-ℚ → is-nonpositive-ℚ q
+  is-nonpositive-leq-zero-ℚ {q} q≤0 =
+    is-nonnegative-leq-zero-ℚ
+      ( tr
+        ( λ y → leq-ℚ y (neg-ℚ q))
+        ( neg-zero-ℚ)
+        ( neg-leq-ℚ q≤0))
+
+is-nonpositive-iff-leq-zero-ℚ : (q : ℚ) → is-nonpositive-ℚ q ↔ leq-ℚ q zero-ℚ
+is-nonpositive-iff-leq-zero-ℚ q =
+  ( leq-zero-is-nonpositive-ℚ ,
+    is-nonpositive-leq-zero-ℚ)
 ```
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index 836af5c3954..93a69da7873 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -11,14 +11,22 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.nonzero-rational-numbers
+open import elementary-number-theory.positive-and-negative-integers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
+open import foundation.binary-transport
+open import foundation.cartesian-product-types
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
+open import foundation.negation
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
@@ -35,6 +43,27 @@ In this file, we outline basic relations between
 
 ## Properties
 
+### Trichotomies
+
+#### A rational number is either negative, zero, or positive
+
+```agda
+abstract
+  trichotomy-sign-ℚ :
+    {l : Level} {A : UU l} (x : ℚ) →
+    ( is-negative-ℚ x → A) →
+    ( x ＝ zero-ℚ → A) →
+    ( is-positive-ℚ x → A) →
+    A
+  trichotomy-sign-ℚ x neg zero pos =
+    trichotomy-le-ℚ
+      ( x)
+      ( zero-ℚ)
+      ( λ x<0 → neg (is-negative-le-zero-ℚ x<0))
+      ( zero)
+      ( λ 0<x → pos (is-positive-le-zero-ℚ 0<x))
+```
+
 ### Dichotomies
 
 #### A rational number is either negative or nonnegative
@@ -46,8 +75,8 @@ abstract
     is-negative-ℚ q + is-nonnegative-ℚ q
   decide-is-negative-is-nonnegative-ℚ q =
     map-coproduct
-      ( is-negative-le-zero-ℚ q)
-      ( is-nonnegative-leq-zero-ℚ q)
+      ( is-negative-le-zero-ℚ)
+      ( is-nonnegative-leq-zero-ℚ)
       ( decide-le-leq-ℚ q zero-ℚ)
 ```
 
@@ -60,80 +89,138 @@ abstract
     is-positive-ℚ q + is-nonpositive-ℚ q
   decide-is-positive-is-nonpositive-ℚ q =
     map-coproduct
-      ( is-positive-le-zero-ℚ q)
-      ( is-nonpositive-leq-zero-ℚ q)
+      ( is-positive-le-zero-ℚ)
+      ( is-nonpositive-leq-zero-ℚ)
       ( decide-le-leq-ℚ zero-ℚ q)
 ```
 
-### Trichotomies
-
-#### A rational number is either negative, zero, or positive
+#### A rational number is nonpositive or nonnegative
 
 ```agda
 abstract
-  trichotomy-sign-ℚ :
-    {l : Level} {A : UU l} (x : ℚ) →
-    ( is-negative-ℚ x → A) →
-    ( x ＝ zero-ℚ → A) →
-    ( is-positive-ℚ x → A) →
-    A
-  trichotomy-sign-ℚ x neg zero pos =
-    trichotomy-le-ℚ
-      ( x)
-      ( zero-ℚ)
-      ( λ x<0 → neg (is-negative-le-zero-ℚ x x<0))
-      ( zero)
-      ( λ 0<x → pos (is-positive-le-zero-ℚ x 0<x))
+  is-nonpositive-or-nonnegative-ℚ :
+    (q : ℚ) →
+    is-nonpositive-ℚ q + is-nonnegative-ℚ q
+  is-nonpositive-or-nonnegative-ℚ q =
+    map-coproduct
+      ( is-nonpositive-leq-zero-ℚ)
+      ( is-nonnegative-leq-zero-ℚ)
+      ( linear-leq-ℚ q zero-ℚ)
 ```
 
-### Inequalities
+### If a rational number is nonzero, it is negative or positive
 
-#### A negative rational number is less than a nonnegative rational number
+```agda
+decide-is-negative-is-positive-is-nonzero-ℚ :
+  {q : ℚ} → is-nonzero-ℚ q → is-negative-ℚ q + is-positive-ℚ q
+decide-is-negative-is-positive-is-nonzero-ℚ {q} q≠0 =
+  trichotomy-sign-ℚ q
+    ( inl)
+    ( ex-falso ∘ q≠0)
+    ( inr)
+```
+
+### Implications
+
+#### Any positive rational number is nonnegative
 
 ```agda
-abstract
-  le-negative-nonnegative-ℚ :
-    (p : ℚ⁻) (q : ℚ⁰⁺) → le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁰⁺ q)
-  le-negative-nonnegative-ℚ (p , neg-p) (q , nonneg-q) =
-    concatenate-le-leq-ℚ p zero-ℚ q
-      ( le-zero-is-negative-ℚ p neg-p)
-      ( leq-zero-is-nonnegative-ℚ q nonneg-q)
+opaque
+  unfolding is-nonnegative-ℚ is-positive-ℚ
+
+  is-nonnegative-is-positive-ℚ : {q : ℚ} → is-positive-ℚ q → is-nonnegative-ℚ q
+  is-nonnegative-is-positive-ℚ = is-nonnegative-is-positive-ℤ
+
+nonnegative-ℚ⁺ : ℚ⁺ → ℚ⁰⁺
+nonnegative-ℚ⁺ (q , H) = (q , is-nonnegative-is-positive-ℚ H)
 ```
 
-#### A nonpositive rational number is less than a positive rational number
+### Any negative rational number is nonpositive
 
 ```agda
-abstract
-  le-nonpositive-positive-ℚ :
-    (p : ℚ⁰⁻) (q : ℚ⁺) → le-ℚ (rational-ℚ⁰⁻ p) (rational-ℚ⁺ q)
-  le-nonpositive-positive-ℚ (p , nonpos-p) (q , pos-q) =
-    concatenate-leq-le-ℚ p zero-ℚ q
-      ( leq-zero-is-nonpositive-ℚ p nonpos-p)
-      ( le-zero-is-positive-ℚ q pos-q)
+opaque
+  unfolding is-negative-ℚ is-nonpositive-ℚ
+
+  is-nonpositive-is-negative-ℚ : {q : ℚ} → is-negative-ℚ q → is-nonpositive-ℚ q
+  is-nonpositive-is-negative-ℚ = is-nonnegative-is-positive-ℚ
+
+nonpositive-ℚ⁻ : ℚ⁻ → ℚ⁰⁻
+nonpositive-ℚ⁻ (q , H) = (q , is-nonpositive-is-negative-ℚ H)
 ```
 
-### If `p < q` and `p` is nonnegative, then `q` is positive
+### Operations
+
+#### The negation of a positive rational number is negative
 
 ```agda
-abstract
-  is-positive-le-ℚ⁰⁺ :
-    (p : ℚ⁰⁺) (q : ℚ) → le-ℚ (rational-ℚ⁰⁺ p) q → is-positive-ℚ q
-  is-positive-le-ℚ⁰⁺ (p , nonneg-p) q p<q =
-    is-positive-le-zero-ℚ
-      ( q)
-      ( concatenate-leq-le-ℚ _ _ _ (leq-zero-is-nonnegative-ℚ p nonneg-p) p<q)
+opaque
+  unfolding is-negative-ℚ
+
+  is-negative-neg-is-positive-ℚ :
+    {q : ℚ} → is-positive-ℚ q → is-negative-ℚ (neg-ℚ q)
+  is-negative-neg-is-positive-ℚ = inv-tr is-positive-ℚ (neg-neg-ℚ _)
+
+neg-ℚ⁺ : ℚ⁺ → ℚ⁻
+neg-ℚ⁺ (q , is-pos-q) =
+  (neg-ℚ q , is-negative-neg-is-positive-ℚ is-pos-q)
 ```
 
-### If `p < q` and `q` is nonpositive, then `p` is negative
+#### The negation of a negative rational number is positive
+
+```agda
+opaque
+  unfolding is-negative-ℚ
+
+  is-positive-neg-is-negative-ℚ :
+    {q : ℚ} → is-negative-ℚ q → is-positive-ℚ (neg-ℚ q)
+  is-positive-neg-is-negative-ℚ = id
+
+neg-ℚ⁻ : ℚ⁻ → ℚ⁺
+neg-ℚ⁻ (q , is-neg-q) = (neg-ℚ q , is-positive-neg-is-negative-ℚ is-neg-q)
+```
+
+#### The negation of a nonnegative rational number is nonpositive
+
+```agda
+opaque
+  unfolding is-nonpositive-ℚ
+
+  is-nonpositive-neg-is-nonnegative-ℚ :
+    {q : ℚ} → is-nonnegative-ℚ q → is-nonpositive-ℚ (neg-ℚ q)
+  is-nonpositive-neg-is-nonnegative-ℚ = inv-tr is-nonnegative-ℚ (neg-neg-ℚ _)
+
+neg-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁻
+neg-ℚ⁰⁺ (q , is-nonneg-q) =
+  ( neg-ℚ q , is-nonpositive-neg-is-nonnegative-ℚ is-nonneg-q)
+```
+
+#### The negation of a nonpositive rational number is nonnegative
+
+```agda
+opaque
+  unfolding is-nonpositive-ℚ
+
+  is-nonnegative-neg-is-nonpositive-ℚ :
+    {q : ℚ} → is-nonpositive-ℚ q → is-nonnegative-ℚ (neg-ℚ q)
+  is-nonnegative-neg-is-nonpositive-ℚ = id
+
+neg-ℚ⁰⁻ : ℚ⁰⁻ → ℚ⁰⁺
+neg-ℚ⁰⁻ (q , is-nonpos-q) =
+  ( neg-ℚ q , is-nonnegative-neg-is-nonpositive-ℚ is-nonpos-q)
+```
+
+### Exclusive cases
+
+#### A rational is not both negative and positive
 
 ```agda
 abstract
-  is-negative-le-ℚ⁰⁻ :
-    (q : ℚ⁰⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁰⁻ q) → is-negative-ℚ p
-  is-negative-le-ℚ⁰⁻ (q , nonpos-q) p p<q =
-    is-negative-le-zero-ℚ
-      ( p)
-      ( concatenate-le-leq-ℚ p q zero-ℚ
-        ( p<q)
-        ( leq-zero-is-nonpositive-ℚ q nonpos-q))
+  not-is-negative-is-positive-ℚ :
+    {x : ℚ} → ¬ (is-negative-ℚ x × is-positive-ℚ x)
+  not-is-negative-is-positive-ℚ {x} (N , P) =
+    not-leq-le-ℚ
+      ( zero-ℚ)
+      ( x)
+      ( le-zero-is-positive-ℚ P)
+      ( leq-zero-is-negative-ℚ N)
 ```
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index 33ca6716243..5144d0075c3 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -26,6 +26,7 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.cartesian-product-types
 open import foundation.coproduct-types
+open import foundation.decidable-propositions
 open import foundation.decidable-subtypes
 open import foundation.dependent-pair-types
 open import foundation.function-types
@@ -66,11 +67,16 @@ module _
   (x : ℚ)
   where
 
-  is-positive-ℚ : UU lzero
-  is-positive-ℚ = is-positive-fraction-ℤ (fraction-ℚ x)
+  opaque
+    is-positive-ℚ : UU lzero
+    is-positive-ℚ = is-positive-fraction-ℤ (fraction-ℚ x)
+
+    is-prop-is-positive-ℚ : is-prop is-positive-ℚ
+    is-prop-is-positive-ℚ = is-prop-is-positive-fraction-ℤ (fraction-ℚ x)
 
-  is-prop-is-positive-ℚ : is-prop is-positive-ℚ
-  is-prop-is-positive-ℚ = is-prop-is-positive-fraction-ℤ (fraction-ℚ x)
+    is-decidable-prop-is-positive-ℚ : is-decidable-prop is-positive-ℚ
+    is-decidable-prop-is-positive-ℚ =
+      ( is-prop-is-positive-ℚ , is-decidable-is-positive-ℤ (numerator-ℚ x))
 
   is-positive-prop-ℚ : Prop lzero
   pr1 is-positive-prop-ℚ = is-positive-ℚ
@@ -78,7 +84,7 @@ module _
 
 decidable-subtype-positive-ℚ : decidable-subtype lzero ℚ
 decidable-subtype-positive-ℚ x =
-  decidable-subtype-positive-fraction-ℤ (fraction-ℚ x)
+  ( is-positive-ℚ x , is-decidable-prop-is-positive-ℚ x)
 ```
 
 ### The type of positive rational numbers
@@ -109,11 +115,14 @@ module _
   is-positive-rational-ℚ⁺ : is-positive-ℚ rational-ℚ⁺
   is-positive-rational-ℚ⁺ = pr2 x
 
-  is-positive-fraction-ℚ⁺ : is-positive-fraction-ℤ fraction-ℚ⁺
-  is-positive-fraction-ℚ⁺ = is-positive-rational-ℚ⁺
+  opaque
+    unfolding is-positive-ℚ
+
+    is-positive-fraction-ℚ⁺ : is-positive-fraction-ℤ fraction-ℚ⁺
+    is-positive-fraction-ℚ⁺ = is-positive-rational-ℚ⁺
 
-  is-positive-numerator-ℚ⁺ : is-positive-ℤ numerator-ℚ⁺
-  is-positive-numerator-ℚ⁺ = is-positive-rational-ℚ⁺
+    is-positive-numerator-ℚ⁺ : is-positive-ℤ numerator-ℚ⁺
+    is-positive-numerator-ℚ⁺ = is-positive-rational-ℚ⁺
 
   is-positive-denominator-ℚ⁺ : is-positive-ℤ denominator-ℚ⁺
   is-positive-denominator-ℚ⁺ = is-positive-denominator-ℚ rational-ℚ⁺
@@ -150,19 +159,22 @@ abstract
 ### The rational image of a positive integer is positive
 
 ```agda
-abstract
+opaque
+  unfolding is-positive-ℚ
+
   is-positive-rational-ℤ :
-    (x : ℤ) → is-positive-ℤ x → is-positive-ℚ (rational-ℤ x)
-  is-positive-rational-ℤ x P = P
+    {x : ℤ} → is-positive-ℤ x → is-positive-ℚ (rational-ℤ x)
+  is-positive-rational-ℤ = id
 
 positive-rational-positive-ℤ : positive-ℤ → ℚ⁺
-positive-rational-positive-ℤ (z , pos-z) = rational-ℤ z , pos-z
+positive-rational-positive-ℤ (z , pos-z) =
+    ( rational-ℤ z , is-positive-rational-ℤ pos-z)
 
 positive-rational-ℤ⁺ : ℤ⁺ → ℚ⁺
 positive-rational-ℤ⁺ = positive-rational-positive-ℤ
 
 one-ℚ⁺ : ℚ⁺
-one-ℚ⁺ = (one-ℚ , is-positive-int-positive-ℤ one-positive-ℤ)
+one-ℚ⁺ = positive-rational-ℤ⁺ one-ℤ⁺
 ```
 
 ### The type of positive rational numbers is inhabited
@@ -184,7 +196,7 @@ positive-rational-ℕ⁺ n = positive-rational-positive-ℤ (positive-int-ℕ⁺
 
 ```agda
 opaque
-  unfolding rational-fraction-ℤ
+  unfolding is-positive-ℚ rational-fraction-ℤ
 
   is-positive-rational-fraction-ℤ :
     {x : fraction-ℤ} (P : is-positive-fraction-ℤ x) →
@@ -195,20 +207,16 @@ opaque
 ### A rational number `x` is positive if and only if `0 < x`
 
 ```agda
-module _
-  (x : ℚ)
-  where
-
-  opaque
-    unfolding le-ℚ-Prop
+opaque
+  unfolding is-positive-ℚ le-ℚ-Prop
 
-    le-zero-is-positive-ℚ : is-positive-ℚ x → le-ℚ zero-ℚ x
-    le-zero-is-positive-ℚ =
-      is-positive-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
+  le-zero-is-positive-ℚ : {x : ℚ} → is-positive-ℚ x → le-ℚ zero-ℚ x
+  le-zero-is-positive-ℚ {x} =
+    is-positive-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
 
-    is-positive-le-zero-ℚ : le-ℚ zero-ℚ x → is-positive-ℚ x
-    is-positive-le-zero-ℚ =
-      is-positive-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
+  is-positive-le-zero-ℚ : {x : ℚ} → le-ℚ zero-ℚ x → is-positive-ℚ x
+  is-positive-le-zero-ℚ {x} =
+    is-positive-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
 ```
 
 ### Zero is not a positive rational number
@@ -217,7 +225,7 @@ module _
 abstract
   is-not-positive-zero-ℚ : ¬ (is-positive-ℚ zero-ℚ)
   is-not-positive-zero-ℚ pos-0 =
-    irreflexive-le-ℚ zero-ℚ (le-zero-is-positive-ℚ zero-ℚ pos-0)
+    irreflexive-le-ℚ zero-ℚ (le-zero-is-positive-ℚ pos-0)
 ```
 
 ### The difference of a rational number with a lesser rational number is positive
@@ -231,7 +239,6 @@ module _
     is-positive-diff-le-ℚ : is-positive-ℚ (y -ℚ x)
     is-positive-diff-le-ℚ =
       is-positive-le-zero-ℚ
-        ( y -ℚ x)
         ( backward-implication
           ( iff-translate-diff-le-zero-ℚ x y)
           ( H))
@@ -252,44 +259,12 @@ module _
     commutative-add-ℚ x (y -ℚ x) ∙ left-law-positive-diff-le-ℚ
 ```
 
-### A nonzero rational number or its negative is positive
-
-```agda
-opaque
-  unfolding neg-ℚ
-
-  decide-is-negative-is-positive-is-nonzero-ℚ :
-    {x : ℚ} → is-nonzero-ℚ x → is-positive-ℚ (neg-ℚ x) + is-positive-ℚ x
-  decide-is-negative-is-positive-is-nonzero-ℚ {x} H =
-    rec-coproduct
-      ( inl ∘ is-positive-neg-is-negative-ℤ)
-      ( inr)
-      ( decide-sign-nonzero-ℤ
-        { numerator-ℚ x}
-        ( is-nonzero-numerator-is-nonzero-ℚ x H))
-```
-
-### A rational and its negative are not both positive
+### Positive rational numbers are nonzero
 
 ```agda
 opaque
-  unfolding neg-ℚ
-
-  not-is-negative-is-positive-ℚ :
-    (x : ℚ) → ¬ (is-positive-ℚ (neg-ℚ x) × is-positive-ℚ x)
-  not-is-negative-is-positive-ℚ x (N , P) =
-    is-not-negative-and-positive-ℤ
-      ( numerator-ℚ x)
-      ( ( is-negative-eq-ℤ
-          (neg-neg-ℤ (numerator-ℚ x))
-          (is-negative-neg-is-positive-ℤ {numerator-ℚ (neg-ℚ x)} N)) ,
-        ( P))
-```
-
-### Positive rational numbers are nonzero
+  unfolding is-positive-ℚ
 
-```agda
-abstract
   is-nonzero-is-positive-ℚ : {x : ℚ} → is-positive-ℚ x → is-nonzero-ℚ x
   is-nonzero-is-positive-ℚ {x} H =
     is-nonzero-is-nonzero-numerator-ℚ x
@@ -309,8 +284,7 @@ abstract
     (p : ℚ⁺) (q : ℚ) → leq-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
   is-positive-leq-ℚ⁺ (p , pos-p) q p≤q =
     is-positive-le-zero-ℚ
-      ( q)
-      ( concatenate-le-leq-ℚ _ _ _ (le-zero-is-positive-ℚ p pos-p) p≤q)
+      ( concatenate-le-leq-ℚ _ _ _ (le-zero-is-positive-ℚ pos-p) p≤q)
 
   is-positive-le-ℚ⁺ :
     (p : ℚ⁺) (q : ℚ) → le-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
diff --git a/src/elementary-number-theory/rational-numbers.lagda.md b/src/elementary-number-theory/rational-numbers.lagda.md
index 2f51198a474..991d52230c5 100644
--- a/src/elementary-number-theory/rational-numbers.lagda.md
+++ b/src/elementary-number-theory/rational-numbers.lagda.md
@@ -149,6 +149,16 @@ opaque
   pr2 (neg-ℚ (x , H)) = is-reduced-neg-fraction-ℤ x H
 ```
 
+### The negation of zero is zero
+
+```agda
+opaque
+  unfolding neg-ℚ
+
+  neg-zero-ℚ : neg-ℚ zero-ℚ ＝ zero-ℚ
+  neg-zero-ℚ = refl
+```
+
 ### The mediant of two rationals
 
 ```agda
@@ -287,9 +297,9 @@ module _
 opaque
   unfolding neg-ℚ
 
-  preserves-neg-rational-ℤ :
+  neg-rational-ℤ :
     (k : ℤ) → rational-ℤ (neg-ℤ k) ＝ neg-ℚ (rational-ℤ k)
-  preserves-neg-rational-ℤ k =
+  neg-rational-ℤ k =
     eq-ℚ (rational-ℤ (neg-ℤ k)) (neg-ℚ (rational-ℤ k)) refl refl
 ```
 
@@ -297,13 +307,12 @@ opaque
 
 ```agda
 opaque
-  unfolding neg-ℚ
-  unfolding rational-fraction-ℤ
+  unfolding neg-ℚ rational-fraction-ℤ
 
-  preserves-neg-rational-fraction-ℤ :
+  neg-rational-fraction-ℤ :
     (x : fraction-ℤ) →
     rational-fraction-ℤ (neg-fraction-ℤ x) ＝ neg-ℚ (rational-fraction-ℤ x)
-  preserves-neg-rational-fraction-ℤ x =
+  neg-rational-fraction-ℤ x =
     ( eq-ℚ-sim-fraction-ℤ
       ( neg-fraction-ℤ x)
       ( fraction-ℚ (neg-ℚ (rational-fraction-ℤ x)))
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index fd41bb63716..154d5e5c068 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -11,7 +11,9 @@ module elementary-number-theory.squares-rational-numbers where
 ```agda
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-nonpositive-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
@@ -52,15 +54,14 @@ square-root-ℚ _ (root , _) = root
 ### Squares in ℚ are nonnegative
 
 ```agda
-is-nonnegative-square-ℚ : (a : ℚ) → is-nonnegative-ℚ (square-ℚ a)
-is-nonnegative-square-ℚ a =
-  rec-coproduct
-    ( λ H →
-      is-nonnegative-is-positive-ℚ
-        ( a *ℚ a)
-        ( is-positive-mul-negative-ℚ {a} {a} H H))
-    ( λ H → is-nonnegative-mul-ℚ a a H H)
-    ( decide-is-negative-is-nonnegative-ℚ a)
+abstract
+  is-nonnegative-square-ℚ : (q : ℚ) → is-nonnegative-ℚ (square-ℚ q)
+  is-nonnegative-square-ℚ q =
+    rec-coproduct
+      ( λ is-nonpos-q →
+        is-nonnegative-mul-nonpositive-ℚ is-nonpos-q is-nonpos-q)
+      ( λ is-nonneg-q → is-nonnegative-mul-ℚ is-nonneg-q is-nonneg-q)
+      ( is-nonpositive-or-nonnegative-ℚ q)
 ```
 
 ### The square of the negation of `x` is the square of `x`
diff --git a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
index a14cf72ef8c..4ef6623999e 100644
--- a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
@@ -37,13 +37,3 @@ le-prop-ℚ⁰⁺ p q = le-ℚ-Prop (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q
 le-ℚ⁰⁺ : (p q : ℚ⁰⁺) → UU lzero
 le-ℚ⁰⁺ p q = type-Prop (le-prop-ℚ⁰⁺ p q)
 ```
-
-## Properties
-
-### Zero is less than positive rational numbers as nonnegative rational numbers
-
-```agda
-abstract
-  le-zero-nonnegative-ℚ⁰⁺ : (q : ℚ⁺) → le-ℚ⁰⁺ zero-ℚ⁰⁺ (nonnegative-ℚ⁺ q)
-  le-zero-nonnegative-ℚ⁰⁺ (q , pos-q) = le-zero-is-positive-ℚ q pos-q
-```
diff --git a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
index c6ca66c15fb..83f68e3629e 100644
--- a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
@@ -87,16 +87,15 @@ opaque
   mediant-zero-ℚ⁺ x =
     ( mediant-ℚ zero-ℚ (rational-ℚ⁺ x) ,
       is-positive-le-zero-ℚ
-        ( mediant-ℚ zero-ℚ (rational-ℚ⁺ x))
         ( le-left-mediant-ℚ
           ( zero-ℚ)
           ( rational-ℚ⁺ x)
-          ( le-zero-is-positive-ℚ (rational-ℚ⁺ x) (is-positive-rational-ℚ⁺ x))))
+          ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x))))
 
   le-mediant-zero-ℚ⁺ : (x : ℚ⁺) → le-ℚ⁺ (mediant-zero-ℚ⁺ x) x
   le-mediant-zero-ℚ⁺ x =
     le-right-mediant-ℚ
       ( zero-ℚ)
       ( rational-ℚ⁺ x)
-      ( le-zero-is-positive-ℚ (rational-ℚ⁺ x) (is-positive-rational-ℚ⁺ x))
+      ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x))
 ```
diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
index e3986f030dc..79fe7312b00 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -519,11 +519,10 @@ abstract
 
 ```agda
 opaque
-  unfolding le-ℚ-Prop
-  unfolding neg-ℚ
+  unfolding le-ℚ-Prop neg-ℚ
 
-  neg-le-ℚ : (x y : ℚ) → le-ℚ x y → le-ℚ (neg-ℚ y) (neg-ℚ x)
-  neg-le-ℚ x y = neg-le-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
+  neg-le-ℚ : {x y : ℚ} → le-ℚ x y → le-ℚ (neg-ℚ y) (neg-ℚ x)
+  neg-le-ℚ {x} {y} = neg-le-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
 ```
 
 ### Transposing additions on strict inequalities of rational numbers
diff --git a/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md b/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md
index b8aa906a321..568cc71d27d 100644
--- a/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md
+++ b/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md
@@ -259,7 +259,7 @@ module _
           ( ε)
           ( concatenate-leq-le-ℚ q zero-ℚ ε
             ( q≤0)
-            ( le-zero-is-positive-ℚ ε is-pos-ε))
+            ( le-zero-is-positive-ℚ is-pos-ε))
           ( ε<q)
 
     leq-zero-upper-real-dist-Metric-Space :
@@ -267,11 +267,11 @@ module _
     leq-zero-upper-real-dist-Metric-Space q =
       rec-trunc-Prop
         ( le-ℚ-Prop zero-ℚ q)
-        ( λ ((ε , Nεxy) , ε<q) →
+        ( λ (((ε , is-pos-ε) , Nεxy) , ε<q) →
           transitive-le-ℚ
             ( zero-ℚ)
-            ( rational-ℚ⁺ ε)
+            ( ε)
             ( q)
             ( ε<q)
-            ( le-zero-is-positive-ℚ (pr1 ε) (pr2 ε)))
+            ( le-zero-is-positive-ℚ is-pos-ε))
 ```
diff --git a/src/metric-spaces/located-metric-spaces.lagda.md b/src/metric-spaces/located-metric-spaces.lagda.md
index 64aa11e62eb..1fef763229a 100644
--- a/src/metric-spaces/located-metric-spaces.lagda.md
+++ b/src/metric-spaces/located-metric-spaces.lagda.md
@@ -136,12 +136,12 @@ module _
         ( λ 0<p →
           let
             r = mediant-ℚ p q
-            p⁺ = (p , is-positive-le-zero-ℚ p 0<p)
+            p⁺ = (p , is-positive-le-zero-ℚ 0<p)
             p<r = le-left-mediant-ℚ p q p<q
             r<q = le-right-mediant-ℚ p q p<q
             r⁺ =
               ( r ,
-                is-positive-le-zero-ℚ r (transitive-le-ℚ zero-ℚ p r p<r 0<p))
+                is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ p r p<r 0<p))
           in
             map-disjunction
               ( λ ¬Npxy p∈U →
diff --git a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
index 3aae8a5cb1c..5bd39143c27 100644
--- a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
+++ b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
@@ -407,8 +407,6 @@ is-cauchy-approximation-rational-ℚ⁺ :
     rational-ℚ⁺
 is-cauchy-approximation-rational-ℚ⁺ ε δ =
   ( leq-le-ℚ
-    { rational-ℚ⁺ δ}
-    { rational-ℚ⁺ ε +ℚ (rational-ℚ⁺ (ε +ℚ⁺ δ))}
     ( transitive-le-ℚ
       ( rational-ℚ⁺ δ)
       ( rational-ℚ⁺ (ε +ℚ⁺ δ))
@@ -418,8 +416,6 @@ is-cauchy-approximation-rational-ℚ⁺ ε δ =
         ( ε +ℚ⁺ δ))
       ( le-right-add-ℚ⁺ ε δ))) ,
   ( leq-le-ℚ
-    { rational-ℚ⁺ ε}
-    { rational-ℚ⁺ δ +ℚ (rational-ℚ⁺ (ε +ℚ⁺ δ))}
     ( transitive-le-ℚ
       ( rational-ℚ⁺ ε)
       ( rational-ℚ⁺ (ε +ℚ⁺ δ))
@@ -444,9 +440,7 @@ is-zero-limit-rational-ℚ⁺ ε δ =
     { zero-ℚ}
     { rational-ℚ⁺ (ε +ℚ⁺ (ε +ℚ⁺ δ))}
     ( le-zero-is-positive-ℚ
-      ( rational-ℚ⁺ (ε +ℚ⁺ (ε +ℚ⁺ δ)))
-      ( is-positive-rational-ℚ⁺
-        (ε +ℚ⁺ (ε +ℚ⁺ δ))))) ,
+      ( is-positive-rational-ℚ⁺ (ε +ℚ⁺ (ε +ℚ⁺ δ))))) ,
   ( leq-le-ℚ
     { rational-ℚ⁺ ε}
     { zero-ℚ +ℚ rational-ℚ⁺ (ε +ℚ⁺ δ)}
diff --git a/src/metric-spaces/rational-sequences-approximating-zero.lagda.md b/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
index dc29336abcd..821404d222f 100644
--- a/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
+++ b/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
@@ -18,6 +18,7 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-natural-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-natural-numbers
diff --git a/src/order-theory.lagda.md b/src/order-theory.lagda.md
index ce2ff9740b2..14e68460e11 100644
--- a/src/order-theory.lagda.md
+++ b/src/order-theory.lagda.md
@@ -111,6 +111,7 @@ open import order-theory.order-preserving-maps-large-posets public
 open import order-theory.order-preserving-maps-large-preorders public
 open import order-theory.order-preserving-maps-posets public
 open import order-theory.order-preserving-maps-preorders public
+open import order-theory.order-preserving-maps-total-orders public
 open import order-theory.ordinals public
 open import order-theory.poset-closed-intervals-lattices public
 open import order-theory.poset-closed-intervals-posets public
diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md
index 1638130acd9..4d127ed3ff8 100644
--- a/src/order-theory/order-preserving-maps-posets.lagda.md
+++ b/src/order-theory/order-preserving-maps-posets.lagda.md
@@ -25,8 +25,10 @@ open import order-theory.posets
 
 ## Idea
 
-A map `f : P → Q` between the underlying types of two posets is said to be
-**order preserving** if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
+A map `f : P → Q` between the underlying types of two
+[posets](order-theory.posets.md) is said to be
+{{#concept "order preserving" Agda=hom-Poset Disambiguation="map between posets"}}
+if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
 
 ## Definition
 
diff --git a/src/order-theory/order-preserving-maps-total-orders.lagda.md b/src/order-theory/order-preserving-maps-total-orders.lagda.md
new file mode 100644
index 00000000000..f3ae69baab9
--- /dev/null
+++ b/src/order-theory/order-preserving-maps-total-orders.lagda.md
@@ -0,0 +1,193 @@
+# Order preserving maps between total orders
+
+```agda
+module order-theory.order-preserving-maps-total-orders where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-functions
+open import foundation.disjunction
+open import foundation.function-types
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import order-theory.order-preserving-maps-posets
+open import order-theory.total-orders
+```
+
+</details>
+
+## Idea
+
+A map `f : P → Q` between the underlying types of two
+[total orders](order-theory.total-orders.md) is said to be
+{{#concept "order preserving" Agda=hom-Total-Order Disambiguation="map between total orders"}}
+if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
+
+## Definition
+
+### Order preserving maps
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (P : Total-Order l1 l2) (Q : Total-Order l3 l4)
+  where
+
+  preserves-order-prop-Total-Order :
+    (type-Total-Order P → type-Total-Order Q) → Prop (l1 ⊔ l2 ⊔ l4)
+  preserves-order-prop-Total-Order =
+    preserves-order-prop-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  preserves-order-Total-Order :
+    (type-Total-Order P → type-Total-Order Q) → UU (l1 ⊔ l2 ⊔ l4)
+  preserves-order-Total-Order =
+    preserves-order-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  is-prop-preserves-order-Total-Order :
+    (f : type-Total-Order P → type-Total-Order Q) →
+    is-prop (preserves-order-Total-Order f)
+  is-prop-preserves-order-Total-Order =
+    is-prop-preserves-order-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  hom-set-Total-Order : Set (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+  hom-set-Total-Order =
+    set-subset
+      ( function-Set (type-Total-Order P) (set-Total-Order Q))
+      ( preserves-order-prop-Total-Order)
+
+  hom-Total-Order : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+  hom-Total-Order = type-Set hom-set-Total-Order
+
+  map-hom-Total-Order :
+    hom-Total-Order → type-Total-Order P → type-Total-Order Q
+  map-hom-Total-Order =
+    map-hom-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  preserves-order-hom-Total-Order :
+    (f : hom-Total-Order) → preserves-order-Total-Order (map-hom-Total-Order f)
+  preserves-order-hom-Total-Order =
+    preserves-order-hom-Poset (poset-Total-Order P) (poset-Total-Order Q)
+```
+
+## Properties
+
+### Order-preserving maps distribute over the minimum operation
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (P : Total-Order l1 l2) (Q : Total-Order l3 l4)
+  (f : hom-Total-Order P Q)
+  where
+
+  abstract
+    distributive-map-hom-min-Total-Order :
+      (x y : type-Total-Order P) →
+      map-hom-Total-Order P Q f (min-Total-Order P x y) ＝
+      min-Total-Order Q
+        ( map-hom-Total-Order P Q f x)
+        ( map-hom-Total-Order P Q f y)
+    distributive-map-hom-min-Total-Order x y =
+      elim-disjunction
+        ( Id-Prop
+          ( set-Total-Order Q)
+          ( map-hom-Total-Order P Q f (min-Total-Order P x y))
+          ( min-Total-Order Q
+            ( map-hom-Total-Order P Q f x)
+            ( map-hom-Total-Order P Q f y)))
+        ( λ x≤y →
+          equational-reasoning
+            map-hom-Total-Order P Q f (min-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f x
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( left-leq-right-min-Total-Order P x y x≤y)
+            ＝
+              min-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( left-leq-right-min-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f x y x≤y)))
+        ( λ y≤x →
+          equational-reasoning
+            map-hom-Total-Order P Q f (min-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f y
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( right-leq-left-min-Total-Order P x y y≤x)
+            ＝
+              min-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( right-leq-left-min-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f y x y≤x)))
+        ( is-total-Total-Order P x y)
+```
+
+### Order-preserving maps distribute over the maximum operation
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (P : Total-Order l1 l2) (Q : Total-Order l3 l4)
+  (f : hom-Total-Order P Q)
+  where
+
+  abstract
+    distributive-map-hom-max-Total-Order :
+      (x y : type-Total-Order P) →
+      map-hom-Total-Order P Q f (max-Total-Order P x y) ＝
+      max-Total-Order Q
+        ( map-hom-Total-Order P Q f x)
+        ( map-hom-Total-Order P Q f y)
+    distributive-map-hom-max-Total-Order x y =
+      elim-disjunction
+        ( Id-Prop
+          ( set-Total-Order Q)
+          ( map-hom-Total-Order P Q f (max-Total-Order P x y))
+          ( max-Total-Order Q
+            ( map-hom-Total-Order P Q f x)
+            ( map-hom-Total-Order P Q f y)))
+        ( λ x≤y →
+          equational-reasoning
+            map-hom-Total-Order P Q f (max-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f y
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( left-leq-right-max-Total-Order P x y x≤y)
+            ＝
+              max-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( left-leq-right-max-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f x y x≤y)))
+        ( λ y≤x →
+          equational-reasoning
+            map-hom-Total-Order P Q f (max-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f x
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( right-leq-left-max-Total-Order P x y y≤x)
+            ＝
+              max-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( right-leq-left-max-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f y x y≤x)))
+        ( is-total-Total-Order P x y)
+```
diff --git a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
index 6d2c68d957e..d970a20b789 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -12,6 +12,8 @@ module real-numbers.addition-lower-dedekind-real-numbers where
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
@@ -213,9 +215,7 @@ module _
   {l : Level} (x : lower-ℝ l)
   where
 
-  opaque
-    unfolding neg-ℚ
-
+  abstract
     right-unit-law-add-lower-ℝ : add-lower-ℝ x (lower-real-ℚ zero-ℚ) ＝ x
     right-unit-law-add-lower-ℝ =
       eq-sim-cut-lower-ℝ
@@ -231,8 +231,7 @@ module _
                   ( is-in-cut-diff-rational-ℚ⁺-lower-ℝ
                     ( x)
                     ( p)
-                    ( neg-ℚ q ,
-                      is-positive-le-zero-ℚ (neg-ℚ q) (neg-le-ℚ q zero-ℚ q<0))
+                    ( neg-ℚ⁻ (q , is-negative-le-zero-ℚ q<0))
                     ( p<x)))) ,
           (λ p p<x →
             elim-exists
@@ -241,12 +240,11 @@ module _
                 intro-exists
                   ( q , p -ℚ q)
                   ( q<x ,
-                    tr
-                      ( λ r → le-ℚ r zero-ℚ)
+                    binary-tr
+                      ( le-ℚ)
                       ( distributive-neg-diff-ℚ q p)
+                      ( neg-zero-ℚ)
                       ( neg-le-ℚ
-                        ( zero-ℚ)
-                        ( q -ℚ p)
                         ( backward-implication
                           ( iff-translate-diff-le-zero-ℚ p q)
                           ( p<q))) ,
diff --git a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
index adf12e8e83d..44bd7f51aee 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -140,7 +140,7 @@ module _
                   ( q -ℚ px)
                   ( neg-ℚ (q -ℚ p))
                   ( neg-ℚ ε)
-                  ( neg-le-ℚ ε (q -ℚ p) (le-mediant-zero-ℚ⁺ q-p⁺))))
+                  ( neg-le-ℚ (le-mediant-zero-ℚ⁺ q-p⁺))))
               ( y<py) ,
             inv ( is-identity-right-conjugation-add-ℚ (px +ℚ ε) q))
       where open do-syntax-trunc-Prop (cut-add-upper-ℝ q)
@@ -224,7 +224,7 @@ module _
                   ( is-in-cut-add-rational-ℚ⁺-upper-ℝ
                     ( x)
                     ( p)
-                    ( q , is-positive-le-zero-ℚ q 0<q)
+                    ( q , is-positive-le-zero-ℚ 0<q)
                     ( x<p)))) ,
           ( λ q x<q →
             elim-exists
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 3fbc4ba1e4f..c67e8c9b7ac 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -381,7 +381,7 @@ abstract
                     neg-ℚ a
                     ≤ neg-ℚ (b -ℚ ε)
                       by
-                        neg-leq-ℚ _ _
+                        neg-leq-ℚ
                           ( leq-transpose-right-add-ℚ _ _ _ (leq-le-ℚ b<a+ε))
                     ≤ ε -ℚ b
                       by leq-eq-ℚ _ _ (distributive-neg-diff-ℚ _ _)
@@ -390,7 +390,7 @@ abstract
                         preserves-leq-right-add-ℚ ε
                           ( neg-ℚ b)
                           ( neg-ℚ p)
-                          ( neg-leq-ℚ _ _
+                          ( neg-leq-ℚ
                             ( leq-lower-upper-cut-ℝ x p b p<x x<b))
                     ≤ rational-abs-ℚ (ε -ℚ p)
                       by leq-abs-ℚ _
@@ -425,7 +425,7 @@ abstract
                 ( chain-of-inequalities
                     neg-ℚ b
                     ≤ neg-ℚ a
-                      by neg-leq-ℚ _ _ (leq-lower-upper-cut-ℝ x a b a<x x<b)
+                      by neg-leq-ℚ (leq-lower-upper-cut-ℝ x a b a<x x<b)
                     ≤ rational-abs-ℚ a
                       by neg-leq-abs-ℚ a
                     ≤ rational-abs-ℚ a +ℚ ε
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index 7eb3b7277c0..81975525891 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -33,6 +33,7 @@ open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.poset-closed-intervals-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
diff --git a/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
index 5ee824c84c8..59162e2554b 100644
--- a/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
@@ -79,7 +79,7 @@ module _
         (λ p → le-ℚ p q × is-in-cut-neg-lower-ℝ p)
         ( neg-ℚ)
         ( λ p (-q<p , p<x) →
-          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ (neg-ℚ q) p -q<p) ,
+          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ -q<p) ,
           tr (is-in-cut-lower-ℝ x) (inv (neg-neg-ℚ p)) p<x)
         ( forward-implication (is-rounded-cut-lower-ℝ x (neg-ℚ q)) -q<x)
     pr2 (is-rounded-cut-neg-lower-ℝ q) =
@@ -88,7 +88,7 @@ module _
         ( λ p (p<q , -q<x) →
           backward-implication
             ( is-rounded-cut-lower-ℝ x (neg-ℚ q))
-            ( intro-exists (neg-ℚ p) (neg-le-ℚ p q p<q , -q<x)))
+            ( intro-exists (neg-ℚ p) (neg-le-ℚ p<q , -q<x)))
 
   neg-lower-ℝ : upper-ℝ l
   pr1 neg-lower-ℝ = cut-neg-lower-ℝ
@@ -130,7 +130,7 @@ module _
           tr
             ( λ x → le-ℚ x (neg-ℚ p))
             ( neg-neg-ℚ q)
-            ( neg-le-ℚ p (neg-ℚ q) p<-q) ,
+            ( neg-le-ℚ p<-q) ,
           tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)) x<p)
         ( forward-implication (is-rounded-cut-upper-ℝ x (neg-ℚ q)) x<-q)
     pr2 (is-rounded-cut-neg-upper-ℝ q) =
@@ -139,7 +139,7 @@ module _
         ( λ r (q<r , x<-r) →
           backward-implication
             ( is-rounded-cut-upper-ℝ x (neg-ℚ q))
-            ( intro-exists (neg-ℚ r) (neg-le-ℚ q r q<r , x<-r)))
+            ( intro-exists (neg-ℚ r) (neg-le-ℚ q<r , x<-r)))
 
   neg-upper-ℝ : lower-ℝ l
   pr1 neg-upper-ℝ = cut-neg-upper-ℝ
@@ -201,9 +201,9 @@ neg-lower-real-ℚ q =
       ( cut-neg-lower-ℝ (lower-real-ℚ q))
       ( cut-upper-real-ℚ (neg-ℚ q))
       ( (λ p -p<q →
-          tr (le-ℚ (neg-ℚ q)) (neg-neg-ℚ p) (neg-le-ℚ (neg-ℚ p) q -p<q)) ,
+          tr (le-ℚ (neg-ℚ q)) (neg-neg-ℚ p) (neg-le-ℚ -p<q)) ,
         (λ p -q<p →
-          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ (neg-ℚ q) p -q<p))))
+          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ -q<p))))
 ```
 
 ### The negation of a rational projected to an upper real is the projection of its negation as a lower real
@@ -222,10 +222,10 @@ neg-upper-real-ℚ q =
           tr
             ( λ r → le-ℚ r (neg-ℚ q))
             ( neg-neg-ℚ p)
-            ( neg-le-ℚ q (neg-ℚ p) q<-p)) ,
+            ( neg-le-ℚ q<-p)) ,
         (λ p p<-q →
           tr
             ( λ r → le-ℚ r (neg-ℚ p))
             ( neg-neg-ℚ q)
-            ( neg-le-ℚ p (neg-ℚ q) p<-q))))
+            ( neg-le-ℚ p<-q))))
 ```
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 3f84a77108c..a3e57775b74 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -82,7 +82,7 @@ module _
       ( disjunction-Prop (lower-cut-neg-ℝ q) (upper-cut-neg-ℝ r))
       ( inr-disjunction)
       ( inl-disjunction)
-      ( is-located-lower-upper-cut-ℝ x (neg-ℚ r) (neg-ℚ q) (neg-le-ℚ q r q<r))
+      ( is-located-lower-upper-cut-ℝ x (neg-ℚ r) (neg-ℚ q) (neg-le-ℚ q<r))
 
   opaque
     neg-ℝ : ℝ l
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 9f97dc4cc21..7b555aacc04 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -13,6 +13,7 @@ open import elementary-number-theory.addition-nonnegative-rational-numbers
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-nonnegative-rational-numbers
@@ -100,7 +101,7 @@ eq-ℝ⁰⁺ _ _ = eq-type-subtype is-nonnegative-prop-ℝ
 abstract
   is-nonnegative-real-ℚ⁰⁺ : (q : ℚ⁰⁺) → is-nonnegative-ℝ (real-ℚ⁰⁺ q)
   is-nonnegative-real-ℚ⁰⁺ (q , nonneg-q) =
-    preserves-leq-real-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ q nonneg-q)
+    preserves-leq-real-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ nonneg-q)
 
 nonnegative-real-ℚ⁰⁺ : ℚ⁰⁺ → ℝ⁰⁺ lzero
 nonnegative-real-ℚ⁰⁺ q = (real-ℚ⁰⁺ q , is-nonnegative-real-ℚ⁰⁺ q)
@@ -270,7 +271,6 @@ module _
         intro-exists
           ( q ,
             is-positive-le-zero-ℚ
-              ( q)
               ( reflects-le-real-ℚ zero-ℚ q
                 ( concatenate-leq-le-ℝ
                   ( zero-ℝ)
@@ -443,7 +443,6 @@ module _
       le-ℝ (real-ℝ⁰⁺ x) (real-ℚ q) → is-positive-ℚ q
     is-positive-le-nonnegative-real-ℚ x<q =
       is-positive-le-zero-ℚ
-        ( q)
         ( reflects-le-real-ℚ _ _
           ( concatenate-leq-le-ℝ _ _ _ (is-nonnegative-real-ℝ⁰⁺ x) x<q))
 ```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 8922e6023bc..4fc3b49bc8b 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -106,7 +106,7 @@ module _
     exists-ℚ⁺-in-lower-cut-is-positive-ℝ =
       elim-exists
         ( ∃ ℚ⁺ (λ p → lower-cut-ℝ x (rational-ℚ⁺ p)))
-        ( λ p (0<p , p<x) → intro-exists (p , is-positive-le-zero-ℚ p 0<p) p<x)
+        ( λ p (0<p , p<x) → intro-exists (p , is-positive-le-zero-ℚ 0<p) p<x)
 
     is-positive-exists-ℚ⁺-in-lower-cut-ℝ :
       exists ℚ⁺ (λ p → lower-cut-ℝ x (rational-ℚ⁺ p)) →
@@ -115,7 +115,7 @@ module _
       elim-exists
         ( is-positive-prop-ℝ x)
         ( λ (p , pos-p) p<x →
-          intro-exists p (le-zero-is-positive-ℚ p pos-p , p<x))
+          intro-exists p (le-zero-is-positive-ℚ pos-p , p<x))
 
   is-positive-iff-exists-ℚ⁺-in-lower-cut-ℝ :
     is-positive-ℝ x ↔ exists ℚ⁺ (λ p → lower-cut-ℝ x (rational-ℚ⁺ p))
@@ -187,13 +187,13 @@ module _
   positive-diff-le-ℝ = (y -ℝ x , is-positive-diff-le-ℝ)
 ```
 
-### Positive rational numbers are positive real numbers
+### The canonical embedding of rational numbers preserves positivity
 
 ```agda
-is-positive-real-positive-ℚ :
+preserves-is-positive-real-ℚ :
   (q : ℚ) → is-positive-ℚ q → is-positive-ℝ (real-ℚ q)
-is-positive-real-positive-ℚ q pos-q =
-  preserves-le-real-ℚ zero-ℚ q (le-zero-is-positive-ℚ q pos-q)
+preserves-is-positive-real-ℚ q pos-q =
+  preserves-le-real-ℚ zero-ℚ q (le-zero-is-positive-ℚ pos-q)
 
 opaque
   unfolding le-ℝ real-ℚ
@@ -204,8 +204,8 @@ opaque
     elim-exists
       ( is-positive-prop-ℚ q)
       ( λ r (0<r , r<q) →
-        is-positive-le-zero-ℚ q (transitive-le-ℚ zero-ℚ r q r<q 0<r))
+        is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ r q r<q 0<r))
 
 positive-real-ℚ⁺ : ℚ⁺ → ℝ⁺ lzero
-positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , is-positive-real-positive-ℚ q pos-q)
+positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , preserves-is-positive-real-ℚ q pos-q)
 ```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index fb1b378d78f..a68d6e8cbd1 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -751,7 +751,7 @@ module _
               ( right-inverse-law-add-ℝ x)
               ( preserves-le-right-add-ℝ (neg-ℝ x) x y x<y))
         intro-exists
-          ( q , is-positive-le-zero-ℚ q (reflects-le-real-ℚ _ _ 0<q))
+          ( q , is-positive-le-zero-ℚ (reflects-le-real-ℚ _ _ 0<q))
           ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
 ```
 

From 124c9f689e58dc348553c33bbb7ae304d0254bf5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 10 Oct 2025 16:00:52 -0700
Subject: [PATCH 002/134] make pre-commit

---
 .../archimedean-property-rational-numbers.lagda.md              | 2 +-
 ...inequalities-positive-and-negative-rational-numbers.lagda.md | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
index f2fdcf6a303..f9f12927fbf 100644
--- a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
@@ -15,7 +15,7 @@ open import elementary-number-theory.integers
 open import elementary-number-theory.multiplication-integer-fractions
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.natural-numbers
--- open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
index 3c35c0ff087..ac81b04c088 100644
--- a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -23,7 +23,7 @@ open import foundation.dependent-pair-types
 ## Idea
 
 This page describes
-[inequalities](elementary-number-theory.inequalities-rational-numbers.md) and
+[inequalities](elementary-number-theory.inequality-rational-numbers.md) and
 [strict inequalities](elementary-number-theory.strict-inequality-rational-numbers.md)
 between [positive](elementary-number-theory.positive-rational-numbers.md),
 [negative](elementary-number-theory.negative-rational-numbers.md),

From 9dc460aca8f3d50aede96fcb7222cc61ea54219b Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 11 Oct 2025 21:23:22 -0700
Subject: [PATCH 003/134] Update

---
 ...ive-and-negative-rational-numbers.lagda.md | 15 ++++++++
 .../negative-rational-numbers.lagda.md        |  6 ++--
 .../nonnegative-rational-numbers.lagda.md     | 25 ++++++++++++++
 .../nonpositive-rational-numbers.lagda.md     | 13 +++++++
 ...ive-and-negative-rational-numbers.lagda.md | 20 +++++------
 .../rational-numbers.lagda.md                 |  3 +-
 ...e-roots-positive-rational-numbers.lagda.md |  2 +-
 .../squares-rational-numbers.lagda.md         |  6 ++--
 .../nonnegative-real-numbers.lagda.md         |  7 ++--
 ...re-roots-nonnegative-real-numbers.lagda.md | 34 ++++++++++---------
 10 files changed, 93 insertions(+), 38 deletions(-)

diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
index ac81b04c088..414fa5ecdc3 100644
--- a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -11,6 +11,7 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -55,6 +56,20 @@ abstract
   leq-negative-nonnegative-ℚ p q = leq-le-ℚ (le-negative-nonnegative-ℚ p q)
 ```
 
+#### Any negative rational number is less than any positive rational number
+
+```agda
+abstract
+  le-negative-positive-ℚ :
+    (p : ℚ⁻) (q : ℚ⁺) → le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁺ q)
+  le-negative-positive-ℚ p q =
+    le-negative-nonnegative-ℚ p (nonnegative-ℚ⁺ q)
+
+  leq-negative-positive-ℚ :
+    (p : ℚ⁻) (q : ℚ⁺) → leq-ℚ (rational-ℚ⁻ p) (rational-ℚ⁺ q)
+  leq-negative-positive-ℚ p q = leq-le-ℚ (le-negative-positive-ℚ p q)
+```
+
 #### A nonpositive rational number is less than a positive rational number
 
 ```agda
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 2dc1da8460c..20a8ca5963f 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -264,13 +264,13 @@ opaque
   mediant-zero-ℚ⁻ : ℚ⁻ → ℚ⁻
   mediant-zero-ℚ⁻ (q , is-neg-q) =
     ( mediant-ℚ q zero-ℚ ,
-      is-negative-le-zero-ℚ _
-        ( le-right-mediant-ℚ _ _ (le-zero-is-negative-ℚ q is-neg-q)))
+      is-negative-le-zero-ℚ
+      ( le-right-mediant-ℚ _ _ (le-zero-is-negative-ℚ is-neg-q)))
 
   le-mediant-zero-ℚ⁻ :
     (q : ℚ⁻) → le-ℚ (rational-ℚ⁻ q) (rational-ℚ⁻ (mediant-zero-ℚ⁻ q))
   le-mediant-zero-ℚ⁻ (q , is-neg-q) =
-    le-left-mediant-ℚ _ _ (le-zero-is-negative-ℚ q is-neg-q)
+    le-left-mediant-ℚ _ _ (le-zero-is-negative-ℚ is-neg-q)
 
 rational-mediant-zero-ℚ⁻ : ℚ⁻ → ℚ
 rational-mediant-zero-ℚ⁻ q = rational-ℚ⁻ (mediant-zero-ℚ⁻ q)
diff --git a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
index 621a2787146..b7d23dc65ad 100644
--- a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
@@ -249,3 +249,28 @@ abstract
 nonnegative-diff-leq-ℚ : (x y : ℚ) → leq-ℚ x y → ℚ⁰⁺
 nonnegative-diff-leq-ℚ x y x≤y = (y -ℚ x , is-nonnegative-diff-leq-ℚ x y x≤y)
 ```
+
+### If `x ≤ y` and `x` is nonnegative, then `y` is nonnegative
+
+```agda
+abstract
+  is-nonnegative-leq-ℚ⁰⁺ :
+    (x : ℚ⁰⁺) (y : ℚ) → leq-ℚ (rational-ℚ⁰⁺ x) y →
+    is-nonnegative-ℚ y
+  is-nonnegative-leq-ℚ⁰⁺ (x , is-nonneg-x) y x≤y =
+    is-nonnegative-leq-zero-ℚ
+      ( transitive-leq-ℚ zero-ℚ x y
+        ( x≤y)
+        ( leq-zero-is-nonnegative-ℚ is-nonneg-x))
+```
+
+### If `x < y` and `x` is nonnegative, then `y` is nonnegative
+
+```agda
+abstract
+  is-nonnegative-le-ℚ⁰⁺ :
+    (x : ℚ⁰⁺) (y : ℚ) → le-ℚ (rational-ℚ⁰⁺ x) y →
+    is-nonnegative-ℚ y
+  is-nonnegative-le-ℚ⁰⁺ x y x<y =
+    is-nonnegative-leq-ℚ⁰⁺ x y (leq-le-ℚ x<y)
+```
diff --git a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
index b0506e21798..fa039b16795 100644
--- a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
@@ -120,3 +120,16 @@ is-nonpositive-iff-leq-zero-ℚ q =
   ( leq-zero-is-nonpositive-ℚ ,
     is-nonpositive-leq-zero-ℚ)
 ```
+
+### If `p ≤ q` and `q` is nonpositive, then `p` is nonpositive
+
+```agda
+abstract
+  is-nonpositive-leq-ℚ⁰⁻ :
+    (q : ℚ⁰⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁰⁻ q) → is-nonpositive-ℚ p
+  is-nonpositive-leq-ℚ⁰⁻ (q , nonpos-q) p p≤q =
+    is-nonpositive-leq-zero-ℚ
+      ( transitive-leq-ℚ p q zero-ℚ
+        ( leq-zero-is-nonpositive-ℚ nonpos-q)
+        ( p≤q))
+```
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index 8a11f9eb930..a2f18c7d166 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -211,7 +211,7 @@ neg-ℚ⁰⁻ (q , is-nonpos-q) =
 
 ### Exclusive cases
 
-#### A rational is not both negative and positive
+#### A rational number is not both negative and positive
 
 ```agda
 abstract
@@ -225,16 +225,16 @@ abstract
       ( leq-zero-is-negative-ℚ N)
 ```
 
-### If `p ≤ q` and `q` is nonpositive, then `p` is nonpositive
+#### A rational number is not both negative and nonnegative
 
 ```agda
 abstract
-  is-nonpositive-leq-ℚ⁰⁻ :
-    (q : ℚ⁰⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁰⁻ q) → is-nonpositive-ℚ p
-  is-nonpositive-leq-ℚ⁰⁻ (q , nonpos-q) p p≤q =
-    is-nonpositive-leq-zero-ℚ
-      ( p)
-      ( transitive-leq-ℚ p q zero-ℚ
-        ( leq-zero-is-nonpositive-ℚ q nonpos-q)
-        ( p≤q))
+  not-is-negative-is-nonnegative-ℚ :
+    {x : ℚ} → ¬ (is-negative-ℚ x × is-nonnegative-ℚ x)
+  not-is-negative-is-nonnegative-ℚ {x} (N , NN) =
+    not-leq-le-ℚ
+      ( x)
+      ( zero-ℚ)
+      ( le-zero-is-negative-ℚ N)
+      ( leq-zero-is-nonnegative-ℚ NN)
 ```
diff --git a/src/elementary-number-theory/rational-numbers.lagda.md b/src/elementary-number-theory/rational-numbers.lagda.md
index 936003357b2..ed81dcc1dc9 100644
--- a/src/elementary-number-theory/rational-numbers.lagda.md
+++ b/src/elementary-number-theory/rational-numbers.lagda.md
@@ -308,8 +308,7 @@ opaque
 ```agda
 abstract
   eq-neg-one-ℚ : neg-ℚ one-ℚ ＝ neg-one-ℚ
-  eq-neg-one-ℚ =
-    inv (preserves-neg-rational-ℤ one-ℤ)
+  eq-neg-one-ℚ = inv (neg-rational-ℤ one-ℤ)
 ```
 
 ### The reduced fraction of the negative of an integer fraction is the negative of the reduced fraction
diff --git a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
index e9327a4d02f..5864700e784 100644
--- a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
@@ -260,7 +260,7 @@ abstract
               ( is-identity-right-conjugation-add-ℚ (p *ℚ p) q))
             ( refl)
             ( preserves-le-right-add-ℚ (p *ℚ p) _ _
-              ( neg-le-ℚ _ _
+              ( neg-le-ℚ
                 ( tr
                   ( le-ℚ _)
                   ( ap
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index f40ac8fef9d..16de8f520d9 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -11,6 +11,9 @@ module elementary-number-theory.squares-rational-numbers where
 ```agda
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers
@@ -61,9 +64,8 @@ abstract
     rec-coproduct
       ( λ H →
         is-nonnegative-is-positive-ℚ
-          ( a *ℚ a)
           ( is-positive-mul-negative-ℚ {a} {a} H H))
-      ( λ H → is-nonnegative-mul-ℚ a a H H)
+      ( λ H → is-nonnegative-mul-ℚ H H)
       ( decide-is-negative-is-nonnegative-ℚ a)
 
 nonnegative-square-ℚ : ℚ → ℚ⁰⁺
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index e3f09710870..2dd03cdc110 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -288,7 +288,6 @@ abstract
     is-positive-ℚ q
   is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) q x<q =
     is-positive-le-zero-ℚ
-      ( q)
       ( reflects-le-real-ℚ
         ( zero-ℚ)
         ( q)
@@ -303,7 +302,7 @@ opaque
     {l : Level} → (x : ℝ l) → (upper-cut-ℝ x ⊆ is-positive-prop-ℚ) →
     is-nonnegative-ℝ x
   is-nonnegative-is-positive-upper-cut-ℝ x Uₓ⊆ℚ⁺ =
-    leq-leq'-ℝ zero-ℝ x (λ q q∈Uₓ → le-zero-is-positive-ℚ q (Uₓ⊆ℚ⁺ q q∈Uₓ))
+    leq-leq'-ℝ zero-ℝ x (λ q q∈Uₓ → le-zero-is-positive-ℚ (Uₓ⊆ℚ⁺ q q∈Uₓ))
 ```
 
 ### A real number is nonnegative if and only if every negative rational number is in its lower cut
@@ -316,13 +315,13 @@ opaque
     {l : Level} (x : ℝ l) → (is-negative-prop-ℚ ⊆ lower-cut-ℝ x) →
     is-nonnegative-ℝ x
   is-nonnegative-leq-negative-lower-cut-ℝ x ℚ⁻⊆Lₓ q q<0 =
-    ℚ⁻⊆Lₓ q (is-negative-le-zero-ℚ q q<0)
+    ℚ⁻⊆Lₓ q (is-negative-le-zero-ℚ q<0)
 
   leq-negative-lower-cut-is-nonnegative-ℝ :
     {l : Level} (x : ℝ l) → is-nonnegative-ℝ x →
     (is-negative-prop-ℚ ⊆ lower-cut-ℝ x)
   leq-negative-lower-cut-is-nonnegative-ℝ x 0≤x q is-neg-q =
-    0≤x q (le-zero-is-negative-ℚ q is-neg-q)
+    0≤x q (le-zero-is-negative-ℚ is-neg-q)
 ```
 
 ### Every nonnegative real number has a positive rational number in its upper cut
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index eba2790f9fc..0bcd3acc129 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -9,6 +9,7 @@ module real-numbers.square-roots-nonnegative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
@@ -88,8 +89,7 @@ module _
         ( λ is-nonneg-neg-one-ℚ →
           ex-falso
             ( not-is-negative-is-nonnegative-ℚ
-              ( is-nonneg-neg-one-ℚ)
-              ( is-negative-rational-ℚ⁻ neg-one-ℚ⁻)))
+              ( is-negative-rational-ℚ⁻ neg-one-ℚ⁻ , is-nonneg-neg-one-ℚ)))
 
     is-inhabited-upper-cut-sqrt-ℝ⁰⁺ : is-inhabited-subtype upper-cut-sqrt-ℝ⁰⁺
     is-inhabited-upper-cut-sqrt-ℝ⁰⁺ =
@@ -122,8 +122,8 @@ module _
                   λ is-nonneg-q' →
                     ex-falso
                       ( not-is-negative-is-nonnegative-ℚ
-                        ( is-nonneg-q')
-                        ( is-negative-rational-ℚ⁻ (mediant-zero-ℚ⁻ q⁻)))))
+                        ( is-negative-rational-ℚ⁻ (mediant-zero-ℚ⁻ q⁻) ,
+                          is-nonneg-q'))))
           ( λ q=0 →
             do
               ( q' , 0<q' , q'<x) ←
@@ -132,9 +132,13 @@ module _
                   ( tr
                     ( is-in-lower-cut-ℝ⁰⁺ x)
                     ( ap-mul-ℚ q=0 q=0 ∙ right-zero-law-mul-ℚ zero-ℚ)
-                    ( q²<x (inv-tr is-nonnegative-ℚ q=0 _)))
+                    ( q²<x
+                      ( inv-tr
+                        ( is-nonnegative-ℚ)
+                        ( q=0)
+                        ( is-nonnegative-rational-ℚ⁰⁺ zero-ℚ⁰⁺))))
               let
-                is-pos-q' = is-positive-le-zero-ℚ q' 0<q'
+                is-pos-q' = is-positive-le-zero-ℚ 0<q'
                 q'⁺ = (q' , is-pos-q')
               ( p⁺@(p , is-pos-p) , p²<q') ← square-le-ℚ⁺ q'⁺
               intro-exists
@@ -142,7 +146,7 @@ module _
                 ( inv-tr
                     ( λ r → le-ℚ r p)
                     ( q=0)
-                    ( le-zero-is-positive-ℚ p is-pos-p) ,
+                    ( le-zero-is-positive-ℚ is-pos-p) ,
                   λ _ →
                     le-lower-cut-ℝ (real-ℝ⁰⁺ x) (p *ℚ p) q' p²<q' q'<x))
           ( λ is-pos-q →
@@ -150,7 +154,7 @@ module _
               (p , q²<p , p<x) ←
                 forward-implication
                   ( is-rounded-lower-cut-ℝ⁰⁺ x (q *ℚ q))
-                  ( q²<x (is-nonnegative-is-positive-ℚ q is-pos-q))
+                  ( q²<x (is-nonnegative-is-positive-ℚ is-pos-q))
               let
                 is-pos-p =
                   is-positive-le-ℚ⁺
@@ -194,7 +198,6 @@ module _
         let
           is-nonneg-r =
             is-nonnegative-is-positive-ℚ
-              ( r)
               ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) r q<r)
         le-lower-cut-ℝ
           ( real-ℝ⁰⁺ x)
@@ -217,8 +220,8 @@ module _
             ( q *ℚ q)
             ( r *ℚ r)
             ( preserves-le-square-ℚ⁰⁺
-              ( q , is-nonnegative-is-positive-ℚ q is-pos-q)
-              ( r , is-nonnegative-is-positive-ℚ r is-pos-r)
+              ( q , is-nonnegative-is-positive-ℚ is-pos-q)
+              ( r , is-nonnegative-is-positive-ℚ is-pos-r)
               q<r)
             ( x<q²))
 
@@ -244,7 +247,7 @@ module _
       is-disjoint-cut-ℝ
         ( real-ℝ⁰⁺ x)
         ( q *ℚ q)
-        ( q<√x (is-nonnegative-is-positive-ℚ q is-pos-q) ,
+        ( q<√x (is-nonnegative-is-positive-ℚ is-pos-q) ,
           x<q²)
 
     is-located-lower-upper-cut-sqrt-ℝ⁰⁺ :
@@ -252,10 +255,11 @@ module _
       type-disjunction-Prop (lower-cut-sqrt-ℝ⁰⁺ p) (upper-cut-sqrt-ℝ⁰⁺ q)
     is-located-lower-upper-cut-sqrt-ℝ⁰⁺ p q p<q =
       rec-coproduct
-        ( λ p<0 →
+        ( λ is-neg-p →
           inl-disjunction
             ( λ is-nonneg-p →
-              ex-falso (not-is-negative-is-nonnegative-ℚ is-nonneg-p p<0)))
+              ex-falso
+                ( not-is-negative-is-nonnegative-ℚ (is-neg-p , is-nonneg-p))))
         ( λ is-nonneg-p →
           map-disjunction
             ( λ p²<x _ → p²<x)
@@ -343,7 +347,6 @@ module _
                   ( λ is-neg-a →
                     ex-falso
                       ( not-is-negative-is-positive-ℚ
-                        ( a *ℚ d)
                         ( is-negative-mul-negative-positive-ℚ
                             ( is-neg-a)
                             ( is-pos-d) ,
@@ -359,7 +362,6 @@ module _
                   ( λ is-neg-c →
                     ex-falso
                       ( not-is-negative-is-positive-ℚ
-                        ( b *ℚ c)
                         ( is-negative-mul-positive-negative-ℚ
                             ( is-pos-b)
                             ( is-neg-c) ,

From e7c64234e700065eaf367b0d1f8a334ad87dad41 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 14:29:14 -0700
Subject: [PATCH 004/134] Progress

---
 ...wers-of-positive-rational-numbers.lagda.md |  0
 ...wers-of-positive-rational-numbers.lagda.md | 49 +++++++++++++++++++
 2 files changed, 49 insertions(+)
 create mode 100644 src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md
new file mode 100644
index 00000000000..e69de29bb2d
diff --git a/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
new file mode 100644
index 00000000000..33cb3b88f16
--- /dev/null
+++ b/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
@@ -0,0 +1,49 @@
+# Powers of positive rational numbers
+
+```agda
+module elementary-number-theory.powers-of-positive-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.bernoullis-inequality-positive-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import group-theory.powers-of-elements-groups
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+on the [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
+which is defined by [iteratively](foundation.iterating-functions.md) multiplying
+`x` with itself `n` times.
+
+## Definition
+
+```agda
+power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
+power-ℚ⁺ = power-Group group-mul-ℚ⁺
+```
+
+## Properties
+
+### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+
+```agda
+
+```
+
+## See also
+
+- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)

From 9963c472ee1f912b12ac290301d97d71fb48dd33 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 14:49:48 -0700
Subject: [PATCH 005/134] Updates

---
 .../maximum-nonnegative-rational-numbers.lagda.md  |  2 +-
 .../minimum-positive-rational-numbers.lagda.md     |  6 +++---
 ...positive-and-negative-rational-numbers.lagda.md |  6 ++++--
 ...positive-and-negative-rational-numbers.lagda.md | 14 ++++++++++++++
 ...icative-inverses-positive-real-numbers.lagda.md |  5 ++++-
 src/real-numbers/positive-real-numbers.lagda.md    |  1 -
 6 files changed, 26 insertions(+), 8 deletions(-)

diff --git a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
index e58e1036a68..2160af8be1b 100644
--- a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
@@ -37,7 +37,7 @@ abstract
     {p q : ℚ} → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
     is-nonnegative-ℚ (max-ℚ p q)
   is-nonnegative-max-ℚ {p} {q} is-nonneg-p _ =
-    is-nonnegative-leq-zero-ℚ _
+    is-nonnegative-leq-zero-ℚ
       ( transitive-leq-ℚ
         ( zero-ℚ)
         ( p)
diff --git a/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md b/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md
index 47c9745f5f1..b5ccc9867d9 100644
--- a/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md
@@ -39,10 +39,10 @@ abstract
   is-positive-min-ℚ :
     {p q : ℚ} → is-positive-ℚ p → is-positive-ℚ q → is-positive-ℚ (min-ℚ p q)
   is-positive-min-ℚ {p} {q} is-pos-p is-pos-q =
-    is-positive-le-zero-ℚ _
+    is-positive-le-zero-ℚ
       ( le-min-le-both-ℚ zero-ℚ p q
-        ( le-zero-is-positive-ℚ p is-pos-p)
-        ( le-zero-is-positive-ℚ q is-pos-q))
+        ( le-zero-is-positive-ℚ is-pos-p)
+        ( le-zero-is-positive-ℚ is-pos-q))
 
 min-ℚ⁺ : ℚ⁺ → ℚ⁺ → ℚ⁺
 min-ℚ⁺ (p , is-pos-p) (q , is-pos-q) =
diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index 72a497095a0..8cbfc9610b2 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -71,7 +71,7 @@ mul-negative-positive-ℚ (p , neg-p) (q , pos-q) =
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-nonnegative-ℚ is-positive-ℚ mul-ℚ
 
   is-nonnegative-mul-nonnegative-positive-ℚ :
     {x y : ℚ} → is-nonnegative-ℚ x → is-positive-ℚ y → is-nonnegative-ℚ (x *ℚ y)
@@ -83,7 +83,9 @@ opaque
 ### The product of a nonpositive and a positive rational number is nonpositive
 
 ```agda
-abstract
+opaque
+  unfolding is-nonpositive-ℚ
+
   is-nonpositive-mul-nonpositive-positive-ℚ :
     {x y : ℚ} → is-nonpositive-ℚ x → is-positive-ℚ y → is-nonpositive-ℚ (x *ℚ y)
   is-nonpositive-mul-nonpositive-positive-ℚ is-nonneg-neg-x is-pos-y =
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index a2f18c7d166..af6fead7f8f 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -238,3 +238,17 @@ abstract
       ( le-zero-is-negative-ℚ N)
       ( leq-zero-is-nonnegative-ℚ NN)
 ```
+
+### A rational number is not both positive and nonpositive
+
+```agda
+abstract
+  not-is-positive-is-nonpositive-ℚ :
+    {x : ℚ} → ¬ (is-positive-ℚ x × is-nonpositive-ℚ x)
+  not-is-positive-is-nonpositive-ℚ {x} (P , NP) =
+    not-leq-le-ℚ
+      ( zero-ℚ)
+      ( x)
+      ( le-zero-is-positive-ℚ P)
+      ( leq-zero-is-nonpositive-ℚ NP)
+```
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index b8a957c53fe..384e7af4c8a 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -19,6 +19,7 @@ open import elementary-number-theory.multiplication-closed-intervals-rational-nu
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
@@ -443,7 +444,9 @@ module _
                 ( inv-le-ℚ⁺ p⁺ q⁺ p<q)))
         ( λ is-nonpos-p →
           inl-disjunction
-            ( ex-falso ∘ not-is-positive-is-nonpositive-ℚ is-nonpos-p))
+            ( λ is-pos-p →
+              ex-falso
+                ( not-is-positive-is-nonpositive-ℚ (is-pos-p , is-nonpos-p))))
         ( decide-is-positive-is-nonpositive-ℚ p)
 
   opaque
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index a2ce70292e5..9181fcb5459 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -275,7 +275,6 @@ abstract
     {l : Level} (x : ℝ⁺ l) (q : ℚ) → is-in-upper-cut-ℝ⁺ x q → is-positive-ℚ q
   is-positive-is-in-upper-cut-ℝ⁺ x⁺@(x , _) q x<q =
     is-positive-le-zero-ℚ
-      ( q)
       ( le-lower-upper-cut-ℝ x zero-ℚ q (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
 ```
 

From 817f878ff421c7857fb5c279797ee718e95a0300 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 16:39:53 -0700
Subject: [PATCH 006/134] More properties

---
 src/elementary-number-theory.lagda.md         |   2 +
 ...roup-of-positive-rational-numbers.lagda.md |   8 +
 ...wers-of-positive-rational-numbers.lagda.md |  49 ----
 .../powers-positive-rational-numbers.lagda.md | 256 ++++++++++++++++++
 .../powers-of-elements-groups.lagda.md        |  25 ++
 ...ve-inverses-positive-real-numbers.lagda.md |   2 +-
 6 files changed, 292 insertions(+), 50 deletions(-)
 delete mode 100644 src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/powers-positive-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 542cd4fce73..f53d1f7c1a8 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -108,6 +108,7 @@ open import elementary-number-theory.infinitude-of-primes public
 open import elementary-number-theory.initial-segments-natural-numbers public
 open import elementary-number-theory.integer-fractions public
 open import elementary-number-theory.integer-partitions public
+open import elementary-number-theory.integer-powers-of-positive-rational-numbers public
 open import elementary-number-theory.integers public
 open import elementary-number-theory.interior-closed-intervals-rational-numbers public
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers public
@@ -179,6 +180,7 @@ open import elementary-number-theory.positive-integers public
 open import elementary-number-theory.positive-rational-numbers public
 open import elementary-number-theory.powers-integers public
 open import elementary-number-theory.powers-of-two public
+open import elementary-number-theory.powers-positive-rational-numbers public
 open import elementary-number-theory.prime-numbers public
 open import elementary-number-theory.products-of-natural-numbers public
 open import elementary-number-theory.proper-closed-intervals-rational-numbers public
diff --git a/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md
index 9ccaea277e7..3b2ca16a732 100644
--- a/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md
@@ -236,3 +236,11 @@ abstract
       ( is-retraction-left-div-ℚ⁺ p⁺ r)
       ( preserves-le-left-mul-ℚ⁺ (inv-ℚ⁺ p⁺) _ _ pq<pr)
 ```
+
+### The inverse of 1 is 1
+
+```agda
+abstract
+  inv-one-ℚ⁺ : inv-ℚ⁺ one-ℚ⁺ ＝ one-ℚ⁺
+  inv-one-ℚ⁺ = inv-unit-Group group-mul-ℚ⁺
+```
diff --git a/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
deleted file mode 100644
index 33cb3b88f16..00000000000
--- a/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
+++ /dev/null
@@ -1,49 +0,0 @@
-# Powers of positive rational numbers
-
-```agda
-module elementary-number-theory.powers-of-positive-rational-numbers where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import elementary-number-theory.multiplication-positive-rational-numbers
-open import elementary-number-theory.rational-numbers
-open import elementary-number-theory.positive-rational-numbers
-open import elementary-number-theory.bernoullis-inequality-positive-rational-numbers
-open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
-open import elementary-number-theory.natural-numbers
-open import elementary-number-theory.addition-natural-numbers
-open import elementary-number-theory.multiplication-natural-numbers
-open import group-theory.powers-of-elements-groups
-```
-
-</details>
-
-## Idea
-
-The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
-on the [positive](elementary-number-theory.positive-rational-numbers.md)
-[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
-which is defined by [iteratively](foundation.iterating-functions.md) multiplying
-`x` with itself `n` times.
-
-## Definition
-
-```agda
-power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
-power-ℚ⁺ = power-Group group-mul-ℚ⁺
-```
-
-## Properties
-
-### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
-
-```agda
-
-```
-
-## See also
-
-- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
new file mode 100644
index 00000000000..7815783fb91
--- /dev/null
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -0,0 +1,256 @@
+# Powers of positive rational numbers
+
+```agda
+module elementary-number-theory.powers-positive-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
+open import elementary-number-theory.archimedean-property-rational-numbers
+open import elementary-number-theory.arithmetic-sequences-positive-rational-numbers
+open import elementary-number-theory.bernoullis-inequality-positive-rational-numbers
+open import elementary-number-theory.geometric-sequences-positive-rational-numbers
+open import elementary-number-theory.inequality-positive-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.nonzero-natural-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-positive-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.existential-quantification
+open import foundation.function-types
+open import foundation.identity-types
+open import foundation.propositional-truncations
+open import foundation.transport-along-identifications
+
+open import group-theory.multiples-of-elements-abelian-groups
+open import group-theory.powers-of-elements-groups
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+on the [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
+which is defined by [iteratively](foundation.iterating-functions.md) multiplying
+`x` with itself `n` times.
+
+## Definition
+
+```agda
+power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
+power-ℚ⁺ = power-Group group-mul-ℚ⁺
+```
+
+## Properties
+
+### `1ⁿ = 1`
+
+```agda
+power-one-ℚ⁺ : (n : ℕ) → power-ℚ⁺ n one-ℚ⁺ ＝ one-ℚ⁺
+power-one-ℚ⁺ = power-unit-Group group-mul-ℚ⁺
+```
+
+### `qⁿ⁺¹ = qⁿq`
+
+```agda
+power-succ-ℚ⁺ : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ power-ℚ⁺ n q *ℚ⁺ q
+power-succ-ℚ⁺ = power-succ-Group group-mul-ℚ⁺
+```
+
+### `qⁿ = qqⁿ`
+
+```agda
+power-succ-ℚ⁺' : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ q *ℚ⁺ power-ℚ⁺ n q
+power-succ-ℚ⁺' = power-succ-Group' group-mul-ℚ⁺
+```
+
+### `qᵐ⁺ⁿ = qᵐqⁿ`
+
+```agda
+distributive-power-add-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m +ℕ n) q ＝ power-ℚ⁺ m q *ℚ⁺ power-ℚ⁺ n q
+distributive-power-add-ℚ⁺ m n _ = distributive-power-add-Group group-mul-ℚ⁺ m n
+```
+
+### `(pq)ⁿ=pⁿqⁿ`
+
+```agda
+left-distributive-power-mul-ℚ⁺ :
+  (n : ℕ) (p q : ℚ⁺) → power-ℚ⁺ n (p *ℚ⁺ q) ＝ power-ℚ⁺ n p *ℚ⁺ power-ℚ⁺ n q
+left-distributive-power-mul-ℚ⁺ n p q =
+  left-distributive-multiple-add-Ab abelian-group-mul-ℚ⁺ n
+```
+
+### `pᵐⁿ = (pᵐ)ⁿ`
+
+```agda
+power-mul-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m *ℕ n) q ＝ power-ℚ⁺ n (power-ℚ⁺ m q)
+power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
+```
+
+### If `p` and `q` are positive rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+
+```agda
+abstract
+  preserves-le-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
+    le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
+  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
+  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
+    transitive-le-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-le-right-mul-ℚ⁺ q _ _
+        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
+      ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
+```
+
+### If `p` and `q` are positive rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+
+```agda
+abstract
+  preserves-leq-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → leq-ℚ⁺ p q → leq-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-leq-power-ℚ⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
+  preserves-leq-power-ℚ⁺ 1 p q p≤q = p≤q
+  preserves-leq-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
+    transitive-leq-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-leq-right-mul-ℚ⁺ q _ _ (preserves-leq-power-ℚ⁺ n p q p≤q))
+      ( preserves-leq-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p≤q)
+```
+
+### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  bound-unbounded-power-one-plus-ℚ⁺ ε⁺@(ε , is-pos-ε) b =
+    let
+      (n , b<nε) = bound-archimedean-property-ℚ ε b is-pos-ε
+    in
+      ( n ,
+        transitive-le-ℚ _ _ _
+          ( concatenate-le-leq-ℚ _ _ _
+            ( le-left-add-rational-ℚ⁺ _ one-ℚ⁺)
+            ( binary-tr
+              ( leq-ℚ)
+              ( inv (compute-standard-arithmetic-sequence-ℚ⁺ one-ℚ⁺ ε⁺ n))
+              ( ap
+                ( rational-ℚ⁺)
+                ( equational-reasoning
+                  seq-standard-geometric-sequence-ℚ⁺
+                      ( one-ℚ⁺)
+                      ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                      ( n)
+                  ＝ one-ℚ⁺ *ℚ⁺ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by
+                      inv
+                        ( compute-standard-geometric-sequence-ℚ⁺
+                          ( one-ℚ⁺)
+                          ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                          ( n))
+                  ＝ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by left-unit-law-mul-ℚ⁺ _))
+              ( bernoullis-inequality-ℚ⁺ ε⁺ n)))
+          ( b<nε))
+
+  unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  unbounded-power-one-plus-ℚ⁺ ε b =
+    unit-trunc-Prop (bound-unbounded-power-one-plus-ℚ⁺ ε b)
+```
+
+### If `1 < q`, `qⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  bound-unbounded-power-greater-than-one-ℚ⁺ q⁺@(q , _) b 1<q =
+    let
+      q-1⁺ = positive-diff-le-ℚ one-ℚ q 1<q
+      (n , b<⟨1+q-1⟩ⁿ) = bound-unbounded-power-one-plus-ℚ⁺ q-1⁺ b
+    in
+      ( n ,
+        tr
+          ( le-ℚ b)
+          ( ap
+            ( rational-ℚ⁺ ∘ power-ℚ⁺ n)
+            ( eq-ℚ⁺ (is-identity-right-conjugation-add-ℚ one-ℚ q)))
+          ( b<⟨1+q-1⟩ⁿ))
+
+  unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  unbounded-power-greater-than-one-ℚ⁺ q b 1<q =
+    unit-trunc-Prop (bound-unbounded-power-greater-than-one-ℚ⁺ q b 1<q)
+```
+
+### If `ε` is a positive rational number less than one, `εⁿ` becomes arbitrarily small
+
+```agda
+abstract
+  bound-arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    Σ ℕ (λ n → le-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ ε<1 =
+    let
+      1/δ = inv-ℚ⁺ δ
+      1/ε = inv-ℚ⁺ ε
+      1<1/ε =
+        tr
+          ( λ q → le-ℚ⁺ q 1/ε)
+          ( inv-one-ℚ⁺)
+          ( inv-le-ℚ⁺ ε one-ℚ⁺ ε<1)
+      (n , 1/δ<⟨1/ε⟩ⁿ) =
+        bound-unbounded-power-greater-than-one-ℚ⁺ 1/ε (rational-ℚ⁺ 1/δ) 1<1/ε
+    in
+      ( n ,
+        binary-tr
+          ( le-ℚ⁺)
+          ( equational-reasoning
+            inv-ℚ⁺ (power-ℚ⁺ n 1/ε)
+            ＝ inv-ℚ⁺ (inv-ℚ⁺ (power-ℚ⁺ n ε))
+              by ap inv-ℚ⁺ (power-inv-Group group-mul-ℚ⁺ n ε)
+            ＝ power-ℚ⁺ n ε
+              by inv-inv-ℚ⁺ (power-ℚ⁺ n ε))
+          ( inv-inv-ℚ⁺ δ)
+          ( inv-le-ℚ⁺ 1/δ (power-ℚ⁺ n 1/ε) 1/δ<⟨1/ε⟩ⁿ))
+
+  arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    exists ℕ (λ n → le-prop-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε =
+    unit-trunc-Prop (bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε)
+```
+
+## See also
+
+- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)
diff --git a/src/group-theory/powers-of-elements-groups.lagda.md b/src/group-theory/powers-of-elements-groups.lagda.md
index a9e28db03ab..f2e9511b258 100644
--- a/src/group-theory/powers-of-elements-groups.lagda.md
+++ b/src/group-theory/powers-of-elements-groups.lagda.md
@@ -11,6 +11,7 @@ open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.multiplication-natural-numbers
 open import elementary-number-theory.natural-numbers
 
+open import foundation.action-on-identifications-functions
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
@@ -175,6 +176,30 @@ module _
   power-mul-Group = power-mul-Monoid (monoid-Group G)
 ```
 
+### The inverse of `x`, raised to the power `n`, is the inverse of `x` raised to the power `n`
+
+```agda
+module _
+  {l : Level} (G : Group l)
+  where
+
+  abstract
+    power-inv-Group :
+      (n : ℕ) (x : type-Group G) →
+      power-Group G n (inv-Group G x) ＝ inv-Group G (power-Group G n x)
+    power-inv-Group 0 _ = inv (inv-unit-Group G)
+    power-inv-Group 1 _ = refl
+    power-inv-Group (succ-ℕ n@(succ-ℕ _)) x =
+      equational-reasoning
+        mul-Group G (power-Group G n (inv-Group G x)) (inv-Group G x)
+        ＝ mul-Group G (inv-Group G (power-Group G n x)) (inv-Group G x)
+          by ap-mul-Group G (power-inv-Group n x) refl
+        ＝ inv-Group G (mul-Group G x (power-Group G n x))
+          by inv (distributive-inv-mul-Group G)
+        ＝ inv-Group G (power-Group G (succ-ℕ n) x)
+          by ap (inv-Group G) (inv (power-succ-Group' G n x))
+```
+
 ### Group homomorphisms preserve powers
 
 ```agda
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 384e7af4c8a..06b96659065 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -10,6 +10,7 @@ module real-numbers.multiplicative-inverses-positive-real-numbers where
 
 ```agda
 open import elementary-number-theory.closed-intervals-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-positive-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
@@ -19,7 +20,6 @@ open import elementary-number-theory.multiplication-closed-intervals-rational-nu
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers

From 984d8d40f546f7aef53a05614035a3d2615a048c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 16:43:12 -0700
Subject: [PATCH 007/134] Back out integer powers

---
 .../integer-powers-of-positive-rational-numbers.lagda.md          | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 delete mode 100644 src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md
deleted file mode 100644
index e69de29bb2d..00000000000

From a69f76695b653a7178ad6277aa99b9713adfd65f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 20:45:33 -0700
Subject: [PATCH 008/134] Fixes

---
 src/elementary-number-theory.lagda.md                           | 1 -
 .../powers-positive-rational-numbers.lagda.md                   | 2 +-
 2 files changed, 1 insertion(+), 2 deletions(-)

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index f53d1f7c1a8..be3b1e0ad01 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -108,7 +108,6 @@ open import elementary-number-theory.infinitude-of-primes public
 open import elementary-number-theory.initial-segments-natural-numbers public
 open import elementary-number-theory.integer-fractions public
 open import elementary-number-theory.integer-partitions public
-open import elementary-number-theory.integer-powers-of-positive-rational-numbers public
 open import elementary-number-theory.integers public
 open import elementary-number-theory.interior-closed-intervals-rational-numbers public
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers public
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 7815783fb91..811235c5420 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -45,7 +45,7 @@ open import group-theory.powers-of-elements-groups
 ## Idea
 
 The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
 on the [positive](elementary-number-theory.positive-rational-numbers.md)
 [rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
 which is defined by [iteratively](foundation.iterating-functions.md) multiplying

From ea3fa12c81c28d11c10da66a04d9e42d6a7baf98 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 15 Oct 2025 09:15:11 -0700
Subject: [PATCH 009/134] Fix merge

---
 ...ion-positive-and-negative-rational-numbers.lagda.md |  6 ++----
 .../positive-and-negative-rational-numbers.lagda.md    | 10 ----------
 ...ltiplicative-inverses-nonzero-real-numbers.lagda.md |  4 +---
 src/real-numbers/negative-real-numbers.lagda.md        |  4 ++--
 .../square-roots-nonnegative-real-numbers.lagda.md     | 10 ++--------
 5 files changed, 7 insertions(+), 27 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index e8b894794eb..eda0123b5e7 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -148,8 +148,7 @@ abstract
             ( y≠0)
             ( λ is-pos-y →
               ex-falso
-                ( is-not-negative-and-positive-ℚ
-                  ( x *ℚ y)
+                ( not-is-negative-is-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-x is-pos-y ,
                     is-pos-xy))))
         ( λ x=0 →
@@ -164,8 +163,7 @@ abstract
             ( y)
             ( λ is-neg-y →
               ex-falso
-                ( is-not-negative-and-positive-ℚ
-                  ( x *ℚ y)
+                ( not-is-negative-is-positive-ℚ
                   ( is-negative-mul-positive-negative-ℚ is-pos-x is-neg-y ,
                     is-pos-xy)))
             ( y≠0)
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index ed64fcf3fdf..00e1063ad38 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -239,16 +239,6 @@ abstract
       ( leq-zero-is-nonnegative-ℚ NN)
 ```
 
-### A positive rational number is not negative
-
-```agda
-abstract
-  not-is-negative-is-positive-ℚ :
-    {q : ℚ} → is-positive-ℚ q → ¬ (is-negative-ℚ q)
-  not-is-negative-is-positive-ℚ {q} is-pos-q =
-    not-is-negative-is-nonnegative-ℚ (is-nonnegative-is-positive-ℚ q is-pos-q)
-```
-
 ### If `p < q` and `q` is nonpositive, then `p` is negative
 
 ```agda
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index 79a1985bb71..43c55227e30 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -151,7 +151,6 @@ opaque
       let
         is-positive-mul p q {r} lb1 lb2 =
           is-positive-le-zero-ℚ
-            ( p *ℚ q)
             ( concatenate-le-leq-ℚ
               ( zero-ℚ)
               ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -173,8 +172,7 @@ opaque
                   ( intro-exists (b , is-neg-b) x<b)))
             ( λ (is-pos-b , is-pos-d) →
               ex-falso
-                ( is-not-negative-and-positive-ℚ
-                  ( a *ℚ d)
+                ( not-is-negative-is-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-a is-pos-d ,
                     is-positive-mul a d
                       ( leq-right-min-ℚ _ _)
diff --git a/src/real-numbers/negative-real-numbers.lagda.md b/src/real-numbers/negative-real-numbers.lagda.md
index f33b4ee1af5..fd87ef7b8c7 100644
--- a/src/real-numbers/negative-real-numbers.lagda.md
+++ b/src/real-numbers/negative-real-numbers.lagda.md
@@ -77,7 +77,7 @@ module _
     exists-ℚ⁻-in-upper-cut-is-negative-ℝ =
       elim-exists
         ( ∃ ℚ⁻ (λ p → upper-cut-ℝ x (rational-ℚ⁻ p)))
-        ( λ p (x<p , p<0) → intro-exists (p , is-negative-le-zero-ℚ p p<0) x<p)
+        ( λ p (x<p , p<0) → intro-exists (p , is-negative-le-zero-ℚ p<0) x<p)
 
     is-negative-exists-ℚ⁻-in-upper-cut-ℝ :
       exists ℚ⁻ (λ p → upper-cut-ℝ x (rational-ℚ⁻ p)) → is-negative-ℝ x
@@ -85,7 +85,7 @@ module _
       elim-exists
         ( is-negative-prop-ℝ x)
         ( λ (p , is-neg-p) x<p →
-          intro-exists p (x<p , le-zero-is-negative-ℚ p is-neg-p))
+          intro-exists p (x<p , le-zero-is-negative-ℚ is-neg-p))
 
     is-negative-iff-exists-ℚ⁻-in-upper-cut-ℝ :
       is-negative-ℝ x ↔ exists ℚ⁻ (λ p → upper-cut-ℝ x (rational-ℚ⁻ p))
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index a10b9ce4c7d..0bcd3acc129 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -354,10 +354,7 @@ module _
                             ( a)
                             ( d)
                             ( leq-right-min-ℚ _ _)
-                            ( leq-left-min-ℚ _ _))
-                        ( is-negative-mul-negative-positive-ℚ
-                            ( is-neg-a)
-                            ( is-pos-d))))
+                            ( leq-left-min-ℚ _ _))))
                   ( id)
                   ( decide-is-negative-is-nonnegative-ℚ a)
               is-nonneg-c =
@@ -372,10 +369,7 @@ module _
                             ( b)
                             ( c)
                             ( leq-left-min-ℚ _ _)
-                            ( leq-right-min-ℚ _ _))
-                        ( is-negative-mul-positive-negative-ℚ
-                            ( is-pos-b)
-                            ( is-neg-c))))
+                            ( leq-right-min-ℚ _ _))))
                   ( id)
                   ( decide-is-negative-is-nonnegative-ℚ c)
               a' = max-ℚ a c

From c7176d6136c1caff408391a579cde7cf849712ea Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 15 Oct 2025 09:29:06 -0700
Subject: [PATCH 010/134] make pre-commit

---
 .../multiplicative-inverses-positive-real-numbers.lagda.md      | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 384e7af4c8a..06b96659065 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -10,6 +10,7 @@ module real-numbers.multiplicative-inverses-positive-real-numbers where
 
 ```agda
 open import elementary-number-theory.closed-intervals-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-positive-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
@@ -19,7 +20,6 @@ open import elementary-number-theory.multiplication-closed-intervals-rational-nu
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers

From df308e2425fb8b1863565200a6a735d7e5afc7cd Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 15 Oct 2025 13:58:21 -0700
Subject: [PATCH 011/134] Fix merge

---
 ...ication-negative-rational-numbers.lagda.md | 19 +++++++++----------
 .../squares-real-numbers.lagda.md             |  4 ++--
 2 files changed, 11 insertions(+), 12 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
index 5a41de6f305..bd53f0e9c03 100644
--- a/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
@@ -21,18 +21,9 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
+open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
-open import elementary-number-theory.multiplication-positive-rational-numbers
-open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.negative-integer-fractions
-open import elementary-number-theory.negative-rational-numbers
-open import elementary-number-theory.positive-and-negative-rational-numbers
-open import elementary-number-theory.positive-integer-fractions
-open import elementary-number-theory.positive-rational-numbers
-open import elementary-number-theory.rational-numbers
-
-open import foundation.dependent-pair-types
 ```
 
 </details>
@@ -60,6 +51,14 @@ opaque
       ( is-positive-ℚ)
       ( negative-law-mul-ℚ x y)
       ( is-positive-mul-ℚ P Q)
+
+mul-ℚ⁻ : ℚ⁻ → ℚ⁻ → ℚ⁺
+mul-ℚ⁻ (p , is-neg-p) (q , is-neg-q) =
+  (p *ℚ q , is-positive-mul-negative-ℚ is-neg-p is-neg-q)
+
+infixl 40 _*ℚ⁻_
+_*ℚ⁻_ : ℚ⁻ → ℚ⁻ → ℚ⁺
+_*ℚ⁻_ = mul-ℚ⁻
 ```
 
 ### Multiplication by a negative rational number reverses inequality
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 9420a081e95..c6fedb34990 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -98,7 +98,7 @@ opaque
             ( is-positive-prop-ℚ q)
             ( λ a'<0 →
               let
-                a'⁻ = (a' , is-negative-le-zero-ℚ a' a'<0)
+                a'⁻ = (a' , is-negative-le-zero-ℚ a'<0)
               in
                 is-positive-le-ℚ⁺
                   ( a'⁻ *ℚ⁻ a'⁻)
@@ -113,7 +113,7 @@ opaque
                     ( [a',b'][a',b']<q)))
             ( λ 0<b' →
               let
-                b'⁺ = (b' , is-positive-le-zero-ℚ b' 0<b')
+                b'⁺ = (b' , is-positive-le-zero-ℚ 0<b')
               in
                 is-positive-le-ℚ⁺
                   ( b'⁺ *ℚ⁺ b'⁺)

From 607eba5ec1a6559aeea74938da2646f7f2ce780e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 09:39:18 -0700
Subject: [PATCH 012/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 ...lities-positive-and-negative-rational-numbers.lagda.md | 8 ++++----
 ...cation-positive-and-negative-rational-numbers.lagda.md | 3 +--
 2 files changed, 5 insertions(+), 6 deletions(-)

diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
index 414fa5ecdc3..88ac145a32a 100644
--- a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -36,8 +36,8 @@ between [positive](elementary-number-theory.positive-rational-numbers.md),
 
 ### Inequalities between rational numbers with different signs
 
-Some inequalities can be deduced directly from the sign of a rational number:
-for example, every negative rational number is less than every nonnegative
+Some inequalities can be deduced directly from the sign of a rational number.
+For example, every negative rational number is less than every nonnegative
 rational number.
 
 #### Any negative rational number is less than any nonnegative rational number
@@ -70,7 +70,7 @@ abstract
   leq-negative-positive-ℚ p q = leq-le-ℚ (le-negative-positive-ℚ p q)
 ```
 
-#### A nonpositive rational number is less than a positive rational number
+#### A nonpositive rational number is less than any positive rational number
 
 ```agda
 abstract
@@ -82,7 +82,7 @@ abstract
       ( le-zero-is-positive-ℚ pos-q)
 ```
 
-#### A nonpositive rational number is less than or equal to a nonnegative rational number
+#### A nonpositive rational number is less than or equal to any nonnegative rational number
 
 ```agda
 abstract
diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index eda0123b5e7..39db46c6512 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -30,8 +30,7 @@ open import foundation.transport-along-identifications
 ## Idea
 
 When we have information about the sign of the factors of a
-[rational](elementary-number-theory.rational-numbers.md)
-[product](elementary-number-theory.multiplication-rational-numbers.md), we can
+[product](elementary-number-theory.multiplication-rational-numbers.md) of [rational numbers](elementary-number-theory.rational-numbers.md), we can
 deduce the sign of their product too.
 
 ## Lemmas

From daa9fc084e696aa8775564973cdcc2804bd7e3aa Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 09:43:42 -0700
Subject: [PATCH 013/134] Respond to review comments

---
 ...positive-and-negative-rational-numbers.lagda.md |  9 +++++----
 ...positive-and-negative-rational-numbers.lagda.md | 14 +++++++-------
 .../order-preserving-maps-posets.lagda.md          |  2 +-
 .../order-preserving-maps-total-orders.lagda.md    |  2 +-
 ...licative-inverses-nonzero-real-numbers.lagda.md |  2 +-
 ...icative-inverses-positive-real-numbers.lagda.md |  2 +-
 .../square-roots-nonnegative-real-numbers.lagda.md | 10 +++++-----
 7 files changed, 21 insertions(+), 20 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index 39db46c6512..4cbe0a91112 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -30,8 +30,9 @@ open import foundation.transport-along-identifications
 ## Idea
 
 When we have information about the sign of the factors of a
-[product](elementary-number-theory.multiplication-rational-numbers.md) of [rational numbers](elementary-number-theory.rational-numbers.md), we can
-deduce the sign of their product too.
+[product](elementary-number-theory.multiplication-rational-numbers.md) of
+[rational numbers](elementary-number-theory.rational-numbers.md), we can deduce
+the sign of their product too.
 
 ## Lemmas
 
@@ -147,7 +148,7 @@ abstract
             ( y≠0)
             ( λ is-pos-y →
               ex-falso
-                ( not-is-negative-is-positive-ℚ
+                ( is-not-negative-and-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-x is-pos-y ,
                     is-pos-xy))))
         ( λ x=0 →
@@ -162,7 +163,7 @@ abstract
             ( y)
             ( λ is-neg-y →
               ex-falso
-                ( not-is-negative-is-positive-ℚ
+                ( is-not-negative-and-positive-ℚ
                   ( is-negative-mul-positive-negative-ℚ is-pos-x is-neg-y ,
                     is-pos-xy)))
             ( y≠0)
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index 00e1063ad38..e0447a64bbc 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -215,9 +215,9 @@ neg-ℚ⁰⁻ (q , is-nonpos-q) =
 
 ```agda
 abstract
-  not-is-negative-is-positive-ℚ :
+  is-not-negative-and-positive-ℚ :
     {x : ℚ} → ¬ (is-negative-ℚ x × is-positive-ℚ x)
-  not-is-negative-is-positive-ℚ {x} (N , P) =
+  is-not-negative-and-positive-ℚ {x} (N , P) =
     not-leq-le-ℚ
       ( zero-ℚ)
       ( x)
@@ -229,9 +229,9 @@ abstract
 
 ```agda
 abstract
-  not-is-negative-is-nonnegative-ℚ :
+  is-not-negative-and-nonnegativeℚ :
     {x : ℚ} → ¬ (is-negative-ℚ x × is-nonnegative-ℚ x)
-  not-is-negative-is-nonnegative-ℚ {x} (N , NN) =
+  is-not-negative-and-nonnegativeℚ {x} (N , NN) =
     not-leq-le-ℚ
       ( x)
       ( zero-ℚ)
@@ -239,13 +239,13 @@ abstract
       ( leq-zero-is-nonnegative-ℚ NN)
 ```
 
-### If `p < q` and `q` is nonpositive, then `p` is negative
+### A rational number is not both positive and nonpositive
 
 ```agda
 abstract
-  not-is-positive-is-nonpositive-ℚ :
+  is-not-positive-and-nonpositive-ℚ :
     {x : ℚ} → ¬ (is-positive-ℚ x × is-nonpositive-ℚ x)
-  not-is-positive-is-nonpositive-ℚ {x} (P , NP) =
+  is-not-positive-and-nonpositive-ℚ {x} (P , NP) =
     not-leq-le-ℚ
       ( zero-ℚ)
       ( x)
diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md
index 4d127ed3ff8..bcd5ba75286 100644
--- a/src/order-theory/order-preserving-maps-posets.lagda.md
+++ b/src/order-theory/order-preserving-maps-posets.lagda.md
@@ -27,7 +27,7 @@ open import order-theory.posets
 
 A map `f : P → Q` between the underlying types of two
 [posets](order-theory.posets.md) is said to be
-{{#concept "order preserving" Agda=hom-Poset Disambiguation="map between posets"}}
+{{#concept "order preserving" WD="increasing function" WDID=Q3075182 Agda=hom-Poset Disambiguation="map between posets"}}
 if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
 
 ## Definition
diff --git a/src/order-theory/order-preserving-maps-total-orders.lagda.md b/src/order-theory/order-preserving-maps-total-orders.lagda.md
index f3ae69baab9..161a0dafd86 100644
--- a/src/order-theory/order-preserving-maps-total-orders.lagda.md
+++ b/src/order-theory/order-preserving-maps-total-orders.lagda.md
@@ -26,7 +26,7 @@ open import order-theory.total-orders
 
 A map `f : P → Q` between the underlying types of two
 [total orders](order-theory.total-orders.md) is said to be
-{{#concept "order preserving" Agda=hom-Total-Order Disambiguation="map between total orders"}}
+{{#concept "order preserving" WD="increasing function" WDID=Q3075182 Agda=hom-Total-Order Disambiguation="map between total orders"}}
 if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
 
 ## Definition
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index 43c55227e30..f91c1d2bfcf 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -172,7 +172,7 @@ opaque
                   ( intro-exists (b , is-neg-b) x<b)))
             ( λ (is-pos-b , is-pos-d) →
               ex-falso
-                ( not-is-negative-is-positive-ℚ
+                ( is-not-negative-and-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-a is-pos-d ,
                     is-positive-mul a d
                       ( leq-right-min-ℚ _ _)
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 06b96659065..73cd01dce63 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -446,7 +446,7 @@ module _
           inl-disjunction
             ( λ is-pos-p →
               ex-falso
-                ( not-is-positive-is-nonpositive-ℚ (is-pos-p , is-nonpos-p))))
+                ( is-not-positive-and-nonpositive-ℚ (is-pos-p , is-nonpos-p))))
         ( decide-is-positive-is-nonpositive-ℚ p)
 
   opaque
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index aa52cf66ce6..a9576b242d4 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -92,7 +92,7 @@ module _
         ( neg-one-ℚ)
         ( λ is-nonneg-neg-one-ℚ →
           ex-falso
-            ( not-is-negative-is-nonnegative-ℚ
+            ( is-not-negative-and-nonnegativeℚ
               ( is-negative-rational-ℚ⁻ neg-one-ℚ⁻ , is-nonneg-neg-one-ℚ)))
 
     is-inhabited-upper-cut-sqrt-ℝ⁰⁺ : is-inhabited-subtype upper-cut-sqrt-ℝ⁰⁺
@@ -125,7 +125,7 @@ module _
                 ( le-mediant-zero-ℚ⁻ q⁻ ,
                   λ is-nonneg-q' →
                     ex-falso
-                      ( not-is-negative-is-nonnegative-ℚ
+                      ( is-not-negative-and-nonnegativeℚ
                         ( is-negative-rational-ℚ⁻ (mediant-zero-ℚ⁻ q⁻) ,
                           is-nonneg-q'))))
           ( λ q=0 →
@@ -263,7 +263,7 @@ module _
           inl-disjunction
             ( λ is-nonneg-p →
               ex-falso
-                ( not-is-negative-is-nonnegative-ℚ (is-neg-p , is-nonneg-p))))
+                ( is-not-negative-and-nonnegativeℚ (is-neg-p , is-nonneg-p))))
         ( λ is-nonneg-p →
           map-disjunction
             ( λ p²<x _ → p²<x)
@@ -350,7 +350,7 @@ module _
                 rec-coproduct
                   ( λ is-neg-a →
                     ex-falso
-                      ( not-is-negative-is-positive-ℚ
+                      ( is-not-negative-and-positive-ℚ
                         ( is-negative-mul-negative-positive-ℚ
                             ( is-neg-a)
                             ( is-pos-d) ,
@@ -365,7 +365,7 @@ module _
                 rec-coproduct
                   ( λ is-neg-c →
                     ex-falso
-                      ( not-is-negative-is-positive-ℚ
+                      ( is-not-negative-and-positive-ℚ
                         ( is-negative-mul-positive-negative-ℚ
                             ( is-pos-b)
                             ( is-neg-c) ,

From 0905e0fd91186800a05f80be2911e727c1562d80 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:07:16 -0700
Subject: [PATCH 014/134] Progress

---
 .../powers-rational-numbers.lagda.md          | 228 ++++++++++++++++++
 1 file changed, 228 insertions(+)
 create mode 100644 src/elementary-number-theory/powers-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
new file mode 100644
index 00000000000..ee922d0fad0
--- /dev/null
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -0,0 +1,228 @@
+# Powers of rational numbers
+
+```agda
+module elementary-number-theory.powers-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
+open import group-theory.powers-of-elements-monoids
+open import foundation.identity-types
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
+on the [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
+which is defined by [iteratively](foundation.iterating-functions.md) multiplying
+`x` with itself `n` times.
+
+## Definition
+
+```agda
+power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
+power-ℚ⁺ = power-Group group-mul-ℚ⁺
+```
+
+## Properties
+
+### `1ⁿ = 1`
+
+```agda
+power-one-ℚ⁺ : (n : ℕ) → power-ℚ⁺ n one-ℚ⁺ ＝ one-ℚ⁺
+power-one-ℚ⁺ = power-unit-Group group-mul-ℚ⁺
+```
+
+### `qⁿ⁺¹ = qⁿq`
+
+```agda
+power-succ-ℚ⁺ : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ power-ℚ⁺ n q *ℚ⁺ q
+power-succ-ℚ⁺ = power-succ-Group group-mul-ℚ⁺
+```
+
+### `qⁿ = qqⁿ`
+
+```agda
+power-succ-ℚ⁺' : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ q *ℚ⁺ power-ℚ⁺ n q
+power-succ-ℚ⁺' = power-succ-Group' group-mul-ℚ⁺
+```
+
+### `qᵐ⁺ⁿ = qᵐqⁿ`
+
+```agda
+distributive-power-add-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m +ℕ n) q ＝ power-ℚ⁺ m q *ℚ⁺ power-ℚ⁺ n q
+distributive-power-add-ℚ⁺ m n _ = distributive-power-add-Group group-mul-ℚ⁺ m n
+```
+
+### `(pq)ⁿ=pⁿqⁿ`
+
+```agda
+left-distributive-power-mul-ℚ⁺ :
+  (n : ℕ) (p q : ℚ⁺) → power-ℚ⁺ n (p *ℚ⁺ q) ＝ power-ℚ⁺ n p *ℚ⁺ power-ℚ⁺ n q
+left-distributive-power-mul-ℚ⁺ n p q =
+  left-distributive-multiple-add-Ab abelian-group-mul-ℚ⁺ n
+```
+
+### `pᵐⁿ = (pᵐ)ⁿ`
+
+```agda
+power-mul-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m *ℕ n) q ＝ power-ℚ⁺ n (power-ℚ⁺ m q)
+power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
+```
+
+### If `p` and `q` are positive rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+
+```agda
+abstract
+  preserves-le-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
+    le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
+  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
+  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
+    transitive-le-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-le-right-mul-ℚ⁺ q _ _
+        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
+      ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
+```
+
+### If `p` and `q` are positive rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+
+```agda
+abstract
+  preserves-leq-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → leq-ℚ⁺ p q → leq-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-leq-power-ℚ⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
+  preserves-leq-power-ℚ⁺ 1 p q p≤q = p≤q
+  preserves-leq-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
+    transitive-leq-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-leq-right-mul-ℚ⁺ q _ _ (preserves-leq-power-ℚ⁺ n p q p≤q))
+      ( preserves-leq-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p≤q)
+```
+
+### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  bound-unbounded-power-one-plus-ℚ⁺ ε⁺@(ε , is-pos-ε) b =
+    let
+      (n , b<nε) = bound-archimedean-property-ℚ ε b is-pos-ε
+    in
+      ( n ,
+        transitive-le-ℚ _ _ _
+          ( concatenate-le-leq-ℚ _ _ _
+            ( le-left-add-rational-ℚ⁺ _ one-ℚ⁺)
+            ( binary-tr
+              ( leq-ℚ)
+              ( inv (compute-standard-arithmetic-sequence-ℚ⁺ one-ℚ⁺ ε⁺ n))
+              ( ap
+                ( rational-ℚ⁺)
+                ( equational-reasoning
+                  seq-standard-geometric-sequence-ℚ⁺
+                      ( one-ℚ⁺)
+                      ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                      ( n)
+                  ＝ one-ℚ⁺ *ℚ⁺ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by
+                      inv
+                        ( compute-standard-geometric-sequence-ℚ⁺
+                          ( one-ℚ⁺)
+                          ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                          ( n))
+                  ＝ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by left-unit-law-mul-ℚ⁺ _))
+              ( bernoullis-inequality-ℚ⁺ ε⁺ n)))
+          ( b<nε))
+
+  unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  unbounded-power-one-plus-ℚ⁺ ε b =
+    unit-trunc-Prop (bound-unbounded-power-one-plus-ℚ⁺ ε b)
+```
+
+### If `1 < q`, `qⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  bound-unbounded-power-greater-than-one-ℚ⁺ q⁺@(q , _) b 1<q =
+    let
+      q-1⁺ = positive-diff-le-ℚ one-ℚ q 1<q
+      (n , b<⟨1+q-1⟩ⁿ) = bound-unbounded-power-one-plus-ℚ⁺ q-1⁺ b
+    in
+      ( n ,
+        tr
+          ( le-ℚ b)
+          ( ap
+            ( rational-ℚ⁺ ∘ power-ℚ⁺ n)
+            ( eq-ℚ⁺ (is-identity-right-conjugation-add-ℚ one-ℚ q)))
+          ( b<⟨1+q-1⟩ⁿ))
+
+  unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  unbounded-power-greater-than-one-ℚ⁺ q b 1<q =
+    unit-trunc-Prop (bound-unbounded-power-greater-than-one-ℚ⁺ q b 1<q)
+```
+
+### If `ε` is a positive rational number less than one, `εⁿ` becomes arbitrarily small
+
+```agda
+abstract
+  bound-arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    Σ ℕ (λ n → le-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ ε<1 =
+    let
+      1/δ = inv-ℚ⁺ δ
+      1/ε = inv-ℚ⁺ ε
+      1<1/ε =
+        tr
+          ( λ q → le-ℚ⁺ q 1/ε)
+          ( inv-one-ℚ⁺)
+          ( inv-le-ℚ⁺ ε one-ℚ⁺ ε<1)
+      (n , 1/δ<⟨1/ε⟩ⁿ) =
+        bound-unbounded-power-greater-than-one-ℚ⁺ 1/ε (rational-ℚ⁺ 1/δ) 1<1/ε
+    in
+      ( n ,
+        binary-tr
+          ( le-ℚ⁺)
+          ( equational-reasoning
+            inv-ℚ⁺ (power-ℚ⁺ n 1/ε)
+            ＝ inv-ℚ⁺ (inv-ℚ⁺ (power-ℚ⁺ n ε))
+              by ap inv-ℚ⁺ (power-inv-Group group-mul-ℚ⁺ n ε)
+            ＝ power-ℚ⁺ n ε
+              by inv-inv-ℚ⁺ (power-ℚ⁺ n ε))
+          ( inv-inv-ℚ⁺ δ)
+          ( inv-le-ℚ⁺ 1/δ (power-ℚ⁺ n 1/ε) 1/δ<⟨1/ε⟩ⁿ))
+
+  arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    exists ℕ (λ n → le-prop-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε =
+    unit-trunc-Prop (bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε)
+```
+
+## See also
+
+- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)

From 06518412e816c73dff5818342930a5783b9ce9f4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:07:41 -0700
Subject: [PATCH 015/134] Clarify concept disambiguation

---
 .../powers-positive-rational-numbers.lagda.md                   | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 811235c5420..789446dfa67 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -45,7 +45,7 @@ open import group-theory.powers-of-elements-groups
 ## Idea
 
 The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
+{{#concept "power operation" Disambiguation="a positive rational number to a natural number power" Agda=power-ℚ⁺}}
 on the [positive](elementary-number-theory.positive-rational-numbers.md)
 [rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
 which is defined by [iteratively](foundation.iterating-functions.md) multiplying

From e0a24b825f0e63177d5c9fe62c33c3a092b47e2d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:08:06 -0700
Subject: [PATCH 016/134] Progress

---
 .../powers-rational-numbers.lagda.md                     | 9 ++++-----
 1 file changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index ee922d0fad0..2f5c88228e5 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -17,11 +17,10 @@ open import foundation.identity-types
 ## Idea
 
 The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
-on the [positive](elementary-number-theory.positive-rational-numbers.md)
-[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
-which is defined by [iteratively](foundation.iterating-functions.md) multiplying
-`x` with itself `n` times.
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+on the [rational numbers](elementary-number-theory.rational-numbers.md)
+`n x ↦ xⁿ`, is defined by [iteratively](foundation.iterating-functions.md)
+multiplying `x` with itself `n` times.
 
 ## Definition
 

From c5b3be8a204d11ef7e5b5439126af5ecc24b5cd4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:08:45 -0700
Subject: [PATCH 017/134] Further clarify doc

---
 .../powers-positive-rational-numbers.lagda.md             | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 789446dfa67..61a906395d3 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -47,9 +47,11 @@ open import group-theory.powers-of-elements-groups
 The
 {{#concept "power operation" Disambiguation="a positive rational number to a natural number power" Agda=power-ℚ⁺}}
 on the [positive](elementary-number-theory.positive-rational-numbers.md)
-[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
-which is defined by [iteratively](foundation.iterating-functions.md) multiplying
-`x` with itself `n` times.
+[rational numbers](elementary-number-theory.rational-numbers.md) is the
+operation `n x ↦ xⁿ`, which is defined by
+[iteratively](foundation.iterating-functions.md) multiplying `x` with itself `n`
+times. This file covers the case where `n` is a
+[natural number](elementary-number-theory.natural-numbers.md).
 
 ## Definition
 

From a8c35cd424f51abc92b27cfe1e0f2330b3df1025 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:55:55 -0700
Subject: [PATCH 018/134] Progress

---
 src/elementary-number-theory.lagda.md         |   1 +
 .../parity-natural-numbers.lagda.md           |  10 +-
 .../powers-rational-numbers.lagda.md          | 271 +++++++++---------
 .../squares-rational-numbers.lagda.md         |  24 ++
 ...s-of-elements-commutative-monoids.lagda.md |   8 +-
 5 files changed, 168 insertions(+), 146 deletions(-)

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index be3b1e0ad01..0e518ce0112 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -180,6 +180,7 @@ open import elementary-number-theory.positive-rational-numbers public
 open import elementary-number-theory.powers-integers public
 open import elementary-number-theory.powers-of-two public
 open import elementary-number-theory.powers-positive-rational-numbers public
+open import elementary-number-theory.powers-rational-numbers public
 open import elementary-number-theory.prime-numbers public
 open import elementary-number-theory.products-of-natural-numbers public
 open import elementary-number-theory.proper-closed-intervals-rational-numbers public
diff --git a/src/elementary-number-theory/parity-natural-numbers.lagda.md b/src/elementary-number-theory/parity-natural-numbers.lagda.md
index 503b5245425..9fa410e7f87 100644
--- a/src/elementary-number-theory/parity-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/parity-natural-numbers.lagda.md
@@ -29,9 +29,13 @@ open import foundation.universe-levels
 
 ## Idea
 
-Parity partitions the natural numbers into two classes: the even and the odd
-natural numbers. Even natural numbers are those that are divisible by two, and
-odd natural numbers are those that aren't.
+{{#concept "Parity" WDID=Q230967 WD="parity"}} partitions the
+[natural numbers](elementary-number-theory.natural-numbers.md) into two classes:
+the {{#concept "even" WDID=Q13366104 WD="even number" Agda=is-even-ℕ}} and the
+{{#concept "odd" WDID=Q13366129 WD="odd number" Agda=is-odd-ℕ}} natural numbers.
+Even natural numbers are those that are
+[divisible](elementary-number-theory.divisibility-natural-numbers.md) by two,
+and odd natural numbers are those that [aren't](foundation.negation.md).
 
 ## Definition
 
diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index 2f5c88228e5..83f95cea235 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -7,9 +7,32 @@ module elementary-number-theory.powers-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.absolute-value-rational-numbers
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
-open import group-theory.powers-of-elements-monoids
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.parity-natural-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.squares-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
 open import foundation.identity-types
+open import foundation.transport-along-identifications
+
+open import group-theory.powers-of-elements-commutative-monoids
+open import group-theory.powers-of-elements-monoids
 ```
 
 </details>
@@ -25,8 +48,8 @@ multiplying `x` with itself `n` times.
 ## Definition
 
 ```agda
-power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
-power-ℚ⁺ = power-Group group-mul-ℚ⁺
+power-ℚ : ℕ → ℚ → ℚ
+power-ℚ = power-Monoid monoid-mul-ℚ
 ```
 
 ## Properties
@@ -34,194 +57,162 @@ power-ℚ⁺ = power-Group group-mul-ℚ⁺
 ### `1ⁿ = 1`
 
 ```agda
-power-one-ℚ⁺ : (n : ℕ) → power-ℚ⁺ n one-ℚ⁺ ＝ one-ℚ⁺
-power-one-ℚ⁺ = power-unit-Group group-mul-ℚ⁺
+abstract
+  power-one-ℚ : (n : ℕ) → power-ℚ n one-ℚ ＝ one-ℚ
+  power-one-ℚ = power-unit-Monoid monoid-mul-ℚ
 ```
 
-### `qⁿ⁺¹ = qⁿq`
+### `qⁿ¹ = qⁿq`
 
 ```agda
-power-succ-ℚ⁺ : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ power-ℚ⁺ n q *ℚ⁺ q
-power-succ-ℚ⁺ = power-succ-Group group-mul-ℚ⁺
+abstract
+  power-succ-ℚ : (n : ℕ) (q : ℚ) → power-ℚ (succ-ℕ n) q ＝ power-ℚ n q *ℚ q
+  power-succ-ℚ = power-succ-Monoid monoid-mul-ℚ
 ```
 
 ### `qⁿ = qqⁿ`
 
 ```agda
-power-succ-ℚ⁺' : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ q *ℚ⁺ power-ℚ⁺ n q
-power-succ-ℚ⁺' = power-succ-Group' group-mul-ℚ⁺
+abstract
+  power-succ-ℚ' : (n : ℕ) (q : ℚ) → power-ℚ (succ-ℕ n) q ＝ q *ℚ power-ℚ n q
+  power-succ-ℚ' = power-succ-Monoid' monoid-mul-ℚ
 ```
 
-### `qᵐ⁺ⁿ = qᵐqⁿ`
+### `qᵐⁿ = qᵐqⁿ`
 
 ```agda
-distributive-power-add-ℚ⁺ :
-  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m +ℕ n) q ＝ power-ℚ⁺ m q *ℚ⁺ power-ℚ⁺ n q
-distributive-power-add-ℚ⁺ m n _ = distributive-power-add-Group group-mul-ℚ⁺ m n
+abstract
+  distributive-power-add-ℚ :
+    (m n : ℕ) (q : ℚ) → power-ℚ (m +ℕ n) q ＝ power-ℚ m q *ℚ power-ℚ n q
+  distributive-power-add-ℚ m n _ =
+    distributive-power-add-Monoid monoid-mul-ℚ m n
 ```
 
 ### `(pq)ⁿ=pⁿqⁿ`
 
 ```agda
-left-distributive-power-mul-ℚ⁺ :
-  (n : ℕ) (p q : ℚ⁺) → power-ℚ⁺ n (p *ℚ⁺ q) ＝ power-ℚ⁺ n p *ℚ⁺ power-ℚ⁺ n q
-left-distributive-power-mul-ℚ⁺ n p q =
-  left-distributive-multiple-add-Ab abelian-group-mul-ℚ⁺ n
+abstract
+  left-distributive-power-mul-ℚ :
+    (n : ℕ) (p q : ℚ) → power-ℚ n (p *ℚ q) ＝ power-ℚ n p *ℚ power-ℚ n q
+  left-distributive-power-mul-ℚ n p q =
+    distributive-power-mul-Commutative-Monoid commutative-monoid-mul-ℚ n
 ```
 
 ### `pᵐⁿ = (pᵐ)ⁿ`
 
 ```agda
-power-mul-ℚ⁺ :
-  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m *ℕ n) q ＝ power-ℚ⁺ n (power-ℚ⁺ m q)
-power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
+abstract
+  power-mul-ℚ :
+    (m n : ℕ) (q : ℚ) → power-ℚ (m *ℕ n) q ＝ power-ℚ n (power-ℚ m q)
+  power-mul-ℚ m n q = power-mul-Monoid monoid-mul-ℚ m n
 ```
 
-### If `p` and `q` are positive rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+### Even powers of rational numbers are nonnegative
 
 ```agda
 abstract
-  preserves-le-power-ℚ⁺ :
-    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
-    le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
-  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
-  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
-  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
-    transitive-le-ℚ⁺
-      ( power-ℚ⁺ (succ-ℕ n) p)
-      ( power-ℚ⁺ n p *ℚ⁺ q)
-      ( power-ℚ⁺ (succ-ℕ n) q)
-      ( preserves-le-right-mul-ℚ⁺ q _ _
-        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
-      ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
+  is-nonnegative-even-power-ℚ :
+    (n : ℕ) (q : ℚ) → is-even-ℕ n → is-nonnegative-ℚ (power-ℚ n q)
+  is-nonnegative-even-power-ℚ _ q (k , refl) =
+    inv-tr
+      ( is-nonnegative-ℚ)
+      ( power-mul-ℚ k 2 q)
+      ( is-nonnegative-square-ℚ (power-ℚ k q))
+
+nonnegative-even-power-ℚ :
+  (n : ℕ) (q : ℚ) → is-even-ℕ n → ℚ⁰⁺
+nonnegative-even-power-ℚ n q even-n =
+  ( power-ℚ n q , is-nonnegative-even-power-ℚ n q even-n)
 ```
 
-### If `p` and `q` are positive rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+### Powers of positive rational numbers are positive
 
 ```agda
 abstract
-  preserves-leq-power-ℚ⁺ :
-    (n : ℕ) (p q : ℚ⁺) → leq-ℚ⁺ p q → leq-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
-  preserves-leq-power-ℚ⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
-  preserves-leq-power-ℚ⁺ 1 p q p≤q = p≤q
-  preserves-leq-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
-    transitive-leq-ℚ⁺
-      ( power-ℚ⁺ (succ-ℕ n) p)
-      ( power-ℚ⁺ n p *ℚ⁺ q)
-      ( power-ℚ⁺ (succ-ℕ n) q)
-      ( preserves-leq-right-mul-ℚ⁺ q _ _ (preserves-leq-power-ℚ⁺ n p q p≤q))
-      ( preserves-leq-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p≤q)
+  is-positive-power-ℚ⁺ :
+    (n : ℕ) (q : ℚ⁺) → is-positive-ℚ (power-ℚ n (rational-ℚ⁺ q))
+  is-positive-power-ℚ⁺ 0 q = is-positive-rational-ℚ⁺ one-ℚ⁺
+  is-positive-power-ℚ⁺ (succ-ℕ n) q⁺@(q , is-pos-q) =
+    inv-tr
+      ( is-positive-ℚ)
+      ( power-succ-ℚ n q)
+      ( is-positive-mul-ℚ (is-positive-power-ℚ⁺ n q⁺) is-pos-q)
 ```
 
-### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+### Even powers of negative rational numbers are positive
 
 ```agda
 abstract
-  bound-unbounded-power-one-plus-ℚ⁺ :
-    (ε : ℚ⁺) (b : ℚ) →
-    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
-  bound-unbounded-power-one-plus-ℚ⁺ ε⁺@(ε , is-pos-ε) b =
-    let
-      (n , b<nε) = bound-archimedean-property-ℚ ε b is-pos-ε
-    in
-      ( n ,
-        transitive-le-ℚ _ _ _
-          ( concatenate-le-leq-ℚ _ _ _
-            ( le-left-add-rational-ℚ⁺ _ one-ℚ⁺)
-            ( binary-tr
-              ( leq-ℚ)
-              ( inv (compute-standard-arithmetic-sequence-ℚ⁺ one-ℚ⁺ ε⁺ n))
-              ( ap
-                ( rational-ℚ⁺)
-                ( equational-reasoning
-                  seq-standard-geometric-sequence-ℚ⁺
-                      ( one-ℚ⁺)
-                      ( one-ℚ⁺ +ℚ⁺ ε⁺)
-                      ( n)
-                  ＝ one-ℚ⁺ *ℚ⁺ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
-                    by
-                      inv
-                        ( compute-standard-geometric-sequence-ℚ⁺
-                          ( one-ℚ⁺)
-                          ( one-ℚ⁺ +ℚ⁺ ε⁺)
-                          ( n))
-                  ＝ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
-                    by left-unit-law-mul-ℚ⁺ _))
-              ( bernoullis-inequality-ℚ⁺ ε⁺ n)))
-          ( b<nε))
-
-  unbounded-power-one-plus-ℚ⁺ :
-    (ε : ℚ⁺) (b : ℚ) →
-    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
-  unbounded-power-one-plus-ℚ⁺ ε b =
-    unit-trunc-Prop (bound-unbounded-power-one-plus-ℚ⁺ ε b)
+  is-positive-even-power-ℚ⁻ :
+    (n : ℕ) (q : ℚ⁻) → is-even-ℕ n → is-positive-ℚ (power-ℚ n (rational-ℚ⁻ q))
+  is-positive-even-power-ℚ⁻ _ q⁻@(q , is-neg-q) (k , refl) =
+    inv-tr
+      ( is-positive-ℚ)
+      ( equational-reasoning
+        power-ℚ (k *ℕ 2) q
+        ＝ power-ℚ (2 *ℕ k) q
+          by ap (λ m → power-ℚ m q) (commutative-mul-ℕ k 2)
+        ＝ power-ℚ k (square-ℚ q)
+          by power-mul-ℚ 2 k q)
+      ( is-positive-power-ℚ⁺ k (square-ℚ⁻ q⁻))
 ```
 
-### If `1 < q`, `qⁿ` grows without bound
+### Odd powers of negative rational numbers are negative
 
 ```agda
 abstract
-  bound-unbounded-power-greater-than-one-ℚ⁺ :
-    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
-    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n q)))
-  bound-unbounded-power-greater-than-one-ℚ⁺ q⁺@(q , _) b 1<q =
+  is-negative-odd-power-ℚ⁻ :
+    (n : ℕ) (q : ℚ⁻) → is-odd-ℕ n → is-negative-ℚ (power-ℚ n (rational-ℚ⁻ q))
+  is-negative-odd-power-ℚ⁻ n q⁻@(q , is-neg-q) odd-n =
     let
-      q-1⁺ = positive-diff-le-ℚ one-ℚ q 1<q
-      (n , b<⟨1+q-1⟩ⁿ) = bound-unbounded-power-one-plus-ℚ⁺ q-1⁺ b
+      (k , k2+1=n) = has-odd-expansion-is-odd n odd-n
     in
-      ( n ,
-        tr
-          ( le-ℚ b)
-          ( ap
-            ( rational-ℚ⁺ ∘ power-ℚ⁺ n)
-            ( eq-ℚ⁺ (is-identity-right-conjugation-add-ℚ one-ℚ q)))
-          ( b<⟨1+q-1⟩ⁿ))
-
-  unbounded-power-greater-than-one-ℚ⁺ :
-    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
-    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n q)))
-  unbounded-power-greater-than-one-ℚ⁺ q b 1<q =
-    unit-trunc-Prop (bound-unbounded-power-greater-than-one-ℚ⁺ q b 1<q)
+      tr
+        ( is-negative-ℚ)
+        ( equational-reasoning
+          power-ℚ (k *ℕ 2) q *ℚ q
+          ＝ power-ℚ (succ-ℕ (k *ℕ 2)) q
+            by inv (power-succ-ℚ (k *ℕ 2) q)
+          ＝ power-ℚ n q
+            by ap (λ m → power-ℚ m q) k2+1=n)
+        ( is-negative-mul-positive-negative-ℚ
+          ( is-positive-even-power-ℚ⁻ (k *ℕ 2) q⁻ (k , refl))
+          ( is-neg-q))
 ```
 
-### If `ε` is a positive rational number less than one, `εⁿ` becomes arbitrarily small
+### If `|p| ≤ |q|`, `|pⁿ| ≤ |qⁿ|`
 
 ```agda
 abstract
-  bound-arbitrarily-small-power-le-one-ℚ⁺ :
-    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
-    Σ ℕ (λ n → le-ℚ⁺ (power-ℚ⁺ n ε) δ)
-  bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ ε<1 =
-    let
-      1/δ = inv-ℚ⁺ δ
-      1/ε = inv-ℚ⁺ ε
-      1<1/ε =
-        tr
-          ( λ q → le-ℚ⁺ q 1/ε)
-          ( inv-one-ℚ⁺)
-          ( inv-le-ℚ⁺ ε one-ℚ⁺ ε<1)
-      (n , 1/δ<⟨1/ε⟩ⁿ) =
-        bound-unbounded-power-greater-than-one-ℚ⁺ 1/ε (rational-ℚ⁺ 1/δ) 1<1/ε
-    in
-      ( n ,
-        binary-tr
-          ( le-ℚ⁺)
-          ( equational-reasoning
-            inv-ℚ⁺ (power-ℚ⁺ n 1/ε)
-            ＝ inv-ℚ⁺ (inv-ℚ⁺ (power-ℚ⁺ n ε))
-              by ap inv-ℚ⁺ (power-inv-Group group-mul-ℚ⁺ n ε)
-            ＝ power-ℚ⁺ n ε
-              by inv-inv-ℚ⁺ (power-ℚ⁺ n ε))
-          ( inv-inv-ℚ⁺ δ)
-          ( inv-le-ℚ⁺ 1/δ (power-ℚ⁺ n 1/ε) 1/δ<⟨1/ε⟩ⁿ))
-
-  arbitrarily-small-power-le-one-ℚ⁺ :
-    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
-    exists ℕ (λ n → le-prop-ℚ⁺ (power-ℚ⁺ n ε) δ)
-  arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε =
-    unit-trunc-Prop (bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε)
+  preserves-leq-abs-power-ℚ :
+    (n : ℕ) (p q : ℚ) → leq-ℚ⁰⁺ (abs-ℚ p) (abs-ℚ q) →
+    leq-ℚ⁰⁺ (abs-ℚ (power-ℚ n p)) (abs-ℚ (power-ℚ n q))
+  preserves-leq-abs-power-ℚ 0 p q _ = refl-leq-ℚ _
+  preserves-leq-abs-power-ℚ (succ-ℕ n) p q |p|≤|q| =
+    binary-tr
+      ( leq-ℚ)
+      ( equational-reasoning
+        rational-abs-ℚ (power-ℚ n p) *ℚ rational-abs-ℚ p
+        ＝ rational-abs-ℚ (power-ℚ n p *ℚ p)
+          by inv (rational-abs-mul-ℚ (power-ℚ n p) p)
+        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) p)
+          by ap rational-abs-ℚ (inv (power-succ-ℚ n p)))
+      ( equational-reasoning
+        rational-abs-ℚ (power-ℚ n q) *ℚ rational-abs-ℚ q
+        ＝ rational-abs-ℚ (power-ℚ n q *ℚ q)
+          by inv (rational-abs-mul-ℚ (power-ℚ n q) q)
+        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) q)
+          by ap rational-abs-ℚ (inv (power-succ-ℚ n q)))
+      ( preserves-leq-mul-ℚ⁰⁺
+        ( abs-ℚ (power-ℚ n p))
+        ( abs-ℚ (power-ℚ n q))
+        ( abs-ℚ p)
+        ( abs-ℚ q)
+        ( preserves-leq-abs-power-ℚ n p q |p|≤|q|)
+        ( |p|≤|q|))
 ```
 
 ## See also
 
-- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)
+- [Powers of elements of a monoid](group-theory.powers-of-elements-monoids.md)
+- [Powers of elements of a commutative monoid](group-theory.powers-of-elements-commutative-monoids.md)
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index 16de8f520d9..d9c561a0178 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -9,11 +9,13 @@ module elementary-number-theory.squares-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers
@@ -171,6 +173,10 @@ abstract
     (x : ℚ) → is-negative-ℚ x → is-positive-ℚ (square-ℚ x)
   is-positive-square-negative-ℚ x neg-x =
     is-positive-mul-negative-ℚ {x} {x} neg-x neg-x
+
+square-ℚ⁻ : ℚ⁻ → ℚ⁺
+square-ℚ⁻ (q , is-neg-q) =
+  (square-ℚ q , is-positive-square-negative-ℚ q is-neg-q)
 ```
 
 ### If the square of a rational number is 0, it is zero
@@ -274,3 +280,21 @@ abstract
             ( preserves-leq-square-ℚ⁰⁺ q⁰⁺ p⁰⁺ q≤p)))
       ( decide-le-leq-ℚ p q)
 ```
+
+### `|p|² = p²`
+
+```agda
+abstract
+  square-abs-ℚ : (q : ℚ) → square-ℚ (rational-abs-ℚ q) ＝ square-ℚ q
+  square-abs-ℚ q =
+    rec-coproduct
+      ( λ q≤-q →
+        equational-reasoning
+          square-ℚ (rational-abs-ℚ q)
+          ＝ square-ℚ (neg-ℚ q)
+            by ap square-ℚ (left-leq-right-max-ℚ _ _ q≤-q)
+          ＝ square-ℚ q
+            by square-neg-ℚ q)
+      ( λ -q≤q → ap square-ℚ (right-leq-left-max-ℚ _ _ -q≤q))
+      ( linear-leq-ℚ q (neg-ℚ q))
+```
diff --git a/src/group-theory/powers-of-elements-commutative-monoids.lagda.md b/src/group-theory/powers-of-elements-commutative-monoids.lagda.md
index 344471dcb03..54c92815ab0 100644
--- a/src/group-theory/powers-of-elements-commutative-monoids.lagda.md
+++ b/src/group-theory/powers-of-elements-commutative-monoids.lagda.md
@@ -144,13 +144,15 @@ module _
 
   distributive-power-mul-Commutative-Monoid :
     (n : ℕ) {x y : type-Commutative-Monoid M} →
-    (H : mul-Commutative-Monoid M x y ＝ mul-Commutative-Monoid M y x) →
     power-Commutative-Monoid M n (mul-Commutative-Monoid M x y) ＝
     mul-Commutative-Monoid M
       ( power-Commutative-Monoid M n x)
       ( power-Commutative-Monoid M n y)
-  distributive-power-mul-Commutative-Monoid =
-    distributive-power-mul-Monoid (monoid-Commutative-Monoid M)
+  distributive-power-mul-Commutative-Monoid n =
+    distributive-power-mul-Monoid
+      ( monoid-Commutative-Monoid M)
+      ( n)
+      ( commutative-mul-Commutative-Monoid M _ _)
 ```
 
 ### Powers by products of natural numbers are iterated powers

From 906c3fb059f557700d7b67fb336fef916c093875 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 12:21:35 -0700
Subject: [PATCH 019/134] Progress

---
 src/elementary-number-theory.lagda.md         |   2 +
 .../absolute-value-rational-numbers.lagda.md  |  14 +-
 ...lity-nonnegative-rational-numbers.lagda.md |   8 +
 ...d-of-nonnegative-rational-numbers.lagda.md |  55 ++++
 ...cative-monoid-of-rational-numbers.lagda.md |   4 +-
 .../nonnegative-rational-numbers.lagda.md     |   3 +
 .../parity-natural-numbers.lagda.md           |   9 +
 ...wers-nonnegative-rational-numbers.lagda.md | 146 ++++++++++
 .../powers-positive-rational-numbers.lagda.md |  10 +-
 .../powers-rational-numbers.lagda.md          | 258 +++++++++++++++---
 ...lity-nonnegative-rational-numbers.lagda.md |  22 ++
 .../submonoids-commutative-monoids.lagda.md   |   5 +-
 src/group-theory/submonoids.lagda.md          |   6 +-
 13 files changed, 495 insertions(+), 47 deletions(-)
 create mode 100644 src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 0e518ce0112..6655218d7ca 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -148,6 +148,7 @@ open import elementary-number-theory.multiplicative-group-of-positive-rational-n
 open import elementary-number-theory.multiplicative-group-of-rational-numbers public
 open import elementary-number-theory.multiplicative-inverses-positive-integer-fractions public
 open import elementary-number-theory.multiplicative-monoid-of-natural-numbers public
+open import elementary-number-theory.multiplicative-monoid-of-nonnegative-rational-numbers public
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers public
 open import elementary-number-theory.multiplicative-units-integers public
 open import elementary-number-theory.multiplicative-units-standard-cyclic-rings public
@@ -178,6 +179,7 @@ open import elementary-number-theory.positive-integer-fractions public
 open import elementary-number-theory.positive-integers public
 open import elementary-number-theory.positive-rational-numbers public
 open import elementary-number-theory.powers-integers public
+open import elementary-number-theory.powers-nonnegative-rational-numbers public
 open import elementary-number-theory.powers-of-two public
 open import elementary-number-theory.powers-positive-rational-numbers public
 open import elementary-number-theory.powers-rational-numbers public
diff --git a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
index fa1c1a6995c..4b0ad01d885 100644
--- a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
@@ -174,13 +174,15 @@ abstract
         ( neg-neg-ℚ q)
         ( neg-leq-ℚ (neg-leq-abs-ℚ q)))
 
+  rational-abs-zero-ℚ : rational-abs-ℚ zero-ℚ ＝ zero-ℚ
+  rational-abs-zero-ℚ =
+    equational-reasoning
+      max-ℚ zero-ℚ (neg-ℚ zero-ℚ)
+      ＝ max-ℚ zero-ℚ zero-ℚ by ap-max-ℚ refl neg-zero-ℚ
+      ＝ zero-ℚ by idempotent-max-ℚ zero-ℚ
+
   abs-zero-ℚ : abs-ℚ zero-ℚ ＝ zero-ℚ⁰⁺
-  abs-zero-ℚ =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        max-ℚ zero-ℚ (neg-ℚ zero-ℚ)
-        ＝ max-ℚ zero-ℚ zero-ℚ by ap-max-ℚ refl neg-zero-ℚ
-        ＝ zero-ℚ by idempotent-max-ℚ zero-ℚ)
+  abs-zero-ℚ = eq-ℚ⁰⁺ rational-abs-zero-ℚ
 ```
 
 ### The triangle inequality
diff --git a/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md
index 7ec428e6cf1..a78e7658d87 100644
--- a/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md
@@ -51,3 +51,11 @@ decidable-total-order-ℚ⁰⁺ =
     ℚ-Decidable-Total-Order
     is-nonnegative-prop-ℚ
 ```
+
+### Inequality on the nonnegative rational numbers is transitive
+
+```agda
+abstract
+  transitive-leq-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → leq-ℚ⁰⁺ q r → leq-ℚ⁰⁺ p q → leq-ℚ⁰⁺ p r
+  transitive-leq-ℚ⁰⁺ (p , _) (q , _) (r , _) = transitive-leq-ℚ p q r
+```
diff --git a/src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..da8d59e5b14
--- /dev/null
+++ b/src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md
@@ -0,0 +1,55 @@
+# The multiplicative monoid of nonnegative rational numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module elementary-number-theory.multiplicative-monoid-of-nonnegative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import group-theory.commutative-monoids
+open import group-theory.monoids
+open import group-theory.submonoids-commutative-monoids
+```
+
+</details>
+
+## Idea
+
+The type of
+[nonnegative](elementary-number-theory.nonnegative-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) equipped with
+[multiplication](elementary-number-theory.multiplication-nonnegative-rational-numbers.md)
+is a [commutative monoid](group-theory.commutative-monoids.md) with unit
+`one-ℚ⁰⁺`.
+
+## Definitions
+
+### The multiplicative commutative monoid of nonnegative rational numbers
+
+```agda
+nonnegative-submonoid-commutative-monoid-mul-ℚ :
+  Commutative-Submonoid lzero commutative-monoid-mul-ℚ
+nonnegative-submonoid-commutative-monoid-mul-ℚ =
+  ( is-nonnegative-prop-ℚ ,
+    is-nonnegative-one-ℚ ,
+    λ _ _ → is-nonnegative-mul-ℚ)
+
+commutative-monoid-mul-ℚ⁰⁺ : Commutative-Monoid lzero
+commutative-monoid-mul-ℚ⁰⁺ =
+  commutative-monoid-Commutative-Submonoid
+    ( commutative-monoid-mul-ℚ)
+    ( nonnegative-submonoid-commutative-monoid-mul-ℚ)
+
+monoid-mul-ℚ⁰⁺ : Monoid lzero
+monoid-mul-ℚ⁰⁺ = monoid-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
diff --git a/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md
index 781ac3df4c6..a3ac5b301d7 100644
--- a/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md
@@ -27,8 +27,8 @@ open import group-theory.semigroups
 
 The type of [rational numbers](elementary-number-theory.rational-numbers.md)
 equipped with
-[multiplication](elementary-number-theory.addition-rational-numbers.md) is a
-[commutative monoid](group-theory.commutative-monoids.md) with unit `one-ℚ`.
+[multiplication](elementary-number-theory.multiplication-rational-numbers.md) is
+a [commutative monoid](group-theory.commutative-monoids.md) with unit `one-ℚ`.
 
 ## Definitions
 
diff --git a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
index b7d23dc65ad..f16ddf889fb 100644
--- a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
@@ -166,6 +166,9 @@ zero-ℚ⁰⁺ = nonnegative-rational-ℕ zero-ℕ
 
 one-ℚ⁰⁺ : ℚ⁰⁺
 one-ℚ⁰⁺ = nonnegative-rational-ℕ 1
+
+is-nonnegative-one-ℚ : is-nonnegative-ℚ one-ℚ
+is-nonnegative-one-ℚ = is-nonnegative-rational-ℕ 1
 ```
 
 ### The rational image of a nonnegative integer fraction is nonnegative
diff --git a/src/elementary-number-theory/parity-natural-numbers.lagda.md b/src/elementary-number-theory/parity-natural-numbers.lagda.md
index 9fa410e7f87..5a7c82123c1 100644
--- a/src/elementary-number-theory/parity-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/parity-natural-numbers.lagda.md
@@ -12,6 +12,7 @@ open import elementary-number-theory.equality-natural-numbers
 open import elementary-number-theory.modular-arithmetic-standard-finite-types
 open import elementary-number-theory.multiplication-natural-numbers
 open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.nonzero-natural-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.decidable-types
@@ -69,6 +70,14 @@ is-odd-one-ℕ : is-odd-ℕ 1
 is-odd-one-ℕ H = Eq-eq-ℕ (is-one-div-one-ℕ 2 H)
 ```
 
+### An odd natural number is nonzero
+
+```agda
+abstract
+  is-nonzero-is-odd-ℕ : {n : ℕ} → is-odd-ℕ n → is-nonzero-ℕ n
+  is-nonzero-is-odd-ℕ odd-n refl = odd-n is-even-zero-ℕ
+```
+
 ### A natural number `x` is even if and only if `x + 2` is even
 
 ```agda
diff --git a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..9ad7fa97c80
--- /dev/null
+++ b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
@@ -0,0 +1,146 @@
+# Powers of nonnegative rational numbers
+
+```agda
+module elementary-number-theory.powers-nonnegative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplicative-monoid-of-nonnegative-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-nonnegative-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.identity-types
+
+open import group-theory.powers-of-elements-commutative-monoids
+open import group-theory.powers-of-elements-monoids
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="a nonnegative rational number to a natural number power" Agda=power-ℚ⁰⁺}}
+on the [nonnegative](elementary-number-theory.nonnegative-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) is the
+operation `n x ↦ xⁿ`, which is defined by
+[iteratively](foundation.iterating-functions.md) multiplying `x` with itself `n`
+times. This file covers the case where `n` is a
+[natural number](elementary-number-theory.natural-numbers.md).
+
+## Definition
+
+```agda
+power-ℚ⁰⁺ : ℕ → ℚ⁰⁺ → ℚ⁰⁺
+power-ℚ⁰⁺ = power-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
+
+## Properties
+
+### `1ⁿ = 1`
+
+```agda
+power-one-ℚ⁰⁺ : (n : ℕ) → power-ℚ⁰⁺ n one-ℚ⁰⁺ ＝ one-ℚ⁰⁺
+power-one-ℚ⁰⁺ = power-unit-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
+
+### `qⁿ⁺¹ = qⁿq`
+
+```agda
+power-succ-ℚ⁰⁺ :
+  (n : ℕ) (q : ℚ⁰⁺) → power-ℚ⁰⁺ (succ-ℕ n) q ＝ power-ℚ⁰⁺ n q *ℚ⁰⁺ q
+power-succ-ℚ⁰⁺ = power-succ-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
+
+### `qⁿ = qqⁿ`
+
+```agda
+power-succ-ℚ⁰⁺' :
+  (n : ℕ) (q : ℚ⁰⁺) → power-ℚ⁰⁺ (succ-ℕ n) q ＝ q *ℚ⁰⁺ power-ℚ⁰⁺ n q
+power-succ-ℚ⁰⁺' = power-succ-Commutative-Monoid' commutative-monoid-mul-ℚ⁰⁺
+```
+
+### `qᵐ⁺ⁿ = qᵐqⁿ`
+
+```agda
+abstract
+  distributive-power-add-ℚ⁰⁺ :
+    (m n : ℕ) (q : ℚ⁰⁺) →
+    power-ℚ⁰⁺ (m +ℕ n) q ＝ power-ℚ⁰⁺ m q *ℚ⁰⁺ power-ℚ⁰⁺ n q
+  distributive-power-add-ℚ⁰⁺ m n _ =
+    distributive-power-add-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺ m n
+```
+
+### `(pq)ⁿ=pⁿqⁿ`
+
+```agda
+abstract
+  left-distributive-power-mul-ℚ⁰⁺ :
+    (n : ℕ) (p q : ℚ⁰⁺) →
+    power-ℚ⁰⁺ n (p *ℚ⁰⁺ q) ＝ power-ℚ⁰⁺ n p *ℚ⁰⁺ power-ℚ⁰⁺ n q
+  left-distributive-power-mul-ℚ⁰⁺ n p q =
+    distributive-power-mul-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺ n
+```
+
+### `pᵐⁿ = (pᵐ)ⁿ`
+
+```agda
+power-mul-ℚ⁰⁺ :
+  (m n : ℕ) (q : ℚ⁰⁺) → power-ℚ⁰⁺ (m *ℕ n) q ＝ power-ℚ⁰⁺ n (power-ℚ⁰⁺ m q)
+power-mul-ℚ⁰⁺ m n q =
+  power-mul-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺ m n
+```
+
+### If `p` and `q` are nonnegative rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+
+```agda
+abstract
+  preserves-le-power-ℚ⁰⁺ :
+    (n : ℕ) → is-nonzero-ℕ n → (p q : ℚ⁰⁺) → le-ℚ⁰⁺ p q →
+    le-ℚ⁰⁺ (power-ℚ⁰⁺ n p) (power-ℚ⁰⁺ n q)
+  preserves-le-power-ℚ⁰⁺ 0 H p q p<q = ex-falso (H refl)
+  preserves-le-power-ℚ⁰⁺ 1 _ p q p<q = p<q
+  preserves-le-power-ℚ⁰⁺ (succ-ℕ n@(succ-ℕ _)) _ p q p<q =
+    concatenate-leq-le-ℚ⁰⁺
+      ( power-ℚ⁰⁺ (succ-ℕ n) p)
+      ( power-ℚ⁰⁺ n p *ℚ⁰⁺ q)
+      ( power-ℚ⁰⁺ (succ-ℕ n) q)
+      ( preserves-leq-left-mul-ℚ⁰⁺ (power-ℚ⁰⁺ n p) _ _ (leq-le-ℚ p<q))
+      ( preserves-le-right-mul-ℚ⁺
+        ( rational-ℚ⁰⁺ q , is-positive-le-ℚ⁰⁺ p _ p<q)
+        ( rational-ℚ⁰⁺ (power-ℚ⁰⁺ n p))
+        ( rational-ℚ⁰⁺ (power-ℚ⁰⁺ n q))
+        ( preserves-le-power-ℚ⁰⁺ n (is-nonzero-succ-ℕ _) p q p<q))
+```
+
+### If `p` and `q` are nonnegative rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+
+```agda
+abstract
+  preserves-leq-power-ℚ⁰⁺ :
+    (n : ℕ) (p q : ℚ⁰⁺) → leq-ℚ⁰⁺ p q → leq-ℚ⁰⁺ (power-ℚ⁰⁺ n p) (power-ℚ⁰⁺ n q)
+  preserves-leq-power-ℚ⁰⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
+  preserves-leq-power-ℚ⁰⁺ 1 p q p≤q = p≤q
+  preserves-leq-power-ℚ⁰⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
+    transitive-leq-ℚ⁰⁺
+      ( power-ℚ⁰⁺ (succ-ℕ n) p)
+      ( power-ℚ⁰⁺ n p *ℚ⁰⁺ q)
+      ( power-ℚ⁰⁺ (succ-ℕ n) q)
+      ( preserves-leq-right-mul-ℚ⁰⁺ q _ _ (preserves-leq-power-ℚ⁰⁺ n p q p≤q))
+      ( preserves-leq-left-mul-ℚ⁰⁺ (power-ℚ⁰⁺ n p) _ _ p≤q)
+```
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 61a906395d3..3ca56e989fa 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -113,17 +113,17 @@ power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
 ```agda
 abstract
   preserves-le-power-ℚ⁺ :
-    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
+    (n : ℕ) → is-nonzero-ℕ n → (p q : ℚ⁺) → le-ℚ⁺ p q →
     le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
-  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
-  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
-  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
+  preserves-le-power-ℚ⁺ 0 H p q p<q = ex-falso (H refl)
+  preserves-le-power-ℚ⁺ 1 _ p q p<q = p<q
+  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) _ p q p<q =
     transitive-le-ℚ⁺
       ( power-ℚ⁺ (succ-ℕ n) p)
       ( power-ℚ⁺ n p *ℚ⁺ q)
       ( power-ℚ⁺ (succ-ℕ n) q)
       ( preserves-le-right-mul-ℚ⁺ q _ _
-        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
+        ( preserves-le-power-ℚ⁺ n (is-nonzero-succ-ℕ _) p q p<q))
       ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
 ```
 
diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index 83f95cea235..1374740c371 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -9,6 +9,7 @@ module elementary-number-theory.powers-rational-numbers where
 ```agda
 open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.multiplication-natural-numbers
@@ -21,12 +22,17 @@ open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.parity-natural-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.powers-nonnegative-rational-numbers
+open import elementary-number-theory.powers-positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.squares-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
+open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
@@ -54,6 +60,31 @@ power-ℚ = power-Monoid monoid-mul-ℚ
 
 ## Properties
 
+### The power operation on rational numbers agrees with the power operation on positive rational numbers
+
+```agda
+abstract
+  power-rational-ℚ⁺ :
+    (n : ℕ) (q : ℚ⁺) → power-ℚ n (rational-ℚ⁺ q) ＝ rational-ℚ⁺ (power-ℚ⁺ n q)
+  power-rational-ℚ⁺ 0 _ = refl
+  power-rational-ℚ⁺ 1 _ = refl
+  power-rational-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) q =
+    ap-mul-ℚ (power-rational-ℚ⁺ n q) refl
+```
+
+### The power operation on rational numbers agrees with the power operation on nonnegative rational numbers
+
+```agda
+abstract
+  power-rational-ℚ⁰⁺ :
+    (n : ℕ) (q : ℚ⁰⁺) →
+    power-ℚ n (rational-ℚ⁰⁺ q) ＝ rational-ℚ⁰⁺ (power-ℚ⁰⁺ n q)
+  power-rational-ℚ⁰⁺ 0 _ = refl
+  power-rational-ℚ⁰⁺ 1 _ = refl
+  power-rational-ℚ⁰⁺ (succ-ℕ n@(succ-ℕ _)) q =
+    ap-mul-ℚ (power-rational-ℚ⁰⁺ n q) refl
+```
+
 ### `1ⁿ = 1`
 
 ```agda
@@ -105,6 +136,11 @@ abstract
   power-mul-ℚ :
     (m n : ℕ) (q : ℚ) → power-ℚ (m *ℕ n) q ＝ power-ℚ n (power-ℚ m q)
   power-mul-ℚ m n q = power-mul-Monoid monoid-mul-ℚ m n
+
+  power-mul-ℚ' :
+    (m n : ℕ) (q : ℚ) → power-ℚ (m *ℕ n) q ＝ power-ℚ m (power-ℚ n q)
+  power-mul-ℚ' m n q =
+    ap (λ k → power-ℚ k q) (commutative-mul-ℕ m n) ∙ power-mul-ℚ n m q
 ```
 
 ### Even powers of rational numbers are nonnegative
@@ -131,12 +167,11 @@ nonnegative-even-power-ℚ n q even-n =
 abstract
   is-positive-power-ℚ⁺ :
     (n : ℕ) (q : ℚ⁺) → is-positive-ℚ (power-ℚ n (rational-ℚ⁺ q))
-  is-positive-power-ℚ⁺ 0 q = is-positive-rational-ℚ⁺ one-ℚ⁺
-  is-positive-power-ℚ⁺ (succ-ℕ n) q⁺@(q , is-pos-q) =
+  is-positive-power-ℚ⁺ n q =
     inv-tr
       ( is-positive-ℚ)
-      ( power-succ-ℚ n q)
-      ( is-positive-mul-ℚ (is-positive-power-ℚ⁺ n q⁺) is-pos-q)
+      ( power-rational-ℚ⁺ n q)
+      ( is-positive-rational-ℚ⁺ (power-ℚ⁺ n q))
 ```
 
 ### Even powers of negative rational numbers are positive
@@ -148,12 +183,7 @@ abstract
   is-positive-even-power-ℚ⁻ _ q⁻@(q , is-neg-q) (k , refl) =
     inv-tr
       ( is-positive-ℚ)
-      ( equational-reasoning
-        power-ℚ (k *ℕ 2) q
-        ＝ power-ℚ (2 *ℕ k) q
-          by ap (λ m → power-ℚ m q) (commutative-mul-ℕ k 2)
-        ＝ power-ℚ k (square-ℚ q)
-          by power-mul-ℚ 2 k q)
+      ( power-mul-ℚ' k 2 q)
       ( is-positive-power-ℚ⁺ k (square-ℚ⁻ q⁻))
 ```
 
@@ -180,6 +210,66 @@ abstract
           ( is-neg-q))
 ```
 
+### For even `n`, `(-q)ⁿ = qⁿ`
+
+```agda
+abstract
+  even-power-neg-ℚ :
+    (n : ℕ) (q : ℚ) → is-even-ℕ n → power-ℚ n (neg-ℚ q) ＝ power-ℚ n q
+  even-power-neg-ℚ _ q (k , refl) =
+    equational-reasoning
+      power-ℚ (k *ℕ 2) (neg-ℚ q)
+      ＝ power-ℚ k (square-ℚ (neg-ℚ q))
+        by power-mul-ℚ' k 2 (neg-ℚ q)
+      ＝ power-ℚ k (square-ℚ q)
+        by ap (power-ℚ k) (square-neg-ℚ q)
+      ＝ power-ℚ (k *ℕ 2) q
+        by inv (power-mul-ℚ' k 2 q)
+```
+
+### For odd `n`, `(-q)ⁿ = -(qⁿ)`
+
+```agda
+abstract
+  odd-power-neg-ℚ :
+    (n : ℕ) (q : ℚ) → is-odd-ℕ n → power-ℚ n (neg-ℚ q) ＝ neg-ℚ (power-ℚ n q)
+  odd-power-neg-ℚ n q odd-n =
+    let (k , k2+1=n) = has-odd-expansion-is-odd n odd-n
+    in
+      equational-reasoning
+        power-ℚ n (neg-ℚ q)
+        ＝ power-ℚ (succ-ℕ (k *ℕ 2)) (neg-ℚ q)
+          by ap (λ m → power-ℚ m (neg-ℚ q)) (inv k2+1=n)
+        ＝ power-ℚ (k *ℕ 2) (neg-ℚ q) *ℚ neg-ℚ q
+          by power-succ-ℚ (k *ℕ 2) (neg-ℚ q)
+        ＝ power-ℚ (k *ℕ 2) q *ℚ neg-ℚ q
+          by ap-mul-ℚ (even-power-neg-ℚ _ q (k , refl)) refl
+        ＝ neg-ℚ (power-ℚ (k *ℕ 2) q *ℚ q)
+          by right-negative-law-mul-ℚ _ _
+        ＝ neg-ℚ (power-ℚ (succ-ℕ (k *ℕ 2)) q)
+          by ap neg-ℚ (inv (power-succ-ℚ (k *ℕ 2) q))
+        ＝ neg-ℚ (power-ℚ n q)
+          by ap (λ m → neg-ℚ (power-ℚ m q)) k2+1=n
+```
+
+### `|q|ⁿ=|qⁿ|`
+
+```agda
+abstract
+  power-rational-abs-ℚ :
+    (n : ℕ) (q : ℚ) →
+    power-ℚ n (rational-abs-ℚ q) ＝ rational-abs-ℚ (power-ℚ n q)
+  power-rational-abs-ℚ 0 q = inv (ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ one-ℚ⁰⁺))
+  power-rational-abs-ℚ 1 q = refl
+  power-rational-abs-ℚ (succ-ℕ n@(succ-ℕ _)) q =
+    equational-reasoning
+      power-ℚ n (rational-abs-ℚ q) *ℚ rational-abs-ℚ q
+      ＝ rational-abs-ℚ (power-ℚ n q) *ℚ rational-abs-ℚ q
+        by ap-mul-ℚ (power-rational-abs-ℚ n q) refl
+      ＝ rational-abs-ℚ (power-ℚ n q *ℚ q)
+        by inv (rational-abs-mul-ℚ _ _)
+```
+
 ### If `|p| ≤ |q|`, `|pⁿ| ≤ |qⁿ|`
 
 ```agda
@@ -187,29 +277,135 @@ abstract
   preserves-leq-abs-power-ℚ :
     (n : ℕ) (p q : ℚ) → leq-ℚ⁰⁺ (abs-ℚ p) (abs-ℚ q) →
     leq-ℚ⁰⁺ (abs-ℚ (power-ℚ n p)) (abs-ℚ (power-ℚ n q))
-  preserves-leq-abs-power-ℚ 0 p q _ = refl-leq-ℚ _
-  preserves-leq-abs-power-ℚ (succ-ℕ n) p q |p|≤|q| =
+  preserves-leq-abs-power-ℚ n p q |p|≤|q| =
     binary-tr
       ( leq-ℚ)
-      ( equational-reasoning
-        rational-abs-ℚ (power-ℚ n p) *ℚ rational-abs-ℚ p
-        ＝ rational-abs-ℚ (power-ℚ n p *ℚ p)
-          by inv (rational-abs-mul-ℚ (power-ℚ n p) p)
-        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) p)
-          by ap rational-abs-ℚ (inv (power-succ-ℚ n p)))
-      ( equational-reasoning
-        rational-abs-ℚ (power-ℚ n q) *ℚ rational-abs-ℚ q
-        ＝ rational-abs-ℚ (power-ℚ n q *ℚ q)
-          by inv (rational-abs-mul-ℚ (power-ℚ n q) q)
-        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) q)
-          by ap rational-abs-ℚ (inv (power-succ-ℚ n q)))
-      ( preserves-leq-mul-ℚ⁰⁺
-        ( abs-ℚ (power-ℚ n p))
-        ( abs-ℚ (power-ℚ n q))
-        ( abs-ℚ p)
-        ( abs-ℚ q)
-        ( preserves-leq-abs-power-ℚ n p q |p|≤|q|)
-        ( |p|≤|q|))
+      ( inv (power-rational-ℚ⁰⁺ n (abs-ℚ p)) ∙ power-rational-abs-ℚ n p)
+      ( inv (power-rational-ℚ⁰⁺ n (abs-ℚ q)) ∙ power-rational-abs-ℚ n q)
+      ( preserves-leq-power-ℚ⁰⁺ n (abs-ℚ p) (abs-ℚ q) |p|≤|q|)
+```
+
+### Odd powers of rational numbers preserve inequality
+
+```agda
+abstract
+  preserves-leq-odd-power-ℚ :
+    (n : ℕ) (p q : ℚ) → is-odd-ℕ n → leq-ℚ p q →
+    leq-ℚ (power-ℚ n p) (power-ℚ n q)
+  preserves-leq-odd-power-ℚ n p q odd-n p≤q =
+    rec-coproduct
+      ( λ is-neg-p →
+        rec-coproduct
+          ( λ is-neg-q →
+            let
+              p⁻ = (p , is-neg-p)
+              q⁻ = (q , is-neg-q)
+            in
+              binary-tr
+                ( leq-ℚ)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ p))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n p))
+                    by ap neg-ℚ (odd-power-neg-ℚ n p odd-n)
+                  ＝ power-ℚ n p
+                    by neg-neg-ℚ _)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ q))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n q))
+                    by ap neg-ℚ (odd-power-neg-ℚ n q odd-n)
+                  ＝ power-ℚ n q
+                    by neg-neg-ℚ _)
+                ( neg-leq-ℚ
+                  ( preserves-leq-power-ℚ⁺
+                    ( n)
+                    ( neg-ℚ⁻ q⁻)
+                    ( neg-ℚ⁻ p⁻)
+                    ( neg-leq-ℚ p≤q))))
+          ( λ is-nonneg-q →
+            inv-tr
+              ( leq-ℚ (power-ℚ n p))
+              ( power-rational-ℚ⁰⁺ n (q , is-nonneg-q))
+              ( leq-negative-nonnegative-ℚ
+                ( power-ℚ n p , is-negative-odd-power-ℚ⁻ n (p , is-neg-p) odd-n)
+                ( power-ℚ⁰⁺ n (q , is-nonneg-q))))
+          ( decide-is-negative-is-nonnegative-ℚ q))
+      ( λ is-nonneg-p →
+        let
+          p⁰⁺ = (p , is-nonneg-p)
+          q⁰⁺ = (q , is-nonnegative-leq-ℚ⁰⁺ p⁰⁺ q p≤q)
+        in
+          binary-tr
+            ( leq-ℚ)
+            ( inv (power-rational-ℚ⁰⁺ n p⁰⁺))
+            ( inv (power-rational-ℚ⁰⁺ n q⁰⁺))
+            ( preserves-leq-power-ℚ⁰⁺ n p⁰⁺ q⁰⁺ p≤q))
+      ( decide-is-negative-is-nonnegative-ℚ p)
+```
+
+### Odd powers of rational numbers preserve strict inequality
+
+```agda
+abstract
+  preserves-le-odd-power-ℚ :
+    (n : ℕ) (p q : ℚ) → is-odd-ℕ n → le-ℚ p q →
+    le-ℚ (power-ℚ n p) (power-ℚ n q)
+  preserves-le-odd-power-ℚ n p q odd-n p<q =
+    rec-coproduct
+      ( λ is-neg-p →
+        rec-coproduct
+          ( λ is-neg-q →
+            let
+              p⁻ = (p , is-neg-p)
+              q⁻ = (q , is-neg-q)
+            in
+              binary-tr
+                ( le-ℚ)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ p))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n p))
+                    by ap neg-ℚ (odd-power-neg-ℚ n p odd-n)
+                  ＝ power-ℚ n p
+                    by neg-neg-ℚ _)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ q))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n q))
+                    by ap neg-ℚ (odd-power-neg-ℚ n q odd-n)
+                  ＝ power-ℚ n q
+                    by neg-neg-ℚ _)
+                ( neg-le-ℚ
+                  ( preserves-le-power-ℚ⁺
+                    ( n)
+                    ( is-nonzero-is-odd-ℕ odd-n)
+                    ( neg-ℚ⁻ q⁻)
+                    ( neg-ℚ⁻ p⁻)
+                    ( neg-le-ℚ p<q))))
+          ( λ is-nonneg-q →
+            inv-tr
+              ( le-ℚ (power-ℚ n p))
+              ( power-rational-ℚ⁰⁺ n (q , is-nonneg-q))
+              ( le-negative-nonnegative-ℚ
+                ( power-ℚ n p , is-negative-odd-power-ℚ⁻ n (p , is-neg-p) odd-n)
+                ( power-ℚ⁰⁺ n (q , is-nonneg-q))))
+          ( decide-is-negative-is-nonnegative-ℚ q))
+      ( λ is-nonneg-p →
+        let
+          p⁰⁺ = (p , is-nonneg-p)
+          q⁰⁺ = (q , is-nonnegative-le-ℚ⁰⁺ p⁰⁺ q p<q)
+        in
+          binary-tr
+            ( le-ℚ)
+            ( inv (power-rational-ℚ⁰⁺ n p⁰⁺))
+            ( inv (power-rational-ℚ⁰⁺ n q⁰⁺))
+            ( preserves-le-power-ℚ⁰⁺ n (is-nonzero-is-odd-ℕ odd-n) p⁰⁺ q⁰⁺ p<q))
+      ( decide-is-negative-is-nonnegative-ℚ p)
 ```
 
 ## See also
diff --git a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
index 4ef6623999e..c82fd46745e 100644
--- a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
@@ -7,6 +7,7 @@ module elementary-number-theory.strict-inequality-nonnegative-rational-numbers w
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -37,3 +38,24 @@ le-prop-ℚ⁰⁺ p q = le-ℚ-Prop (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q
 le-ℚ⁰⁺ : (p q : ℚ⁰⁺) → UU lzero
 le-ℚ⁰⁺ p q = type-Prop (le-prop-ℚ⁰⁺ p q)
 ```
+
+## Properties
+
+### Strict inequality on the nonnegative rational numbers is transitive
+
+```agda
+abstract
+  transitive-le-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → le-ℚ⁰⁺ q r → le-ℚ⁰⁺ p q → le-ℚ⁰⁺ p r
+  transitive-le-ℚ⁰⁺ (p , _) (q , _) (r , _) = transitive-le-ℚ p q r
+```
+
+### Concatenation laws with inequality
+
+```agda
+abstract
+  concatenate-le-leq-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → le-ℚ⁰⁺ p q → leq-ℚ⁰⁺ q r → le-ℚ⁰⁺ p r
+  concatenate-le-leq-ℚ⁰⁺ (p , _) (q , _) (r , _) = concatenate-le-leq-ℚ p q r
+
+  concatenate-leq-le-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → leq-ℚ⁰⁺ p q → le-ℚ⁰⁺ q r → le-ℚ⁰⁺ p r
+  concatenate-leq-le-ℚ⁰⁺ (p , _) (q , _) (r , _) = concatenate-leq-le-ℚ p q r
+```
diff --git a/src/group-theory/submonoids-commutative-monoids.lagda.md b/src/group-theory/submonoids-commutative-monoids.lagda.md
index 1b93a31af20..6734dc7ba91 100644
--- a/src/group-theory/submonoids-commutative-monoids.lagda.md
+++ b/src/group-theory/submonoids-commutative-monoids.lagda.md
@@ -27,7 +27,10 @@ open import group-theory.subsets-commutative-monoids
 
 ## Idea
 
-A submonoid of a commutative monoid `M` is a subset of `M` that contains the
+A
+{{#concept "submonoid" Disambiguation="of a commutative monoid" Agda=Commutative-Submonoid}}
+of a [commutative monoid](group-theory.commutative-monoids.md) `M` is a
+[subset](group-theory.subsets-commutative-monoids.md) of `M` that contains the
 unit of `M` and is closed under multiplication.
 
 ## Definitions
diff --git a/src/group-theory/submonoids.lagda.md b/src/group-theory/submonoids.lagda.md
index 4245b22e102..28dacdedee4 100644
--- a/src/group-theory/submonoids.lagda.md
+++ b/src/group-theory/submonoids.lagda.md
@@ -27,8 +27,10 @@ open import group-theory.subsets-monoids
 
 ## Idea
 
-A submonoid of a monoid `M` is a subset of `M` that contains the unit of `M` and
-is closed under multiplication.
+A {{#concept "submonoid" WDID=Q121499459 WD="submonoid" Agda=Submonoid}} of a
+[monoid](group-theory.monoids.md) `M` is a
+[subset](group-theory.subsets-monoids.md) of `M` that contains the unit of `M`
+and is closed under multiplication.
 
 ## Definitions
 

From 92b42a5ad47796ab1923535a41077a5cdc97a22c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 12:37:31 -0700
Subject: [PATCH 020/134] Correct superscripts

---
 src/elementary-number-theory/powers-rational-numbers.lagda.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index 1374740c371..392ea1998cf 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -93,7 +93,7 @@ abstract
   power-one-ℚ = power-unit-Monoid monoid-mul-ℚ
 ```
 
-### `qⁿ¹ = qⁿq`
+### `qⁿ⁺¹ = qⁿq`
 
 ```agda
 abstract
@@ -101,7 +101,7 @@ abstract
   power-succ-ℚ = power-succ-Monoid monoid-mul-ℚ
 ```
 
-### `qⁿ = qqⁿ`
+### `qⁿ⁺¹ = qqⁿ`
 
 ```agda
 abstract

From 978b5045b61f96dfd1b8dc09145956ea5da15ce8 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 15:47:48 -0700
Subject: [PATCH 021/134] Progress

---
 ...of-elements-commutative-semirings.lagda.md |  15 ++
 ...icative-group-of-rational-numbers.lagda.md |  21 ++
 .../arithmetic-sequences-semigroups.lagda.md  |  27 +++
 src/lists/sequences.lagda.md                  |   9 +
 src/ring-theory.lagda.md                      |   1 +
 .../geometric-sequences-rings.lagda.md        | 216 ++++++++++++++++++
 .../geometric-sequences-semirings.lagda.md    |  36 ++-
 7 files changed, 323 insertions(+), 2 deletions(-)
 create mode 100644 src/ring-theory/geometric-sequences-rings.lagda.md

diff --git a/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md b/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md
index 0bda772d1a1..a5727d03a8e 100644
--- a/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md
+++ b/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md
@@ -51,6 +51,21 @@ module _
     power-succ-Semiring (semiring-Commutative-Semiring A)
 ```
 
+### `xⁿ⁺¹ = xxⁿ`
+
+```agda
+module _
+  {l : Level} (A : Commutative-Semiring l)
+  where
+
+  power-succ-Commutative-Semiring' :
+    (n : ℕ) (x : type-Commutative-Semiring A) →
+    power-Commutative-Semiring A (succ-ℕ n) x ＝
+    mul-Commutative-Semiring A x (power-Commutative-Semiring A n x)
+  power-succ-Commutative-Semiring' =
+    power-succ-Semiring' (semiring-Commutative-Semiring A)
+```
+
 ### Powers by sums of natural numbers are products of powers
 
 ```agda
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index 4db36a011c9..a35af131e80 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -10,8 +10,10 @@ module elementary-number-theory.multiplicative-group-of-rational-numbers where
 
 ```agda
 open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
+open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
@@ -25,6 +27,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.function-types
+open import foundation.negated-equality
 open import foundation.identity-types
 open import foundation.subtypes
 open import foundation.transport-along-identifications
@@ -86,6 +89,9 @@ _*ℚˣ_ = mul-ℚˣ
 
 inv-ℚˣ : ℚˣ → ℚˣ
 inv-ℚˣ = inv-group-of-units-Ring ring-ℚ
+
+rational-inv-ℚˣ : ℚˣ → ℚ
+rational-inv-ℚˣ q = rational-ℚˣ (inv-ℚˣ q)
 ```
 
 ### Inverse laws in the multiplicative group of rational numbers
@@ -169,3 +175,18 @@ invertible-ℚ⁺ = invertible-nonzero-ℚ ∘ nonzero-ℚ⁺
 invertible-ℚ⁻ : ℚ⁻ → ℚˣ
 invertible-ℚ⁻ = invertible-nonzero-ℚ ∘ nonzero-ℚ⁻
 ```
+
+### If `a ≠ b`, `b - a` is invertible
+
+```agda
+abstract
+  is-invertible-diff-neq-ℚ :
+    (a b : ℚ) → a ≠ b → is-invertible-element-Ring ring-ℚ (b -ℚ a)
+  is-invertible-diff-neq-ℚ a b a≠b =
+    is-invertible-element-ring-is-nonzero-ℚ
+      ( b -ℚ a)
+      ( λ b-a=0 → a≠b (inv (eq-is-unit-right-div-Group group-add-ℚ b-a=0)))
+
+invertible-diff-neq-ℚ : (a b : ℚ) → a ≠ b → ℚˣ
+invertible-diff-neq-ℚ a b a≠b = (b -ℚ a , is-invertible-diff-neq-ℚ a b a≠b)
+```
diff --git a/src/group-theory/arithmetic-sequences-semigroups.lagda.md b/src/group-theory/arithmetic-sequences-semigroups.lagda.md
index f58aa8571c4..0238c5bd311 100644
--- a/src/group-theory/arithmetic-sequences-semigroups.lagda.md
+++ b/src/group-theory/arithmetic-sequences-semigroups.lagda.md
@@ -13,6 +13,7 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
+open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
@@ -188,6 +189,32 @@ module _
       ( refl)
 ```
 
+### The tail of a standard arithmetic sequence is another standard arithmetic sequence
+
+```agda
+module _
+  {l : Level} (G : Semigroup l) (a d : type-Semigroup G)
+  where
+
+  abstract
+    htpy-tail-standard-arithmetric-sequence-Semigroup :
+      tail-sequence
+        ( seq-standard-arithmetic-sequence-Semigroup G a d) ~
+      seq-standard-arithmetic-sequence-Semigroup G (mul-Semigroup G a d) d
+    htpy-tail-standard-arithmetric-sequence-Semigroup 0 = refl
+    htpy-tail-standard-arithmetric-sequence-Semigroup (succ-ℕ n) =
+      ap-mul-Semigroup G
+        ( htpy-tail-standard-arithmetric-sequence-Semigroup n)
+        ( refl)
+
+    eq-tail-standard-arithmetric-sequence-Semigroup :
+      tail-sequence
+        ( seq-standard-arithmetic-sequence-Semigroup G a d) ＝
+      seq-standard-arithmetic-sequence-Semigroup G (mul-Semigroup G a d) d
+    eq-tail-standard-arithmetric-sequence-Semigroup =
+      eq-htpy htpy-tail-standard-arithmetric-sequence-Semigroup
+```
+
 ## External links
 
 - [Arithmetic progressions](https://en.wikipedia.org/wiki/Arithmetic_progression)
diff --git a/src/lists/sequences.lagda.md b/src/lists/sequences.lagda.md
index 2549a666e11..a09c39a608b 100644
--- a/src/lists/sequences.lagda.md
+++ b/src/lists/sequences.lagda.md
@@ -7,6 +7,8 @@ module lists.sequences where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.natural-numbers
+
 open import foundation.universe-levels
 
 open import foundation-core.function-types
@@ -36,6 +38,13 @@ sequence : {l : Level} → UU l → UU l
 sequence A = dependent-sequence (λ _ → A)
 ```
 
+### The tail of a sequence
+
+```agda
+tail-sequence : {l : Level} {A : UU l} → sequence A → sequence A
+tail-sequence u n = u (succ-ℕ n)
+```
+
 ### Functorial action on maps of sequences
 
 ```agda
diff --git a/src/ring-theory.lagda.md b/src/ring-theory.lagda.md
index a16c402dafa..7322fd40478 100644
--- a/src/ring-theory.lagda.md
+++ b/src/ring-theory.lagda.md
@@ -29,6 +29,7 @@ open import ring-theory.full-ideals-rings public
 open import ring-theory.function-rings public
 open import ring-theory.function-semirings public
 open import ring-theory.generating-elements-rings public
+open import ring-theory.geometric-sequences-rings public
 open import ring-theory.geometric-sequences-semirings public
 open import ring-theory.groups-of-units-rings public
 open import ring-theory.homomorphisms-cyclic-rings public
diff --git a/src/ring-theory/geometric-sequences-rings.lagda.md b/src/ring-theory/geometric-sequences-rings.lagda.md
new file mode 100644
index 00000000000..90497c4db21
--- /dev/null
+++ b/src/ring-theory/geometric-sequences-rings.lagda.md
@@ -0,0 +1,216 @@
+# Geometric sequences in rings
+
+```agda
+module ring-theory.geometric-sequences-rings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.natural-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.homotopies
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.universe-levels
+
+open import group-theory.arithmetic-sequences-semigroups
+
+open import lists.sequences
+
+open import ring-theory.geometric-sequences-semirings
+open import ring-theory.powers-of-elements-rings
+open import ring-theory.rings
+```
+
+</details>
+
+## Ideas
+
+A
+{{#concept "geometric sequence" Disambiguation="in a ring" Agda=geometric-sequence-Ring}}
+in a [ring](ring-theory.semirings.md) is an
+[arithmetic sequence](ring-theory.geometric-sequence-semirings.md) in the
+underlying [semiring](ring-theory.semirings.md).
+
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
+semiring.
+
+## Definitions
+
+### Geometric sequences in rings
+
+```agda
+module _
+  {l : Level} (R : Ring l)
+  where
+
+  geometric-sequence-Ring : UU l
+  geometric-sequence-Ring =
+    geometric-sequence-Semiring (semiring-Ring R)
+
+module _
+  {l : Level} (R : Ring l)
+  (u : geometric-sequence-Ring R)
+  where
+
+  seq-geometric-sequence-Ring : ℕ → type-Ring R
+  seq-geometric-sequence-Ring =
+    seq-geometric-sequence-Semiring (semiring-Ring R) u
+
+  common-ratio-geometric-sequence-Ring : type-Ring R
+  common-ratio-geometric-sequence-Ring =
+    common-ratio-geometric-sequence-Semiring (semiring-Ring R) u
+
+  is-common-ratio-geometric-sequence-Ring :
+    ( n : ℕ) →
+    ( seq-geometric-sequence-Ring (succ-ℕ n)) ＝
+    ( mul-Ring
+      ( R)
+      ( seq-geometric-sequence-Ring n)
+      ( common-ratio-geometric-sequence-Ring))
+  is-common-ratio-geometric-sequence-Ring =
+    is-common-difference-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( u)
+
+  initial-term-geometric-sequence-Ring : type-Ring R
+  initial-term-geometric-sequence-Ring =
+    initial-term-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( u)
+```
+
+### The standard geometric sequences in a semiring
+
+The standard geometric sequence with initial term `a` and common factor `r` is
+the sequence `u` defined by:
+
+- `u₀ = a`
+- `uₙ₊₁ = uₙ * r`
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a r : type-Ring R)
+  where
+
+  standard-geometric-sequence-Ring : geometric-sequence-Ring R
+  standard-geometric-sequence-Ring =
+    standard-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( a)
+      ( r)
+
+  seq-standard-geometric-sequence-Ring : ℕ → type-Ring R
+  seq-standard-geometric-sequence-Ring =
+    seq-geometric-sequence-Ring R
+      standard-geometric-sequence-Ring
+```
+
+### The geometric sequences `n ↦ a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a r : type-Ring R)
+  where
+
+  mul-pow-nat-Ring : sequence (type-Ring R)
+  mul-pow-nat-Ring = mul-pow-nat-Semiring (semiring-Ring R) a r
+
+  geometric-mul-pow-nat-Ring : geometric-sequence-Ring R
+  geometric-mul-pow-nat-Ring =
+    geometric-mul-pow-nat-Semiring (semiring-Ring R) a r
+
+  initial-term-mul-pow-nat-Ring : type-Ring R
+  initial-term-mul-pow-nat-Ring =
+    initial-term-mul-pow-nat-Semiring (semiring-Ring R) a r
+
+  eq-initial-term-mul-pow-nat-Ring :
+    initial-term-mul-pow-nat-Ring ＝ a
+  eq-initial-term-mul-pow-nat-Ring =
+    eq-initial-term-mul-pow-nat-Semiring (semiring-Ring R) a r
+```
+
+## Properties
+
+### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Ring l)
+  (u : geometric-sequence-Ring R)
+  where
+
+  htpy-seq-standard-geometric-sequence-Ring :
+    ( seq-geometric-sequence-Ring R
+      ( standard-geometric-sequence-Ring R
+        ( initial-term-geometric-sequence-Ring R u)
+        ( common-ratio-geometric-sequence-Ring R u))) ~
+    ( seq-geometric-sequence-Ring R u)
+  htpy-seq-standard-geometric-sequence-Ring =
+    htpy-seq-standard-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( u)
+```
+
+### The nth term of an geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a r : type-Ring R)
+  where
+
+  htpy-mul-pow-standard-geometric-sequence-Ring :
+    mul-pow-nat-Ring R a r ~ seq-standard-geometric-sequence-Ring R a r
+  htpy-mul-pow-standard-geometric-sequence-Ring =
+    htpy-mul-pow-standard-geometric-sequence-Semiring (semiring-Ring R) a r
+```
+
+```agda
+module _
+  {l : Level} (R : Ring l) (u : geometric-sequence-Ring R)
+  where
+
+  htpy-mul-pow-geometric-sequence-Ring :
+    mul-pow-nat-Ring
+      ( R)
+      ( initial-term-geometric-sequence-Ring R u)
+      ( common-ratio-geometric-sequence-Ring R u) ~
+    seq-geometric-sequence-Ring R u
+  htpy-mul-pow-geometric-sequence-Ring =
+    htpy-mul-pow-geometric-sequence-Semiring (semiring-Ring R) u
+```
+
+### Constant sequences are geometric with common ratio one
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a : type-Ring R)
+  where
+
+  one-is-common-ratio-const-sequence-Ring :
+    is-common-difference-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( λ _ → a)
+      ( one-Ring R)
+  one-is-common-ratio-const-sequence-Ring n =
+    inv (right-unit-law-mul-Ring R a)
+
+  geometric-const-sequence-Ring : geometric-sequence-Ring R
+  geometric-const-sequence-Ring =
+    geometric-const-sequence-Semiring (semiring-Ring R) a
+```
+
+## See also
+
+- [Geometric sequences in a semiring](ring-theory.geometric-sequences-semirings.md)
+
+## External links
+
+- [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)
+  at Wikipedia
diff --git a/src/ring-theory/geometric-sequences-semirings.lagda.md b/src/ring-theory/geometric-sequences-semirings.lagda.md
index 2b0107a2dbd..3d7dff3e3ac 100644
--- a/src/ring-theory/geometric-sequences-semirings.lagda.md
+++ b/src/ring-theory/geometric-sequences-semirings.lagda.md
@@ -13,6 +13,7 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
+open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
@@ -54,6 +55,11 @@ module _
     arithmetic-sequence-Semigroup
       ( multiplicative-semigroup-Semiring R)
 
+  is-geometric-sequence-Semiring : sequence (type-Semiring R) → UU l
+  is-geometric-sequence-Semiring =
+    is-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Semiring R)
+
 module _
   {l : Level} (R : Semiring l)
   (u : geometric-sequence-Semiring R)
@@ -66,8 +72,8 @@ module _
       ( u)
 
   is-geometric-seq-geometric-sequence-Semiring :
-    is-arithmetic-sequence-Semigroup
-      ( multiplicative-semigroup-Semiring R)
+    is-geometric-sequence-Semiring
+      ( R)
       ( seq-geometric-sequence-Semiring)
   is-geometric-seq-geometric-sequence-Semiring =
     is-arithmetic-seq-arithmetic-sequence-Semigroup
@@ -267,6 +273,32 @@ module _
     ( one-Semiring R , one-is-common-ratio-const-sequence-Semiring)
 ```
 
+### The tail of a standard geometric sequence is another standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Semiring l) (a r : type-Semiring R)
+  where
+
+  abstract
+    htpy-tail-standard-geometric-sequence-Semiring :
+      tail-sequence
+        ( seq-standard-geometric-sequence-Semiring R a r) ~
+      seq-standard-geometric-sequence-Semiring R (mul-Semiring R a r) r
+    htpy-tail-standard-geometric-sequence-Semiring =
+      htpy-tail-standard-arithmetric-sequence-Semigroup
+        ( multiplicative-semigroup-Semiring R)
+        ( a)
+        ( r)
+
+    eq-tail-standard-geometric-sequence-Semiring :
+      tail-sequence
+        ( seq-standard-geometric-sequence-Semiring R a r) ＝
+      seq-standard-geometric-sequence-Semiring R (mul-Semiring R a r) r
+    eq-tail-standard-geometric-sequence-Semiring =
+      eq-htpy htpy-tail-standard-geometric-sequence-Semiring
+```
+
 ## External links
 
 - [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)

From e332e1993006ccbeed9f7b35f0e8499bc429155c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 16:26:05 -0700
Subject: [PATCH 022/134] Sums of geometric series for rational numbers

---
 src/commutative-algebra.lagda.md              |   2 +
 ...etric-sequences-commutative-rings.lagda.md | 311 +++++++++++++
 ...c-sequences-commutative-semirings.lagda.md | 439 ++++++++++++++++++
 src/elementary-number-theory.lagda.md         |   1 +
 ...metric-sequences-rational-numbers.lagda.md | 203 ++++++++
 ...icative-group-of-rational-numbers.lagda.md |  23 +-
 .../geometric-sequences-rings.lagda.md        |   2 +-
 .../geometric-sequences-semirings.lagda.md    |   2 +-
 8 files changed, 978 insertions(+), 5 deletions(-)
 create mode 100644 src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
 create mode 100644 src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
 create mode 100644 src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md

diff --git a/src/commutative-algebra.lagda.md b/src/commutative-algebra.lagda.md
index 0ca53e325c4..801d517bd88 100644
--- a/src/commutative-algebra.lagda.md
+++ b/src/commutative-algebra.lagda.md
@@ -22,6 +22,8 @@ open import commutative-algebra.formal-power-series-commutative-semirings public
 open import commutative-algebra.full-ideals-commutative-rings public
 open import commutative-algebra.function-commutative-rings public
 open import commutative-algebra.function-commutative-semirings public
+open import commutative-algebra.geometric-sequences-commutative-rings public
+open import commutative-algebra.geometric-sequences-commutative-semirings public
 open import commutative-algebra.groups-of-units-commutative-rings public
 open import commutative-algebra.homomorphisms-commutative-rings public
 open import commutative-algebra.homomorphisms-commutative-semirings public
diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
new file mode 100644
index 00000000000..ee6b868c9e9
--- /dev/null
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -0,0 +1,311 @@
+# Geometric sequences in commutative rings
+
+```agda
+module commutative-algebra.geometric-sequences-commutative-rings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.commutative-rings
+open import commutative-algebra.commutative-semirings
+open import commutative-algebra.geometric-sequences-commutative-semirings
+open import commutative-algebra.powers-of-elements-commutative-rings
+
+open import elementary-number-theory.natural-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.function-types
+open import foundation.homotopies
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.universe-levels
+
+open import group-theory.arithmetic-sequences-semigroups
+
+open import lists.finite-sequences
+open import lists.sequences
+```
+
+</details>
+
+## Ideas
+
+A
+{{#concept "geometric sequence" Disambiguation="in a commutative semiring" Agda=geometric-sequence-Commutative-Ring}}
+in a [semiring](ring-theory.semirings.md) is an
+[geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
+semiring multiplicative [semigroup](group-theory.semigroups.md).
+
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
+semiring.
+
+## Definitions
+
+### Geometric sequences in semirings
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l)
+  where
+
+  geometric-sequence-Commutative-Ring : UU l
+  geometric-sequence-Commutative-Ring =
+    geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+
+  is-geometric-sequence-Commutative-Ring :
+    sequence (type-Commutative-Ring R) → UU l
+  is-geometric-sequence-Commutative-Ring =
+    is-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+
+module _
+  {l : Level} (R : Commutative-Ring l)
+  (u : geometric-sequence-Commutative-Ring R)
+  where
+
+  seq-geometric-sequence-Commutative-Ring : ℕ → type-Commutative-Ring R
+  seq-geometric-sequence-Commutative-Ring =
+    seq-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  is-geometric-seq-geometric-sequence-Commutative-Ring :
+    is-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( seq-geometric-sequence-Commutative-Ring)
+  is-geometric-seq-geometric-sequence-Commutative-Ring =
+    is-geometric-seq-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  common-ratio-geometric-sequence-Commutative-Ring :
+    type-Commutative-Ring R
+  common-ratio-geometric-sequence-Commutative-Ring =
+    common-ratio-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  is-common-ratio-geometric-sequence-Commutative-Ring :
+    ( n : ℕ) →
+    ( seq-geometric-sequence-Commutative-Ring (succ-ℕ n)) ＝
+    ( mul-Commutative-Ring
+      ( R)
+      ( seq-geometric-sequence-Commutative-Ring n)
+      ( common-ratio-geometric-sequence-Commutative-Ring))
+  is-common-ratio-geometric-sequence-Commutative-Ring =
+    is-common-ratio-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  initial-term-geometric-sequence-Commutative-Ring :
+    type-Commutative-Ring R
+  initial-term-geometric-sequence-Commutative-Ring =
+    initial-term-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+```
+
+### The standard geometric sequences in a semiring
+
+The standard geometric sequence with initial term `a` and common factor `r` is
+the sequence `u` defined by:
+
+- `u₀ = a`
+- `uₙ₊₁ = uₙ * r`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  standard-geometric-sequence-Commutative-Ring :
+    geometric-sequence-Commutative-Ring R
+  standard-geometric-sequence-Commutative-Ring =
+    standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+
+  seq-standard-geometric-sequence-Commutative-Ring :
+    ℕ → type-Commutative-Ring R
+  seq-standard-geometric-sequence-Commutative-Ring =
+    seq-geometric-sequence-Commutative-Ring R
+      standard-geometric-sequence-Commutative-Ring
+
+  is-geometric-standard-geometric-sequence-Commutative-Ring :
+    is-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( seq-standard-geometric-sequence-Commutative-Ring)
+  is-geometric-standard-geometric-sequence-Commutative-Ring =
+    is-geometric-seq-geometric-sequence-Commutative-Ring R
+      standard-geometric-sequence-Commutative-Ring
+```
+
+### The geometric sequences `n ↦ a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  mul-pow-nat-Commutative-Ring : ℕ → type-Commutative-Ring R
+  mul-pow-nat-Commutative-Ring n =
+    mul-Commutative-Ring R a (power-Commutative-Ring R n r)
+
+  geometric-mul-pow-nat-Commutative-Ring : geometric-sequence-Commutative-Ring R
+  geometric-mul-pow-nat-Commutative-Ring =
+    geometric-mul-pow-nat-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+
+  initial-term-mul-pow-nat-Commutative-Ring : type-Commutative-Ring R
+  initial-term-mul-pow-nat-Commutative-Ring =
+    initial-term-geometric-sequence-Commutative-Ring
+      ( R)
+      ( geometric-mul-pow-nat-Commutative-Ring)
+
+  eq-initial-term-mul-pow-nat-Commutative-Ring :
+    initial-term-mul-pow-nat-Commutative-Ring ＝ a
+  eq-initial-term-mul-pow-nat-Commutative-Ring =
+    right-unit-law-mul-Commutative-Ring R a
+```
+
+## Properties
+
+### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l)
+  (u : geometric-sequence-Commutative-Ring R)
+  where
+
+  htpy-seq-standard-geometric-sequence-Commutative-Ring :
+    ( seq-geometric-sequence-Commutative-Ring R
+      ( standard-geometric-sequence-Commutative-Ring R
+        ( initial-term-geometric-sequence-Commutative-Ring R u)
+        ( common-ratio-geometric-sequence-Commutative-Ring R u))) ~
+    ( seq-geometric-sequence-Commutative-Ring R u)
+  htpy-seq-standard-geometric-sequence-Commutative-Ring =
+    htpy-seq-standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+```
+
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Ring :
+    mul-pow-nat-Commutative-Ring R a r ~
+    seq-standard-geometric-sequence-Commutative-Ring R a r
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Ring =
+    htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+
+  initial-term-standard-geometric-sequence-Commutative-Ring :
+    seq-standard-geometric-sequence-Commutative-Ring R a r 0 ＝ a
+  initial-term-standard-geometric-sequence-Commutative-Ring =
+    ( inv (htpy-mul-pow-standard-geometric-sequence-Commutative-Ring 0)) ∙
+    ( eq-initial-term-mul-pow-nat-Commutative-Ring R a r)
+```
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l)
+  (u : geometric-sequence-Commutative-Ring R)
+  where
+
+  htpy-mul-pow-geometric-sequence-Commutative-Ring :
+    mul-pow-nat-Commutative-Ring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Ring R u)
+      ( common-ratio-geometric-sequence-Commutative-Ring R u) ~
+    seq-geometric-sequence-Commutative-Ring R u
+  htpy-mul-pow-geometric-sequence-Commutative-Ring n =
+    ( htpy-mul-pow-standard-geometric-sequence-Commutative-Ring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Ring R u)
+      ( common-ratio-geometric-sequence-Commutative-Ring R u)
+      ( n)) ∙
+    ( htpy-seq-standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+      ( n))
+```
+
+### Constant sequences are geometric with common ratio one
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a : type-Commutative-Ring R)
+  where
+
+  geometric-const-sequence-Commutative-Ring :
+    geometric-sequence-Commutative-Ring R
+  geometric-const-sequence-Commutative-Ring =
+    geometric-const-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+```
+
+### Finite geometric sequences
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  standard-geometric-fin-sequence-Commutative-Ring :
+    (n : ℕ) → fin-sequence (type-Commutative-Ring R) n
+  standard-geometric-fin-sequence-Commutative-Ring =
+    fin-sequence-sequence
+      ( seq-standard-geometric-sequence-Commutative-Ring R a r)
+
+  sum-standard-geometric-fin-sequence-Commutative-Ring :
+    (n : ℕ) → type-Commutative-Ring R
+  sum-standard-geometric-fin-sequence-Commutative-Ring =
+    sum-standard-geometric-fin-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+```
+
+### The sum of a finite geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Ring :
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-Commutative-Ring R
+        ( a)
+        ( r)
+        ( succ-ℕ n) ＝
+      add-Commutative-Ring R
+        ( a)
+        ( mul-Commutative-Ring R
+          ( r)
+          ( sum-standard-geometric-fin-sequence-Commutative-Ring R a r n))
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Ring =
+      compute-sum-standard-geometric-fin-sequence-succ-Commutative-Semiring
+        ( commutative-semiring-Commutative-Ring R)
+        ( a)
+        ( r)
+```
diff --git a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
new file mode 100644
index 00000000000..821a30d43c3
--- /dev/null
+++ b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
@@ -0,0 +1,439 @@
+# Geometric sequences in commutative semirings
+
+```agda
+module commutative-algebra.geometric-sequences-commutative-semirings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.commutative-semirings
+open import commutative-algebra.powers-of-elements-commutative-semirings
+open import commutative-algebra.sums-of-finite-sequences-of-elements-commutative-semirings
+
+open import elementary-number-theory.natural-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.function-types
+open import foundation.homotopies
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.universe-levels
+
+open import group-theory.arithmetic-sequences-semigroups
+
+open import lists.finite-sequences
+open import lists.sequences
+
+open import ring-theory.geometric-sequences-semirings
+open import ring-theory.powers-of-elements-semirings
+open import ring-theory.semirings
+
+open import univalent-combinatorics.standard-finite-types
+```
+
+</details>
+
+## Ideas
+
+A
+{{#concept "geometric sequence" Disambiguation="in a commutative semiring" Agda=geometric-sequence-Commutative-Semiring}}
+in a [semiring](ring-theory.semirings.md) is an
+[geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
+semiring multiplicative [semigroup](group-theory.semigroups.md).
+
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
+semiring.
+
+## Definitions
+
+### Geometric sequences in semirings
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  where
+
+  geometric-sequence-Commutative-Semiring : UU l
+  geometric-sequence-Commutative-Semiring =
+    geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+
+  is-geometric-sequence-Commutative-Semiring :
+    sequence (type-Commutative-Semiring R) → UU l
+  is-geometric-sequence-Commutative-Semiring =
+    is-geometric-sequence-Semiring (semiring-Commutative-Semiring R)
+
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  (u : geometric-sequence-Commutative-Semiring R)
+  where
+
+  seq-geometric-sequence-Commutative-Semiring : ℕ → type-Commutative-Semiring R
+  seq-geometric-sequence-Commutative-Semiring =
+    seq-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  is-geometric-seq-geometric-sequence-Commutative-Semiring :
+    is-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( seq-geometric-sequence-Commutative-Semiring)
+  is-geometric-seq-geometric-sequence-Commutative-Semiring =
+    is-geometric-seq-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  common-ratio-geometric-sequence-Commutative-Semiring :
+    type-Commutative-Semiring R
+  common-ratio-geometric-sequence-Commutative-Semiring =
+    common-ratio-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  is-common-ratio-geometric-sequence-Commutative-Semiring :
+    ( n : ℕ) →
+    ( seq-geometric-sequence-Commutative-Semiring (succ-ℕ n)) ＝
+    ( mul-Commutative-Semiring
+      ( R)
+      ( seq-geometric-sequence-Commutative-Semiring n)
+      ( common-ratio-geometric-sequence-Commutative-Semiring))
+  is-common-ratio-geometric-sequence-Commutative-Semiring =
+    is-common-ratio-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  initial-term-geometric-sequence-Commutative-Semiring :
+    type-Commutative-Semiring R
+  initial-term-geometric-sequence-Commutative-Semiring =
+    initial-term-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+```
+
+### The standard geometric sequences in a semiring
+
+The standard geometric sequence with initial term `a` and common factor `r` is
+the sequence `u` defined by:
+
+- `u₀ = a`
+- `uₙ₊₁ = uₙ * r`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  standard-geometric-sequence-Commutative-Semiring :
+    geometric-sequence-Commutative-Semiring R
+  standard-geometric-sequence-Commutative-Semiring =
+    standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( a)
+      ( r)
+
+  seq-standard-geometric-sequence-Commutative-Semiring :
+    ℕ → type-Commutative-Semiring R
+  seq-standard-geometric-sequence-Commutative-Semiring =
+    seq-geometric-sequence-Commutative-Semiring R
+      standard-geometric-sequence-Commutative-Semiring
+
+  is-geometric-standard-geometric-sequence-Commutative-Semiring :
+    is-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( seq-standard-geometric-sequence-Commutative-Semiring)
+  is-geometric-standard-geometric-sequence-Commutative-Semiring =
+    is-geometric-seq-geometric-sequence-Commutative-Semiring R
+      standard-geometric-sequence-Commutative-Semiring
+```
+
+### The geometric sequences `n ↦ a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  mul-pow-nat-Commutative-Semiring : ℕ → type-Commutative-Semiring R
+  mul-pow-nat-Commutative-Semiring n =
+    mul-Commutative-Semiring R a (power-Commutative-Semiring R n r)
+
+  geometric-mul-pow-nat-Commutative-Semiring :
+    geometric-sequence-Commutative-Semiring R
+  geometric-mul-pow-nat-Commutative-Semiring =
+    geometric-mul-pow-nat-Semiring (semiring-Commutative-Semiring R) a r
+
+  initial-term-mul-pow-nat-Commutative-Semiring : type-Commutative-Semiring R
+  initial-term-mul-pow-nat-Commutative-Semiring =
+    initial-term-geometric-sequence-Commutative-Semiring
+      ( R)
+      ( geometric-mul-pow-nat-Commutative-Semiring)
+
+  eq-initial-term-mul-pow-nat-Commutative-Semiring :
+    initial-term-mul-pow-nat-Commutative-Semiring ＝ a
+  eq-initial-term-mul-pow-nat-Commutative-Semiring =
+    right-unit-law-mul-Commutative-Semiring R a
+```
+
+## Properties
+
+### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  (u : geometric-sequence-Commutative-Semiring R)
+  where
+
+  htpy-seq-standard-geometric-sequence-Commutative-Semiring :
+    ( seq-geometric-sequence-Commutative-Semiring R
+      ( standard-geometric-sequence-Commutative-Semiring R
+        ( initial-term-geometric-sequence-Commutative-Semiring R u)
+        ( common-ratio-geometric-sequence-Commutative-Semiring R u))) ~
+    ( seq-geometric-sequence-Commutative-Semiring R u)
+  htpy-seq-standard-geometric-sequence-Commutative-Semiring =
+    htpy-seq-standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+```
+
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring :
+    mul-pow-nat-Commutative-Semiring R a r ~
+    seq-standard-geometric-sequence-Commutative-Semiring R a r
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring =
+    htpy-mul-pow-standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( a)
+      ( r)
+
+  initial-term-standard-geometric-sequence-Commutative-Semiring :
+    seq-standard-geometric-sequence-Commutative-Semiring R a r 0 ＝ a
+  initial-term-standard-geometric-sequence-Commutative-Semiring =
+    ( inv (htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring 0)) ∙
+    ( eq-initial-term-mul-pow-nat-Commutative-Semiring R a r)
+```
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  (u : geometric-sequence-Commutative-Semiring R)
+  where
+
+  htpy-mul-pow-geometric-sequence-Commutative-Semiring :
+    mul-pow-nat-Commutative-Semiring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Semiring R u)
+      ( common-ratio-geometric-sequence-Commutative-Semiring R u) ~
+    seq-geometric-sequence-Commutative-Semiring R u
+  htpy-mul-pow-geometric-sequence-Commutative-Semiring n =
+    ( htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Semiring R u)
+      ( common-ratio-geometric-sequence-Commutative-Semiring R u)
+      ( n)) ∙
+    ( htpy-seq-standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+      ( n))
+```
+
+### Constant sequences are geometric with common ratio one
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a : type-Commutative-Semiring R)
+  where
+
+  geometric-const-sequence-Commutative-Semiring :
+    geometric-sequence-Commutative-Semiring R
+  geometric-const-sequence-Commutative-Semiring =
+    geometric-const-sequence-Semiring (semiring-Commutative-Semiring R) a
+```
+
+### Finite geometric sequences
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  standard-geometric-fin-sequence-Commutative-Semiring :
+    (n : ℕ) → fin-sequence (type-Commutative-Semiring R) n
+  standard-geometric-fin-sequence-Commutative-Semiring =
+    fin-sequence-sequence
+      ( seq-standard-geometric-sequence-Commutative-Semiring R a r)
+
+  sum-standard-geometric-fin-sequence-Commutative-Semiring :
+    (n : ℕ) → type-Commutative-Semiring R
+  sum-standard-geometric-fin-sequence-Commutative-Semiring n =
+    sum-fin-sequence-type-Commutative-Semiring R n
+      ( standard-geometric-fin-sequence-Commutative-Semiring n)
+```
+
+### The sum of a finite geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Semiring :
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-Commutative-Semiring R
+        ( a)
+        ( r)
+        ( succ-ℕ n) ＝
+      add-Commutative-Semiring R
+        ( a)
+        ( mul-Commutative-Semiring R
+          ( r)
+          ( sum-standard-geometric-fin-sequence-Commutative-Semiring R a r n))
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Semiring n =
+      equational-reasoning
+        sum-standard-geometric-fin-sequence-Commutative-Semiring R
+          ( a)
+          ( r)
+          ( succ-ℕ n)
+        ＝
+          add-Commutative-Semiring R
+            ( term-Fin a r (succ-ℕ n) (zero-Fin n))
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i → term-Fin a r (succ-ℕ n) (inr-Fin n i)))
+          by
+            snoc-sum-fin-sequence-type-Commutative-Semiring R n
+              ( standard-geometric-fin-sequence-Commutative-Semiring R
+                ( a)
+                ( r)
+                ( succ-ℕ n))
+              ( refl)
+        ＝
+          add-Commutative-Semiring R
+            ( term a r 0)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i → term a r (succ-ℕ (nat-Fin n i))))
+          by
+            ap-add-Commutative-Semiring R
+              ( ap
+                ( seq-standard-geometric-sequence-Commutative-Semiring R a r)
+                ( is-zero-nat-zero-Fin {n}))
+              ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                ( λ i → ap (term a r) (nat-inr-Fin n i)))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i →
+                mul-Commutative-Semiring R
+                  ( a)
+                  ( power-Commutative-Semiring R (succ-ℕ (nat-Fin n i)) r)))
+            by
+              ap-add-Commutative-Semiring R
+                ( initial-term-standard-geometric-sequence-Commutative-Semiring
+                  ( R)
+                  ( a)
+                  ( r))
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( λ i →
+                    inv
+                      ( htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+                        ( R)
+                        ( a)
+                        ( r)
+                        ( succ-ℕ (nat-Fin n i)))))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i →
+                mul-Commutative-Semiring R
+                  ( a)
+                  ( mul-Commutative-Semiring R
+                    ( r)
+                    ( power-Commutative-Semiring R (nat-Fin n i) r))))
+            by
+              ap-add-Commutative-Semiring R
+                ( refl)
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( λ i →
+                    ap-mul-Commutative-Semiring R
+                      ( refl)
+                      ( power-succ-Commutative-Semiring' R (nat-Fin n i) r)))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i →
+                mul-Commutative-Semiring R
+                  ( r)
+                  ( mul-Commutative-Semiring R
+                    ( a)
+                    ( power-Commutative-Semiring R (nat-Fin n i) r))))
+            by
+              ap-add-Commutative-Semiring R
+                ( refl)
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( λ i → left-swap-mul-Commutative-Semiring R _ _ _))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( mul-Commutative-Semiring R
+              ( r)
+              ( sum-fin-sequence-type-Commutative-Semiring R n
+                ( λ i →
+                  mul-Commutative-Semiring R
+                    ( a)
+                    ( power-Commutative-Semiring R (nat-Fin n i) r))))
+          by
+            ap-add-Commutative-Semiring R
+              ( refl)
+              ( inv
+                ( left-distributive-mul-sum-fin-sequence-type-Commutative-Semiring
+                  ( R)
+                  ( n)
+                  ( r)
+                  ( _)))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( mul-Commutative-Semiring R
+              ( r)
+              ( sum-standard-geometric-fin-sequence-Commutative-Semiring R
+                ( a)
+                ( r)
+                ( n)))
+          by
+            ap-add-Commutative-Semiring R
+              ( refl)
+              ( ap-mul-Commutative-Semiring R
+                ( refl)
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+                      ( R)
+                      ( a)
+                      ( r) ∘
+                    nat-Fin n)))
+      where
+        term :
+          type-Commutative-Semiring R → type-Commutative-Semiring R →
+          sequence (type-Commutative-Semiring R)
+        term a' r' =
+          seq-standard-geometric-sequence-Commutative-Semiring R a' r'
+        term-Fin :
+          type-Commutative-Semiring R → type-Commutative-Semiring R →
+          (n : ℕ) → fin-sequence (type-Commutative-Semiring R) n
+        term-Fin a' r' =
+          standard-geometric-fin-sequence-Commutative-Semiring R a' r'
+```
diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 6655218d7ca..51a5fb4b31e 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -85,6 +85,7 @@ open import elementary-number-theory.finitary-natural-numbers public
 open import elementary-number-theory.finitely-cyclic-maps public
 open import elementary-number-theory.fundamental-theorem-of-arithmetic public
 open import elementary-number-theory.geometric-sequences-positive-rational-numbers public
+open import elementary-number-theory.geometric-sequences-rational-numbers public
 open import elementary-number-theory.goldbach-conjecture public
 open import elementary-number-theory.greatest-common-divisor-integers public
 open import elementary-number-theory.greatest-common-divisor-natural-numbers public
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
new file mode 100644
index 00000000000..747150c04bb
--- /dev/null
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -0,0 +1,203 @@
+# Geometric sequences of rational numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module elementary-number-theory.geometric-sequences-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.geometric-sequences-commutative-rings
+open import commutative-algebra.sums-of-finite-sequences-of-elements-commutative-rings
+
+open import elementary-number-theory.addition-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
+open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.powers-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.ring-of-rational-numbers
+
+open import foundation.identity-types
+open import foundation.negated-equality
+open import foundation.universe-levels
+
+open import group-theory.groups
+
+open import lists.sequences
+
+open import order-theory.strictly-increasing-sequences-strictly-preordered-sets
+```
+
+</details>
+
+## Idea
+
+A
+{{#concept "geometric sequence" Disambiguation="of rational numbers" Agda=geometric-sequence-ℚ}}
+of [rational numbers](elementary-number-theory.positive-rational-numbers.md) is
+a
+[geometric sequence](commutative-algebra.geometric-sequences-commutative-rings.md)
+in the
+[commutative ring of rational numbers](elementary-number-theory.ring-of-rational-numbers.md).
+
+## Definitions
+
+```agda
+geometric-sequence-ℚ : UU lzero
+geometric-sequence-ℚ = geometric-sequence-Commutative-Ring commutative-ring-ℚ
+
+seq-geometric-sequence-ℚ : geometric-sequence-ℚ → sequence ℚ
+seq-geometric-sequence-ℚ =
+  seq-geometric-sequence-Commutative-Ring commutative-ring-ℚ
+
+standard-geometric-sequence-ℚ : ℚ → ℚ → geometric-sequence-ℚ
+standard-geometric-sequence-ℚ =
+  standard-geometric-sequence-Commutative-Ring commutative-ring-ℚ
+
+seq-standard-geometric-sequence-ℚ : ℚ → ℚ → sequence ℚ
+seq-standard-geometric-sequence-ℚ =
+  seq-standard-geometric-sequence-Commutative-Ring commutative-ring-ℚ
+```
+
+## Properties
+
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  (a r : ℚ)
+  where
+
+  compute-standard-geometric-sequence-ℚ :
+    (n : ℕ) → seq-standard-geometric-sequence-ℚ a r n ＝ a *ℚ power-ℚ n r
+  compute-standard-geometric-sequence-ℚ n =
+    inv
+      ( htpy-mul-pow-standard-geometric-sequence-Commutative-Ring
+        ( commutative-ring-ℚ)
+        ( a)
+        ( r)
+        ( n))
+```
+
+### If `r ≠ 1`, the sum of the first `n` elements of the standard geometric sequence `n ↦ arⁿ` is `a(1-rⁿ)/(1-r)`
+
+```agda
+sum-standard-geometric-fin-sequence-ℚ : ℚ → ℚ → ℕ → ℚ
+sum-standard-geometric-fin-sequence-ℚ =
+  sum-standard-geometric-fin-sequence-Commutative-Ring commutative-ring-ℚ
+
+module _
+  (a r : ℚ) (r≠1 : r ≠ one-ℚ)
+  where
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-ℚ :
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-ℚ a r n ＝
+      ( a *ℚ
+        ( (one-ℚ -ℚ power-ℚ n r) *ℚ
+          rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)))
+    compute-sum-standard-geometric-fin-sequence-ℚ 0 =
+      inv
+        ( equational-reasoning
+          a *ℚ ((one-ℚ -ℚ one-ℚ) *ℚ _)
+          ＝ a *ℚ (zero-ℚ *ℚ _)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ (right-inverse-law-add-ℚ one-ℚ) refl)
+          ＝ a *ℚ zero-ℚ
+            by ap-mul-ℚ refl (left-zero-law-mul-ℚ _)
+          ＝ zero-ℚ
+            by right-zero-law-mul-ℚ a)
+    compute-sum-standard-geometric-fin-sequence-ℚ (succ-ℕ n) =
+      let
+        1/1-r = rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
+      in
+        equational-reasoning
+          sum-standard-geometric-fin-sequence-ℚ a r (succ-ℕ n)
+          ＝
+            sum-standard-geometric-fin-sequence-ℚ a r n +ℚ
+            seq-standard-geometric-sequence-ℚ a r n
+            by
+              cons-sum-fin-sequence-type-Commutative-Ring
+                ( commutative-ring-ℚ)
+                ( n)
+                ( _)
+                ( refl)
+          ＝
+            ( a *ℚ
+              ( (one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r)) +ℚ
+            ( a *ℚ power-ℚ n r)
+            by
+              ap-add-ℚ
+                ( compute-sum-standard-geometric-fin-sequence-ℚ n)
+                ( compute-standard-geometric-sequence-ℚ a r n)
+          ＝
+            a *ℚ
+            (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r) +ℚ power-ℚ n r)
+            by inv (left-distributive-mul-add-ℚ a _ _)
+          ＝
+            a *ℚ
+            ( (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r) +ℚ
+              (power-ℚ n r *ℚ (one-ℚ -ℚ r)) *ℚ 1/1-r))
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-add-ℚ
+                  ( refl)
+                  ( inv
+                    ( cancel-right-mul-div-ℚˣ _
+                      ( invertible-diff-neq-ℚ r one-ℚ r≠1))))
+          ＝
+            a *ℚ
+            ( ( (one-ℚ -ℚ power-ℚ n r) +ℚ (power-ℚ n r *ℚ (one-ℚ -ℚ r))) *ℚ
+              1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( inv (right-distributive-mul-add-ℚ _ _ 1/1-r))
+          ＝
+            a *ℚ
+            ( ( one-ℚ -ℚ power-ℚ n r +ℚ
+                ((power-ℚ n r *ℚ one-ℚ) -ℚ (power-ℚ n r *ℚ r))) *ℚ
+              1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ
+                  ( ap-add-ℚ refl (left-distributive-mul-diff-ℚ _ _ _))
+                  ( refl))
+          ＝
+            a *ℚ
+            ( ( one-ℚ -ℚ power-ℚ n r +ℚ
+                ((power-ℚ n r -ℚ power-ℚ (succ-ℕ n) r))) *ℚ
+              1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ
+                  ( ap-add-ℚ
+                    ( refl)
+                    ( ap-diff-ℚ
+                      ( right-unit-law-mul-ℚ _)
+                      ( inv (power-succ-ℚ n r))))
+                  ( refl))
+          ＝ a *ℚ ((one-ℚ -ℚ power-ℚ (succ-ℕ n) r) *ℚ 1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ
+                  ( mul-right-div-Group group-add-ℚ _ _ _)
+                  ( refl))
+```
+
+## External links
+
+- [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)
+  at Wikipedia
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index a35af131e80..3df03c15ddf 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -9,11 +9,11 @@ module elementary-number-theory.multiplicative-group-of-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
+open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
-open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
@@ -27,8 +27,8 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.function-types
-open import foundation.negated-equality
 open import foundation.identity-types
+open import foundation.negated-equality
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -190,3 +190,20 @@ abstract
 invertible-diff-neq-ℚ : (a b : ℚ) → a ≠ b → ℚˣ
 invertible-diff-neq-ℚ a b a≠b = (b -ℚ a , is-invertible-diff-neq-ℚ a b a≠b)
 ```
+
+### Cancellation laws
+
+```agda
+abstract
+  cancel-right-mul-div-ℚˣ :
+    (p : ℚ) (q : ℚˣ) → (p *ℚ rational-ℚˣ q) *ℚ rational-inv-ℚˣ q ＝ p
+  cancel-right-mul-div-ℚˣ p q =
+    equational-reasoning
+      p *ℚ rational-ℚˣ q *ℚ rational-inv-ℚˣ q
+      ＝ p *ℚ (rational-ℚˣ q *ℚ rational-inv-ℚˣ q)
+        by associative-mul-ℚ _ _ _
+      ＝ p *ℚ rational-ℚˣ one-ℚˣ
+        by ap-mul-ℚ refl (ap rational-ℚˣ (right-inverse-law-mul-ℚˣ q))
+      ＝ p
+        by right-unit-law-mul-ℚ p
+```
diff --git a/src/ring-theory/geometric-sequences-rings.lagda.md b/src/ring-theory/geometric-sequences-rings.lagda.md
index 90497c4db21..9458cf5a5fa 100644
--- a/src/ring-theory/geometric-sequences-rings.lagda.md
+++ b/src/ring-theory/geometric-sequences-rings.lagda.md
@@ -158,7 +158,7 @@ module _
       ( u)
 ```
 
-### The nth term of an geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
 
 ```agda
 module _
diff --git a/src/ring-theory/geometric-sequences-semirings.lagda.md b/src/ring-theory/geometric-sequences-semirings.lagda.md
index 3d7dff3e3ac..1e560ceabd0 100644
--- a/src/ring-theory/geometric-sequences-semirings.lagda.md
+++ b/src/ring-theory/geometric-sequences-semirings.lagda.md
@@ -211,7 +211,7 @@ module _
       ( u)
 ```
 
-### The nth term of an geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
 
 ```agda
 module _

From c33f1fba7247a73527f59da257ecc60b7571777e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 20 Oct 2025 16:04:41 -0700
Subject: [PATCH 023/134] Progress

---
 .../apartness-complex-numbers.lagda.md        | 100 ++++++++++++++++++
 .../nonzero-complex-numbers.lagda.md          |  35 ++++++
 2 files changed, 135 insertions(+)
 create mode 100644 src/complex-numbers/apartness-complex-numbers.lagda.md
 create mode 100644 src/complex-numbers/nonzero-complex-numbers.lagda.md

diff --git a/src/complex-numbers/apartness-complex-numbers.lagda.md b/src/complex-numbers/apartness-complex-numbers.lagda.md
new file mode 100644
index 00000000000..762d6da4092
--- /dev/null
+++ b/src/complex-numbers/apartness-complex-numbers.lagda.md
@@ -0,0 +1,100 @@
+# Apartness of complex numbers
+
+```agda
+module complex-numbers.apartness-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.negation
+open import complex-numbers.complex-numbers
+open import foundation.function-types
+open import foundation.large-apartness-relations
+open import foundation.dependent-pair-types
+open import foundation.functoriality-disjunction
+open import real-numbers.apartness-real-numbers
+open import foundation.empty-types
+open import foundation.disjunction
+open import foundation.propositions
+open import foundation.universe-levels
+```
+
+</details>
+
+## Idea
+
+Two [complex numbers](complex-numbers.complex-numbers.md) are
+{{#concept "apart" Agda=apart-ℂ}} if their
+[real](real-numbers.dedekind-real-numbers.md) parts are
+[apart](real-numbers.apartness-real-numbers.md) [or](foundation.disjunction.md)
+their complex parts are [apart].
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2)
+  where
+
+  apart-prop-ℂ : Prop (l1 ⊔ l2)
+  apart-prop-ℂ =
+    apart-prop-ℝ (re-ℂ z) (re-ℂ w) ∨ apart-prop-ℝ (im-ℂ z) (im-ℂ w)
+
+  apart-ℂ : UU (l1 ⊔ l2)
+  apart-ℂ = type-Prop apart-prop-ℂ
+```
+
+## Properties
+
+### Apartness is antireflexive
+
+```agda
+abstract
+  antireflexive-apart-ℂ : {l : Level} (z : ℂ l) → ¬ (apart-ℂ z z)
+  antireflexive-apart-ℂ (a , b) =
+    elim-disjunction
+      ( empty-Prop)
+      ( antireflexive-apart-ℝ a)
+      ( antireflexive-apart-ℝ b)
+```
+
+### Apartness is symmetric
+
+```agda
+abstract
+  symmetric-apart-ℂ :
+    {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2) → apart-ℂ z w → apart-ℂ w z
+  symmetric-apart-ℂ (a , b) (c , d) =
+    map-disjunction (symmetric-apart-ℝ a c) (symmetric-apart-ℝ b d)
+```
+
+### Apartness is cotransitive
+
+```agda
+abstract
+  cotransitive-apart-ℂ :
+    {l1 l2 l3 : Level} (x : ℂ l1) (y : ℂ l2) (z : ℂ l3) →
+    apart-ℂ x y → disjunction-type (apart-ℂ x z) (apart-ℂ z y)
+  cotransitive-apart-ℂ x@(a , b) y@(c , d) z@(e , f) =
+    elim-disjunction
+      ( (apart-prop-ℂ x z) ∨ (apart-prop-ℂ z y))
+      ( map-disjunction inl-disjunction inl-disjunction ∘
+        cotransitive-apart-ℝ a c e)
+      ( map-disjunction inr-disjunction inr-disjunction ∘
+        cotransitive-apart-ℝ b d f)
+```
+
+### Apartness on the complex numbers is a large apartness relation
+
+```agda
+large-apartness-relation-ℂ : Large-Apartness-Relation _⊔_ ℂ
+apart-prop-Large-Apartness-Relation large-apartness-relation-ℂ =
+  apart-prop-ℂ
+antirefl-Large-Apartness-Relation large-apartness-relation-ℂ =
+  antireflexive-apart-ℂ
+symmetric-Large-Apartness-Relation large-apartness-relation-ℂ =
+  symmetric-apart-ℂ
+cotransitive-Large-Apartness-Relation large-apartness-relation-ℂ =
+  cotransitive-apart-ℂ
+```
diff --git a/src/complex-numbers/nonzero-complex-numbers.lagda.md b/src/complex-numbers/nonzero-complex-numbers.lagda.md
new file mode 100644
index 00000000000..ac02560e7cb
--- /dev/null
+++ b/src/complex-numbers/nonzero-complex-numbers.lagda.md
@@ -0,0 +1,35 @@
+# Nonzero complex numbers
+
+```agda
+module complex-numbers.nonzero-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.complex-numbers
+open import foundation.universe-levels
+open import foundation.propositions
+open import complex-numbers.apartness-complex-numbers
+```
+
+</details>
+
+## Idea
+
+A [complex number](complex-numbers.complex-numbers.md) is
+{{#concept "nonzero" Agda=is-nonzero-ℂ}} if it is
+[apart](complex-numbers.apartness-complex-numbers.md) from zero, that is, its
+[real](real-numbers.dedekind-real-numbers.md) part
+[or](foundation.disjunction.md) its imaginary part are
+[nonzero](real-numbers.nonzero-real-numbers.md).
+
+## Definition
+
+```agda
+is-nonzero-prop-ℂ : {l : Level} → ℂ l → Prop l
+is-nonzero-prop-ℂ z = apart-prop-ℂ z zero-ℂ
+
+is-nonzero-ℂ : {l : Level} → ℂ l → UU l
+is-nonzero-ℂ z = type-Prop (is-nonzero-prop-ℂ z)
+```

From 3c95dda19908bcfd2d60f1adb48b23942da5b787 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 4 Nov 2025 20:06:48 -0800
Subject: [PATCH 024/134] Fix build

---
 .../powers-positive-rational-numbers.lagda.md                   | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 03791d8ca4e..1f3932919c6 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -123,7 +123,7 @@ abstract
       ( power-ℚ⁺ n p *ℚ⁺ q)
       ( power-ℚ⁺ (succ-ℕ n) q)
       ( preserves-le-right-mul-ℚ⁺ q _ _
-        ( preserves-le-power-ℚ⁺ n (is-nonzero-succ-ℕ _) p q p<q))
+        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
       ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
 ```
 

From 02238cacf3a411420670afa0d4b931875a9486d2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 4 Nov 2025 20:09:40 -0800
Subject: [PATCH 025/134] Fix link

---
 src/ring-theory/geometric-sequences-rings.lagda.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/ring-theory/geometric-sequences-rings.lagda.md b/src/ring-theory/geometric-sequences-rings.lagda.md
index 9458cf5a5fa..68d24fa3921 100644
--- a/src/ring-theory/geometric-sequences-rings.lagda.md
+++ b/src/ring-theory/geometric-sequences-rings.lagda.md
@@ -34,8 +34,8 @@ open import ring-theory.rings
 
 A
 {{#concept "geometric sequence" Disambiguation="in a ring" Agda=geometric-sequence-Ring}}
-in a [ring](ring-theory.semirings.md) is an
-[arithmetic sequence](ring-theory.geometric-sequence-semirings.md) in the
+in a [ring](ring-theory.semirings.md) is a
+[geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
 underlying [semiring](ring-theory.semirings.md).
 
 These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the

From 764f62ee7057ae3180373f44a9e808b9b803146a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 10:37:07 -0800
Subject: [PATCH 026/134] If |q|<1 , q^n approaches 0

---
 .../absolute-value-rational-numbers.lagda.md  | 11 +++
 ...metric-sequences-rational-numbers.lagda.md |  2 +
 ...quality-positive-rational-numbers.lagda.md |  9 ++
 ...ication-positive-rational-numbers.lagda.md |  5 ++
 .../powers-positive-rational-numbers.lagda.md | 86 +++++++++++++++++++
 .../powers-rational-numbers.lagda.md          | 72 ++++++++++++++++
 ...limits-of-sequences-metric-spaces.lagda.md | 20 +++++
 ...onal-sequences-approximating-zero.lagda.md | 11 +++
 8 files changed, 216 insertions(+)

diff --git a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
index 4b0ad01d885..3a42ec9a40a 100644
--- a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
@@ -18,6 +18,7 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
@@ -142,6 +143,16 @@ abstract
     ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ q))
 ```
 
+### The absolute value of a negative rational number is the negation of the number
+
+```agda
+abstract
+  rational-abs-rational-ℚ⁻ :
+    (q : ℚ⁻) → rational-abs-ℚ (rational-ℚ⁻ q) ＝ neg-ℚ (rational-ℚ⁻ q)
+  rational-abs-rational-ℚ⁻ q⁻@(q , _) =
+    inv (rational-abs-neg-ℚ q) ∙ rational-abs-rational-ℚ⁺ (neg-ℚ⁻ q⁻)
+```
+
 ### `q` is less than or equal to `abs-ℚ q`
 
 ```agda
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 747150c04bb..68abba9e6e0 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -197,6 +197,8 @@ module _
                   ( refl))
 ```
 
+### If `|r| < 1`, the sum of the standard geometric sequence `n ↦ arⁿ` is `a/(1-r)`
+
 ## External links
 
 - [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)
diff --git a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
index ef8913ec1d2..d967080b3a1 100644
--- a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
@@ -11,8 +11,10 @@ open import elementary-number-theory.decidable-total-order-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
+open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
+open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
 
@@ -90,3 +92,10 @@ transitive-leq-ℚ⁺ = transitive-leq-Poset poset-ℚ⁺
 antisymmetric-leq-ℚ⁺ : is-antisymmetric leq-ℚ⁺
 antisymmetric-leq-ℚ⁺ = antisymmetric-leq-Poset poset-ℚ⁺
 ```
+
+### If `x ＝ y`, `x ≤ y`
+
+```agda
+leq-eq-ℚ⁺ : {x y : ℚ⁺} → x ＝ y → leq-ℚ⁺ x y
+leq-eq-ℚ⁺ x=y = leq-eq-ℚ _ _ (ap rational-ℚ⁺ x=y)
+```
diff --git a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
index cda3184cdbc..c77be1001a5 100644
--- a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
@@ -28,6 +28,7 @@ open import elementary-number-theory.strict-inequality-integers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
+open import foundation.action-on-identifications-binary-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.empty-types
@@ -107,6 +108,10 @@ mul-ℚ⁺ = mul-Submonoid monoid-mul-ℚ submonoid-mul-ℚ⁺
 
 infixl 40 _*ℚ⁺_
 _*ℚ⁺_ = mul-ℚ⁺
+
+ap-mul-ℚ⁺ :
+  {x x' : ℚ⁺} → x ＝ x' → {y y' : ℚ⁺} → y ＝ y' → mul-ℚ⁺ x y ＝ mul-ℚ⁺ x' y'
+ap-mul-ℚ⁺ = ap-binary mul-ℚ⁺
 ```
 
 ## Properties
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 1f3932919c6..fc414c812b1 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -7,20 +7,26 @@ module elementary-number-theory.powers-positive-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.archimedean-property-rational-numbers
 open import elementary-number-theory.arithmetic-sequences-positive-rational-numbers
 open import elementary-number-theory.bernoullis-inequality-positive-rational-numbers
+open import elementary-number-theory.distance-rational-numbers
 open import elementary-number-theory.geometric-sequences-positive-rational-numbers
+open import elementary-number-theory.inequality-natural-numbers
 open import elementary-number-theory.inequality-positive-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.multiplication-natural-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-natural-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
@@ -38,6 +44,12 @@ open import foundation.transport-along-identifications
 
 open import group-theory.multiples-of-elements-abelian-groups
 open import group-theory.powers-of-elements-groups
+
+open import metric-spaces.limits-of-sequences-metric-spaces
+open import metric-spaces.metric-space-of-rational-numbers
+open import metric-spaces.rational-sequences-approximating-zero
+
+open import order-theory.posets
 ```
 
 </details>
@@ -58,6 +70,9 @@ times. This file covers the case where `n` is a
 ```agda
 power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
 power-ℚ⁺ = power-Group group-mul-ℚ⁺
+
+rational-power-ℚ⁺ : ℕ → ℚ⁺ → ℚ
+rational-power-ℚ⁺ n q = rational-ℚ⁺ (power-ℚ⁺ n q)
 ```
 
 ## Properties
@@ -253,6 +268,77 @@ abstract
     unit-trunc-Prop (bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε)
 ```
 
+### If `ε` is a positive rational number less than or equal to 1 and `m ≤ n`, then `εⁿ ≤ εᵐ`
+
+```agda
+abstract
+  leq-power-leq-one-ℚ⁺ :
+    (ε : ℚ⁺) → leq-ℚ⁺ ε one-ℚ⁺ → (m n : ℕ) → leq-ℕ m n →
+    leq-ℚ⁺ (power-ℚ⁺ n ε) (power-ℚ⁺ m ε)
+  leq-power-leq-one-ℚ⁺ ε ε≤1 m n m≤n =
+    let
+      (k , k+m=n) = subtraction-leq-ℕ m n m≤n
+      open inequality-reasoning-Poset ℚ-Poset
+    in
+      chain-of-inequalities
+        rational-power-ℚ⁺ n ε
+        ≤ rational-power-ℚ⁺ (k +ℕ m) ε
+          by leq-eq-ℚ⁺ (ap (λ x → power-ℚ⁺ x ε) (inv k+m=n))
+        ≤ rational-power-ℚ⁺ k ε *ℚ rational-power-ℚ⁺ m ε
+          by leq-eq-ℚ⁺ (distributive-power-add-ℚ⁺ k m ε)
+        ≤ rational-power-ℚ⁺ k one-ℚ⁺ *ℚ rational-power-ℚ⁺ m ε
+          by
+            preserves-leq-right-mul-ℚ⁺
+              ( power-ℚ⁺ m ε)
+              ( _)
+              ( _)
+              ( preserves-leq-power-ℚ⁺ k ε one-ℚ⁺ ε≤1)
+        ≤ one-ℚ *ℚ rational-power-ℚ⁺ m ε
+          by leq-eq-ℚ _ _ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
+        ≤ rational-power-ℚ⁺ m ε
+          by leq-eq-ℚ _ _ (left-unit-law-mul-ℚ _)
+```
+
+### If `ε` is a positive rational number less than 1, `εⁿ` approaches 0
+
+```agda
+abstract
+  is-zero-limit-power-le-one-ℚ⁺ :
+    (ε : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    is-zero-limit-sequence-ℚ (λ n → rational-ℚ⁺ (power-ℚ⁺ n ε))
+  is-zero-limit-power-le-one-ℚ⁺ ε ε<1 =
+    is-limit-bound-modulus-sequence-Metric-Space
+      ( metric-space-ℚ)
+      ( _)
+      ( zero-ℚ)
+      ( λ δ →
+        let
+          (m , εᵐ<δ) =
+            bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ ε<1
+          open inequality-reasoning-Poset ℚ-Poset
+        in
+          ( m ,
+            λ n m≤n →
+              neighborhood-leq-dist-ℚ
+                ( δ)
+                ( rational-power-ℚ⁺ n ε)
+                ( zero-ℚ)
+                ( chain-of-inequalities
+                  rational-dist-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε)) zero-ℚ
+                  ≤ rational-abs-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε))
+                    by
+                      leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
+                  ≤ rational-ℚ⁺ (power-ℚ⁺ n ε)
+                    by
+                      leq-eq-ℚ _ _
+                        ( ap rational-ℚ⁰⁺
+                          ( abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ (power-ℚ⁺ n ε))))
+                  ≤ rational-ℚ⁺ (power-ℚ⁺ m ε)
+                    by leq-power-leq-one-ℚ⁺ ε (leq-le-ℚ ε<1) m n m≤n
+                  ≤ rational-ℚ⁺ δ
+                    by leq-le-ℚ εᵐ<δ)))
+```
+
 ## See also
 
 - [Powers of elements of a group](group-theory.powers-of-elements-groups.md)
diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index 898f71cee0e..ea1efba98d3 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -10,6 +10,7 @@ module elementary-number-theory.powers-rational-numbers where
 open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
+open import elementary-number-theory.inequality-natural-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.multiplication-natural-numbers
@@ -34,11 +35,16 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
+open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.transport-along-identifications
 
 open import group-theory.powers-of-elements-commutative-monoids
 open import group-theory.powers-of-elements-monoids
+
+open import metric-spaces.limits-of-sequences-metric-spaces
+open import metric-spaces.metric-space-of-rational-numbers
+open import metric-spaces.rational-sequences-approximating-zero
 ```
 
 </details>
@@ -101,6 +107,14 @@ abstract
   power-succ-ℚ = power-succ-Monoid monoid-mul-ℚ
 ```
 
+### `0ⁿ = 0` when `1 ≤ n`
+
+```agda
+abstract
+  power-zero-ℚ : (n : ℕ) → leq-ℕ 1 n → power-ℚ n zero-ℚ ＝ zero-ℚ
+  power-zero-ℚ (succ-ℕ n) _ = power-succ-ℚ n _ ∙ right-zero-law-mul-ℚ _
+```
+
 ### `qⁿ⁺¹ = qqⁿ`
 
 ```agda
@@ -392,6 +406,64 @@ abstract
       ( decide-is-negative-is-nonnegative-ℚ p)
 ```
 
+### If `|ε| < 1`, `εⁿ` approaches 0
+
+```agda
+abstract
+  is-zero-limit-power-le-one-abs-ℚ :
+    (ε : ℚ) → le-ℚ (rational-abs-ℚ ε) one-ℚ →
+    is-zero-limit-sequence-ℚ (λ n → power-ℚ n ε)
+  is-zero-limit-power-le-one-abs-ℚ ε |ε|<1 =
+    trichotomy-sign-ℚ ε
+      ( λ is-neg-ε →
+        let
+          ε⁻ = (ε , is-neg-ε)
+        in
+          is-zero-limit-is-zero-limit-abs-sequence-ℚ
+            ( _)
+            ( inv-tr
+              ( is-zero-limit-sequence-ℚ)
+              ( eq-htpy
+                ( λ n →
+                  equational-reasoning
+                    rational-abs-ℚ (power-ℚ n ε)
+                    ＝ power-ℚ n (rational-abs-ℚ ε)
+                      by inv (power-rational-abs-ℚ n ε)
+                    ＝ power-ℚ n (neg-ℚ ε)
+                      by
+                        ap (power-ℚ n) (rational-abs-rational-ℚ⁻ ε⁻)
+                    ＝ rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ ε⁻))
+                      by power-rational-ℚ⁺ n (neg-ℚ⁻ ε⁻)))
+            ( is-zero-limit-power-le-one-ℚ⁺
+              ( neg-ℚ⁻ ε⁻)
+              ( tr
+                ( λ q → le-ℚ q one-ℚ)
+                ( rational-abs-rational-ℚ⁻ ε⁻)
+                ( |ε|<1)))))
+      ( λ ε=0 →
+        is-limit-bound-modulus-sequence-Metric-Space
+          ( metric-space-ℚ)
+          ( λ n → power-ℚ n ε)
+          ( zero-ℚ)
+          ( λ δ →
+            ( 1 ,
+              λ n 1≤n →
+                inv-tr
+                  ( λ z → neighborhood-ℚ δ z zero-ℚ)
+                  ( ap (power-ℚ n) ε=0 ∙ power-zero-ℚ n 1≤n)
+                  ( is-reflexive-neighborhood-ℚ δ zero-ℚ))))
+      ( λ is-pos-ε →
+        inv-tr
+          ( is-zero-limit-sequence-ℚ)
+          ( eq-htpy (λ n → power-rational-ℚ⁺ n (ε , is-pos-ε)))
+          ( is-zero-limit-power-le-one-ℚ⁺
+            ( ε , is-pos-ε)
+            ( tr
+              ( λ q → le-ℚ q one-ℚ)
+              ( rational-abs-rational-ℚ⁺ (ε , is-pos-ε))
+              ( |ε|<1))))
+```
+
 ## See also
 
 - [Powers of elements of a monoid](group-theory.powers-of-elements-monoids.md)
diff --git a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
index 41bb75c87e7..219491a035a 100644
--- a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
@@ -15,6 +15,7 @@ open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.cartesian-product-types
 open import foundation.dependent-pair-types
+open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.functoriality-propositional-truncation
@@ -100,6 +101,25 @@ module _
 
 ## Properties
 
+### Proving the limit of a sequence in a metric space
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  (u : sequence-type-Metric-Space M)
+  (lim : type-Metric-Space M)
+  where
+
+  is-limit-bound-modulus-sequence-Metric-Space :
+    ( (δ : ℚ⁺) →
+      Σ ( ℕ)
+        ( λ (N : ℕ) →
+          (n : ℕ) → leq-ℕ N n → neighborhood-Metric-Space M δ (u n) lim)) →
+    is-limit-sequence-Metric-Space M u lim
+  is-limit-bound-modulus-sequence-Metric-Space H =
+    intro-exists (pr1 ∘ H) (pr2 ∘ H)
+```
+
 ### The limit of a sequence in a metric space is unique
 
 ```agda
diff --git a/src/metric-spaces/rational-sequences-approximating-zero.lagda.md b/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
index 821404d222f..fd54c611598 100644
--- a/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
+++ b/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
@@ -215,6 +215,17 @@ module _
       map-is-inhabited
         ( tot tr-is-modulus-leq-abs-zero-limit-sequence-ℚ)
         ( is-zero-limit-seq-zero-limit-sequence-ℚ v)
+
+abstract
+  is-zero-limit-is-zero-limit-abs-sequence-ℚ :
+    ( u : sequence ℚ) →
+    is-zero-limit-sequence-ℚ (rational-abs-ℚ ∘ u) →
+    is-zero-limit-sequence-ℚ u
+  is-zero-limit-is-zero-limit-abs-sequence-ℚ u H =
+    is-zero-limit-leq-abs-zero-limit-sequence-ℚ
+      ( u)
+      ( rational-abs-ℚ ∘ u , H)
+      ( λ n → refl-leq-ℚ (rational-abs-ℚ (u n)))
 ```
 
 ### Any rational approximation of zero defines a sequence approximating zero

From 3c8f1ad31502712382ef34d6235f2450c35e0803 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 11:33:45 -0800
Subject: [PATCH 027/134] The sum of geometric series of rational numbers

---
 ...gent-series-metric-abelian-groups.lagda.md |  4 +-
 ...metric-sequences-rational-numbers.lagda.md | 75 +++++++++++++++++++
 ...icative-group-of-rational-numbers.lagda.md | 32 ++++++++
 ...trict-inequality-rational-numbers.lagda.md |  5 ++
 .../triangular-numbers.lagda.md               |  1 -
 src/foundation/negated-equality.lagda.md      | 14 +++-
 .../metric-space-of-rational-numbers.lagda.md | 17 +++++
 7 files changed, 143 insertions(+), 5 deletions(-)

diff --git a/src/analysis/convergent-series-metric-abelian-groups.lagda.md b/src/analysis/convergent-series-metric-abelian-groups.lagda.md
index 31a90a78358..6e5e9cfe3fd 100644
--- a/src/analysis/convergent-series-metric-abelian-groups.lagda.md
+++ b/src/analysis/convergent-series-metric-abelian-groups.lagda.md
@@ -36,7 +36,7 @@ associated [metric space](metric-spaces.metric-spaces.md).
 
 ```agda
 module _
-  {l1 l2 : Level} (G : Metric-Ab l1 l2) (σ : series-Metric-Ab G)
+  {l1 l2 : Level} {G : Metric-Ab l1 l2} (σ : series-Metric-Ab G)
   where
 
   is-sum-prop-series-Metric-Ab : type-Metric-Ab G → Prop l2
@@ -61,7 +61,7 @@ module _
 convergent-series-Metric-Ab :
   {l1 l2 : Level} (G : Metric-Ab l1 l2) → UU (l1 ⊔ l2)
 convergent-series-Metric-Ab G =
-  type-subtype (is-convergent-prop-series-Metric-Ab G)
+  type-subtype (is-convergent-prop-series-Metric-Ab {G = G})
 ```
 
 ## Properties
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 68abba9e6e0..03de59ab44e 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -9,27 +9,43 @@ module elementary-number-theory.geometric-sequences-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import analysis.convergent-series-metric-abelian-groups
+open import analysis.series-metric-abelian-groups
+
 open import commutative-algebra.geometric-sequences-commutative-rings
 open import commutative-algebra.sums-of-finite-sequences-of-elements-commutative-rings
 
+open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.metric-additive-group-of-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-rational-numbers
 open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.powers-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.ring-of-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
 
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.negated-equality
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.groups
 
 open import lists.sequences
 
+open import metric-spaces.limits-of-sequences-metric-spaces
+open import metric-spaces.metric-space-of-rational-numbers
+open import metric-spaces.uniformly-continuous-functions-metric-spaces
+
 open import order-theory.strictly-increasing-sequences-strictly-preordered-sets
 ```
 
@@ -199,6 +215,65 @@ module _
 
 ### If `|r| < 1`, the sum of the standard geometric sequence `n ↦ arⁿ` is `a/(1-r)`
 
+```agda
+module _
+  (a r : ℚ)
+  where
+
+  standard-geometric-series-ℚ : series-Metric-Ab metric-ab-add-ℚ
+  standard-geometric-series-ℚ =
+    series-terms-Metric-Ab (seq-standard-geometric-sequence-ℚ a r)
+
+  abstract
+    sum-standard-geometric-sequence-ℚ :
+      (|r|<1 : le-ℚ (rational-abs-ℚ r) one-ℚ) →
+      is-sum-series-Metric-Ab
+        ( standard-geometric-series-ℚ)
+        ( a *ℚ rational-inv-ℚˣ (invertible-diff-le-abs-ℚ r one-ℚ⁺ |r|<1))
+    sum-standard-geometric-sequence-ℚ |r|<1 =
+      let
+        r≠1 =
+          nonequal-map
+            ( rational-abs-ℚ)
+            ( inv-tr
+              ( rational-abs-ℚ r ≠_)
+              ( rational-abs-rational-ℚ⁺ one-ℚ⁺)
+              ( nonequal-le-ℚ |r|<1))
+      in
+        binary-tr
+          ( is-limit-sequence-Metric-Space metric-space-ℚ)
+          ( inv
+            ( eq-htpy (compute-sum-standard-geometric-fin-sequence-ℚ a r r≠1)))
+          ( equational-reasoning
+            a *ℚ
+            ( (one-ℚ -ℚ zero-ℚ) *ℚ
+              rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1))
+            ＝
+              a *ℚ
+              ( one-ℚ *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1))
+              by ap-mul-ℚ refl (ap-mul-ℚ (right-zero-law-diff-ℚ one-ℚ) refl)
+            ＝ a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
+              by ap-mul-ℚ refl (left-unit-law-mul-ℚ _))
+          ( uniformly-continuous-map-limit-sequence-Metric-Space
+            ( metric-space-ℚ)
+            ( metric-space-ℚ)
+            ( comp-uniformly-continuous-function-Metric-Space
+              ( metric-space-ℚ)
+              ( metric-space-ℚ)
+              ( metric-space-ℚ)
+              ( uniformly-continuous-left-mul-ℚ a)
+              ( comp-uniformly-continuous-function-Metric-Space
+                ( metric-space-ℚ)
+                ( metric-space-ℚ)
+                ( metric-space-ℚ)
+                ( uniformly-continuous-right-mul-ℚ
+                  ( rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)))
+                ( uniformly-continuous-diff-ℚ one-ℚ)))
+            ( λ n → power-ℚ n r)
+            ( zero-ℚ)
+            ( is-zero-limit-power-le-one-abs-ℚ r |r|<1))
+```
+
 ## External links
 
 - [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index 1345443d5b1..b6fb3598c70 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -9,12 +9,14 @@ module elementary-number-theory.multiplicative-group-of-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -78,6 +80,13 @@ rational-ℚˣ : ℚˣ → ℚ
 rational-ℚˣ = pr1
 ```
 
+### Equality of invertible rational numbers
+
+```agda
+eq-ℚˣ : (x y : ℚˣ) → rational-ℚˣ x ＝ rational-ℚˣ y → x ＝ y
+eq-ℚˣ _ _ = eq-type-subtype (subtype-group-of-units-Ring ring-ℚ)
+```
+
 ### Operations of the multiplicative group of rational numbers
 
 ```agda
@@ -191,6 +200,29 @@ invertible-diff-neq-ℚ : (a b : ℚ) → a ≠ b → ℚˣ
 invertible-diff-neq-ℚ a b a≠b = (b -ℚ a , is-invertible-diff-neq-ℚ a b a≠b)
 ```
 
+### If `|a| < b` for positive `b`, `b - a` is invertible
+
+```agda
+is-invertible-diff-le-abs-ℚ :
+  (a : ℚ) (b : ℚ⁺) → le-ℚ (rational-abs-ℚ a) (rational-ℚ⁺ b) →
+  is-invertible-element-Ring ring-ℚ (rational-ℚ⁺ b -ℚ a)
+is-invertible-diff-le-abs-ℚ a b⁺@(b , _) |a|<b =
+  is-invertible-diff-neq-ℚ
+    ( a)
+    ( b)
+    ( nonequal-map
+      ( rational-abs-ℚ)
+      ( inv-tr
+        ( rational-abs-ℚ a ≠_)
+        ( rational-abs-rational-ℚ⁺ b⁺)
+        ( nonequal-le-ℚ |a|<b)))
+
+invertible-diff-le-abs-ℚ :
+  (a : ℚ) (b : ℚ⁺) → le-ℚ (rational-abs-ℚ a) (rational-ℚ⁺ b) → ℚˣ
+invertible-diff-le-abs-ℚ a b |a|<b =
+  ( rational-ℚ⁺ b -ℚ a , is-invertible-diff-le-abs-ℚ a b |a|<b)
+```
+
 ### Cancellation laws
 
 ```agda
diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
index fad1efcdb1e..f89d5f88c6f 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -47,6 +47,7 @@ open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
+open import foundation.negated-equality
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
@@ -143,6 +144,10 @@ opaque
 irreflexive-le-ℚ : (x : ℚ) → ¬ (le-ℚ x x)
 irreflexive-le-ℚ =
   is-irreflexive-is-asymmetric le-ℚ asymmetric-le-ℚ
+
+abstract
+  nonequal-le-ℚ : {x y : ℚ} → le-ℚ x y → x ≠ y
+  nonequal-le-ℚ {x} x<x refl = irreflexive-le-ℚ x x<x
 ```
 
 ### Strict inequality on the rationals is transitive
diff --git a/src/elementary-number-theory/triangular-numbers.lagda.md b/src/elementary-number-theory/triangular-numbers.lagda.md
index 08e97577a84..451206ca038 100644
--- a/src/elementary-number-theory/triangular-numbers.lagda.md
+++ b/src/elementary-number-theory/triangular-numbers.lagda.md
@@ -276,7 +276,6 @@ This theorem is the [42nd](literature.100-theorems.md#42) theorem on
 abstract
   sum-reciprocal-triangular-number-ℕ :
     is-sum-series-Metric-Ab
-      ( metric-ab-add-ℚ)
       ( series-reciprocal-triangular-number-ℕ)
       ( rational-ℕ 2)
   sum-reciprocal-triangular-number-ℕ =
diff --git a/src/foundation/negated-equality.lagda.md b/src/foundation/negated-equality.lagda.md
index 96ad3cd6a1c..9d4e4d46d66 100644
--- a/src/foundation/negated-equality.lagda.md
+++ b/src/foundation/negated-equality.lagda.md
@@ -88,6 +88,16 @@ module _
   nonequal-Π f g a = map-neg (λ h → htpy-eq h a)
 ```
 
+### If a function is nonequal at two points, the points are nonequal
+
+```agda
+abstract
+  nonequal-map :
+    {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
+    {a a' : A} → f a ≠ f a' → a ≠ a'
+  nonequal-map f = map-neg (ap f)
+```
+
 ### If either component of two pairs are nonequal, the pairs are nonequal
 
 ```agda
@@ -96,14 +106,14 @@ module _
   where
 
   nonequal-pr1 : (u v : Σ A B) → pr1 u ≠ pr1 v → u ≠ v
-  nonequal-pr1 u v = map-neg (ap pr1)
+  nonequal-pr1 u v = nonequal-map pr1
 
 module _
   {l1 l2 : Level} {A : UU l1} {B : UU l2}
   where
 
   nonequal-product-pr2 : (u v : A × B) → pr2 u ≠ pr2 v → u ≠ v
-  nonequal-product-pr2 u v = map-neg (ap pr2)
+  nonequal-product-pr2 u v = nonequal-map pr2
 ```
 
 ### If there is a reflexive relation that does not relate `a` and `b`, then they are nonequal
diff --git a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
index 0b54f4fdebc..6da9948861f 100644
--- a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
+++ b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
@@ -510,10 +510,27 @@ module _
         ( mul-ℚ x)
         ( is-lipschitz-left-mul-ℚ)
 
+    is-uniformly-continuous-right-mul-ℚ :
+      is-uniformly-continuous-function-Metric-Space
+        ( metric-space-ℚ)
+        ( metric-space-ℚ)
+        ( mul-ℚ' x)
+    is-uniformly-continuous-right-mul-ℚ =
+      is-uniformly-continuous-is-lipschitz-function-Metric-Space
+        ( metric-space-ℚ)
+        ( metric-space-ℚ)
+        ( mul-ℚ' x)
+        ( is-lipschitz-right-mul-ℚ)
+
   uniformly-continuous-left-mul-ℚ :
     uniformly-continuous-function-Metric-Space metric-space-ℚ metric-space-ℚ
   uniformly-continuous-left-mul-ℚ =
     ( mul-ℚ x , is-uniformly-continuous-left-mul-ℚ)
+
+  uniformly-continuous-right-mul-ℚ :
+    uniformly-continuous-function-Metric-Space metric-space-ℚ metric-space-ℚ
+  uniformly-continuous-right-mul-ℚ =
+    ( mul-ℚ' x , is-uniformly-continuous-right-mul-ℚ)
 ```
 
 ### The convergent Cauchy approximation of the canonical inclusion of positive rational numbers into the rational numbers

From 2b7d4a26a4aad4f4389e05ad396b67e082831d54 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 13:10:33 -0800
Subject: [PATCH 028/134] Define real powers

---
 src/commutative-algebra.lagda.md              |   1 +
 .../large-commutative-rings.lagda.md          |  26 +++
 ...ers-of-elements-commutative-rings.lagda.md |  21 ++
 ...-elements-large-commutative-rings.lagda.md | 190 ++++++++++++++++++
 .../large-abelian-groups.lagda.md             |  15 ++
 .../powers-of-elements-monoids.lagda.md       |  19 +-
 src/real-numbers.lagda.md                     |   1 +
 ...ge-additive-group-of-real-numbers.lagda.md |  21 ++
 .../large-ring-of-real-numbers.lagda.md       |  28 +++
 src/real-numbers/powers-real-numbers.lagda.md | 121 +++++++++++
 src/ring-theory.lagda.md                      |   1 +
 src/ring-theory/large-rings.lagda.md          |  23 +++
 .../powers-of-elements-large-rings.lagda.md   | 154 ++++++++++++++
 .../powers-of-elements-rings.lagda.md         |  22 ++
 14 files changed, 634 insertions(+), 9 deletions(-)
 create mode 100644 src/commutative-algebra/powers-of-elements-large-commutative-rings.lagda.md
 create mode 100644 src/real-numbers/powers-real-numbers.lagda.md
 create mode 100644 src/ring-theory/powers-of-elements-large-rings.lagda.md

diff --git a/src/commutative-algebra.lagda.md b/src/commutative-algebra.lagda.md
index db76ba0076b..5b04abae124 100644
--- a/src/commutative-algebra.lagda.md
+++ b/src/commutative-algebra.lagda.md
@@ -48,6 +48,7 @@ open import commutative-algebra.poset-of-ideals-commutative-rings public
 open import commutative-algebra.poset-of-radical-ideals-commutative-rings public
 open import commutative-algebra.powers-of-elements-commutative-rings public
 open import commutative-algebra.powers-of-elements-commutative-semirings public
+open import commutative-algebra.powers-of-elements-large-commutative-rings public
 open import commutative-algebra.precategory-of-commutative-rings public
 open import commutative-algebra.precategory-of-commutative-semirings public
 open import commutative-algebra.prime-ideals-commutative-rings public
diff --git a/src/commutative-algebra/large-commutative-rings.lagda.md b/src/commutative-algebra/large-commutative-rings.lagda.md
index aca65379013..8ec7e10fac0 100644
--- a/src/commutative-algebra/large-commutative-rings.lagda.md
+++ b/src/commutative-algebra/large-commutative-rings.lagda.md
@@ -8,6 +8,7 @@ module commutative-algebra.large-commutative-rings where
 
 ```agda
 open import commutative-algebra.commutative-rings
+open import commutative-algebra.homomorphisms-commutative-rings
 
 open import foundation.dependent-pair-types
 open import foundation.identity-types
@@ -121,4 +122,29 @@ module _
       ( multiplicative-large-monoid-Large-Ring
         ( large-ring-Large-Commutative-Ring R))
       ( commutative-mul-Large-Commutative-Ring R)
+
+  raise-one-Large-Commutative-Ring :
+    (l : Level) → type-Large-Commutative-Ring R l
+  raise-one-Large-Commutative-Ring =
+    raise-one-Large-Ring (large-ring-Large-Commutative-Ring R)
+
+  one-Large-Commutative-Ring : type-Large-Commutative-Ring R lzero
+  one-Large-Commutative-Ring =
+    one-Large-Ring (large-ring-Large-Commutative-Ring R)
+```
+
+### The raise operation is a commutative ring homomorphism
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β) (l1 l2 : Level)
+  where
+
+  hom-raise-Large-Commutative-Ring :
+    hom-Commutative-Ring
+      ( commutative-ring-Large-Commutative-Ring R l1)
+      ( commutative-ring-Large-Commutative-Ring R (l1 ⊔ l2))
+  hom-raise-Large-Commutative-Ring =
+    hom-raise-Large-Ring (large-ring-Large-Commutative-Ring R) l1 l2
 ```
diff --git a/src/commutative-algebra/powers-of-elements-commutative-rings.lagda.md b/src/commutative-algebra/powers-of-elements-commutative-rings.lagda.md
index 9cd508163f8..9111fc02d2d 100644
--- a/src/commutative-algebra/powers-of-elements-commutative-rings.lagda.md
+++ b/src/commutative-algebra/powers-of-elements-commutative-rings.lagda.md
@@ -8,6 +8,7 @@ module commutative-algebra.powers-of-elements-commutative-rings where
 
 ```agda
 open import commutative-algebra.commutative-rings
+open import commutative-algebra.homomorphisms-commutative-rings
 
 open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.multiplication-natural-numbers
@@ -144,3 +145,23 @@ module _
   odd-power-neg-Commutative-Ring =
     odd-power-neg-Ring (ring-Commutative-Ring A)
 ```
+
+### Commutative ring homomorphisms preserve powers of elements
+
+```agda
+module _
+  {l1 l2 : Level} (R : Commutative-Ring l1) (S : Commutative-Ring l2)
+  (f : hom-Commutative-Ring R S)
+  where
+
+  abstract
+    preserves-powers-hom-Commutative-Ring :
+      (n : ℕ) (x : type-Commutative-Ring R) →
+      map-hom-Commutative-Ring R S f (power-Commutative-Ring R n x) ＝
+      power-Commutative-Ring S n (map-hom-Commutative-Ring R S f x)
+    preserves-powers-hom-Commutative-Ring =
+      preserves-powers-hom-Ring
+        ( ring-Commutative-Ring R)
+        ( ring-Commutative-Ring S)
+        ( f)
+```
diff --git a/src/commutative-algebra/powers-of-elements-large-commutative-rings.lagda.md b/src/commutative-algebra/powers-of-elements-large-commutative-rings.lagda.md
new file mode 100644
index 00000000000..c4dfe1869c0
--- /dev/null
+++ b/src/commutative-algebra/powers-of-elements-large-commutative-rings.lagda.md
@@ -0,0 +1,190 @@
+# Powers of elements in large commutative rings
+
+```agda
+module commutative-algebra.powers-of-elements-large-commutative-rings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.large-commutative-rings
+
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.natural-numbers
+
+open import foundation.identity-types
+open import foundation.universe-levels
+
+open import group-theory.large-monoids
+open import group-theory.powers-of-elements-large-commutative-monoids
+open import group-theory.powers-of-elements-large-monoids
+
+open import ring-theory.large-rings
+open import ring-theory.powers-of-elements-large-rings
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising an element of a large commutative ring to a natural number power" Agda=power-Large-Commutative-Ring}}
+on a [large commutative ring](ring-theory.large-rings.md) is the map `n x ↦ xⁿ`,
+which is defined by [iteratively](foundation.iterating-functions.md) multiplying
+`x` with itself `n` times.
+
+## Definition
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  power-Large-Commutative-Ring :
+    {l : Level} →
+    ℕ → type-Large-Commutative-Ring R l → type-Large-Commutative-Ring R l
+  power-Large-Commutative-Ring =
+    power-Large-Ring (large-ring-Large-Commutative-Ring R)
+```
+
+## Properties
+
+### The power operation preserves similarity
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  abstract
+    preserves-sim-power-Large-Commutative-Ring :
+      {l1 l2 : Level} (n : ℕ) →
+      (x : type-Large-Commutative-Ring R l1) →
+      (y : type-Large-Commutative-Ring R l2) →
+      sim-Large-Commutative-Ring R x y →
+      sim-Large-Commutative-Ring R
+        ( power-Large-Commutative-Ring R n x)
+        ( power-Large-Commutative-Ring R n y)
+    preserves-sim-power-Large-Commutative-Ring =
+      preserves-sim-power-Large-Ring
+        ( large-ring-Large-Commutative-Ring R)
+```
+
+### `1ⁿ = 1`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  abstract
+    raise-power-one-Large-Commutative-Ring :
+      (l : Level) (n : ℕ) →
+      power-Large-Commutative-Ring R n (raise-one-Large-Commutative-Ring R l) ＝
+      raise-one-Large-Commutative-Ring R l
+    raise-power-one-Large-Commutative-Ring =
+      raise-power-one-Large-Ring (large-ring-Large-Commutative-Ring R)
+
+    power-one-Large-Commutative-Ring :
+      (n : ℕ) →
+      power-Large-Commutative-Ring R n (one-Large-Commutative-Ring R) ＝
+      one-Large-Commutative-Ring R
+    power-one-Large-Commutative-Ring =
+      power-one-Large-Ring (large-ring-Large-Commutative-Ring R)
+```
+
+### `xⁿ⁺¹ = xⁿx`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  abstract
+    power-succ-Large-Commutative-Ring :
+      {l : Level} (n : ℕ) (x : type-Large-Commutative-Ring R l) →
+      power-Large-Commutative-Ring R (succ-ℕ n) x ＝
+      mul-Large-Commutative-Ring R (power-Large-Commutative-Ring R n x) x
+    power-succ-Large-Commutative-Ring =
+      power-succ-Large-Ring (large-ring-Large-Commutative-Ring R)
+```
+
+### `xⁿ⁺¹ = xxⁿ`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  abstract
+    power-succ-Large-Commutative-Ring' :
+      {l : Level} (n : ℕ) (x : type-Large-Commutative-Ring R l) →
+      power-Large-Commutative-Ring R (succ-ℕ n) x ＝
+      mul-Large-Commutative-Ring R x (power-Large-Commutative-Ring R n x)
+    power-succ-Large-Commutative-Ring' =
+      power-succ-Large-Ring' (large-ring-Large-Commutative-Ring R)
+```
+
+### Powers by sums of natural numbers are products of powers
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  abstract
+    distributive-power-add-Large-Commutative-Ring :
+      {l : Level} (m n : ℕ) {x : type-Large-Commutative-Ring R l} →
+      power-Large-Commutative-Ring R (m +ℕ n) x ＝
+      mul-Large-Commutative-Ring R
+        ( power-Large-Commutative-Ring R m x)
+        ( power-Large-Commutative-Ring R n x)
+    distributive-power-add-Large-Commutative-Ring =
+      distributive-power-add-Large-Ring (large-ring-Large-Commutative-Ring R)
+```
+
+### Powers by products of natural numbers are iterated powers
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  abstract
+    power-mul-Large-Commutative-Ring :
+      {l : Level} (m n : ℕ) {x : type-Large-Commutative-Ring R l} →
+      power-Large-Commutative-Ring R (m *ℕ n) x ＝
+      power-Large-Commutative-Ring R n (power-Large-Commutative-Ring R m x)
+    power-mul-Large-Commutative-Ring =
+      power-mul-Large-Ring (large-ring-Large-Commutative-Ring R)
+```
+
+### `(xy)ⁿ = xⁿyⁿ`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (R : Large-Commutative-Ring α β)
+  where
+
+  abstract
+    distributive-power-mul-Large-Commutative-Ring :
+      {l1 l2 : Level} (n : ℕ) →
+      {x : type-Large-Commutative-Ring R l1} →
+      {y : type-Large-Commutative-Ring R l2} →
+      power-Large-Commutative-Ring R n (mul-Large-Commutative-Ring R x y) ＝
+      mul-Large-Commutative-Ring R
+        ( power-Large-Commutative-Ring R n x)
+        ( power-Large-Commutative-Ring R n y)
+    distributive-power-mul-Large-Commutative-Ring =
+      distributive-power-mul-Large-Commutative-Monoid
+        ( multiplicative-large-commutative-monoid-Large-Commutative-Ring R)
+```
diff --git a/src/group-theory/large-abelian-groups.lagda.md b/src/group-theory/large-abelian-groups.lagda.md
index 24301001f9c..dfc6a5a6579 100644
--- a/src/group-theory/large-abelian-groups.lagda.md
+++ b/src/group-theory/large-abelian-groups.lagda.md
@@ -432,3 +432,18 @@ module _
   emb-right-add-Large-Ab =
     emb-right-mul-Large-Group (large-group-Large-Ab G) l2 x
 ```
+
+### The raise operation is an abelian group homomorphism
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (G : Large-Ab α β)
+  (l1 l2 : Level)
+  where
+
+  hom-raise-Large-Ab :
+    hom-Ab
+      ( ab-Large-Ab G l1)
+      ( ab-Large-Ab G (l1 ⊔ l2))
+  hom-raise-Large-Ab = hom-raise-Large-Group (large-group-Large-Ab G) l1 l2
+```
diff --git a/src/group-theory/powers-of-elements-monoids.lagda.md b/src/group-theory/powers-of-elements-monoids.lagda.md
index 98992a85c57..aefcf398840 100644
--- a/src/group-theory/powers-of-elements-monoids.lagda.md
+++ b/src/group-theory/powers-of-elements-monoids.lagda.md
@@ -326,13 +326,14 @@ module _
   {l1 l2 : Level} (M : Monoid l1) (N : Monoid l2) (f : hom-Monoid M N)
   where
 
-  preserves-powers-hom-Monoid :
-    (n : ℕ) (x : type-Monoid M) →
-    map-hom-Monoid M N f (power-Monoid M n x) ＝
-    power-Monoid N n (map-hom-Monoid M N f x)
-  preserves-powers-hom-Monoid zero-ℕ x = preserves-unit-hom-Monoid M N f
-  preserves-powers-hom-Monoid (succ-ℕ zero-ℕ) x = refl
-  preserves-powers-hom-Monoid (succ-ℕ (succ-ℕ n)) x =
-    ( preserves-mul-hom-Monoid M N f) ∙
-    ( ap (mul-Monoid' N _) (preserves-powers-hom-Monoid (succ-ℕ n) x))
+  abstract
+    preserves-powers-hom-Monoid :
+      (n : ℕ) (x : type-Monoid M) →
+      map-hom-Monoid M N f (power-Monoid M n x) ＝
+      power-Monoid N n (map-hom-Monoid M N f x)
+    preserves-powers-hom-Monoid zero-ℕ x = preserves-unit-hom-Monoid M N f
+    preserves-powers-hom-Monoid (succ-ℕ zero-ℕ) x = refl
+    preserves-powers-hom-Monoid (succ-ℕ (succ-ℕ n)) x =
+      ( preserves-mul-hom-Monoid M N f) ∙
+      ( ap (mul-Monoid' N _) (preserves-powers-hom-Monoid (succ-ℕ n) x))
 ```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index dec3db12f13..436312c0d49 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -61,6 +61,7 @@ open import real-numbers.nonnegative-real-numbers public
 open import real-numbers.nonzero-real-numbers public
 open import real-numbers.positive-and-negative-real-numbers public
 open import real-numbers.positive-real-numbers public
+open import real-numbers.powers-real-numbers public
 open import real-numbers.raising-universe-levels-real-numbers public
 open import real-numbers.rational-lower-dedekind-real-numbers public
 open import real-numbers.rational-real-numbers public
diff --git a/src/real-numbers/large-additive-group-of-real-numbers.lagda.md b/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
index 949d0372206..77808a473ab 100644
--- a/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
@@ -9,9 +9,16 @@ module real-numbers.large-additive-group-of-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.dependent-pair-types
+open import foundation.identity-types
 open import foundation.universe-levels
 
 open import group-theory.abelian-groups
+open import group-theory.homomorphisms-abelian-groups
 open import group-theory.large-abelian-groups
 open import group-theory.large-commutative-monoids
 open import group-theory.large-groups
@@ -86,3 +93,17 @@ large-ab-add-ℝ =
 ab-add-ℝ : (l : Level) → Ab (lsuc l)
 ab-add-ℝ = ab-Large-Ab large-ab-add-ℝ
 ```
+
+### The canonical embedding of rational numbers in the real numbers is an abelian group homomorphism
+
+```agda
+hom-ab-add-real-ℚ : hom-Ab abelian-group-add-ℚ (ab-add-ℝ lzero)
+hom-ab-add-real-ℚ = (real-ℚ , inv (add-real-ℚ _ _))
+```
+
+### Raising the universe levels of real numbers is an abelian group homomorphism
+
+```agda
+hom-ab-add-raise-ℝ : (l1 l2 : Level) → hom-Ab (ab-add-ℝ l1) (ab-add-ℝ (l1 ⊔ l2))
+hom-ab-add-raise-ℝ = hom-raise-Large-Ab large-ab-add-ℝ
+```
diff --git a/src/real-numbers/large-ring-of-real-numbers.lagda.md b/src/real-numbers/large-ring-of-real-numbers.lagda.md
index 5e171467b3b..5786d7c56cb 100644
--- a/src/real-numbers/large-ring-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-ring-of-real-numbers.lagda.md
@@ -10,12 +10,22 @@ module real-numbers.large-ring-of-real-numbers where
 
 ```agda
 open import commutative-algebra.commutative-rings
+open import commutative-algebra.homomorphisms-commutative-rings
 open import commutative-algebra.large-commutative-rings
 
+open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.ring-of-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.dependent-pair-types
+open import foundation.identity-types
 open import foundation.universe-levels
 
+open import group-theory.large-monoids
+
 open import real-numbers.large-additive-group-of-real-numbers
 open import real-numbers.multiplication-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 
 open import ring-theory.large-rings
@@ -62,3 +72,21 @@ commutative-ring-ℝ : (l : Level) → Commutative-Ring (lsuc l)
 commutative-ring-ℝ =
   commutative-ring-Large-Commutative-Ring large-commutative-ring-ℝ
 ```
+
+### The canonical embedding of rational numbers in the real numbers is a ring homomorphism
+
+```agda
+hom-ring-real-ℚ :
+  hom-Commutative-Ring commutative-ring-ℚ (commutative-ring-ℝ lzero)
+hom-ring-real-ℚ =
+  ( hom-ab-add-real-ℚ , inv (mul-real-ℚ _ _) , eq-raise-ℝ _)
+```
+
+### Raising the universe level of real numbers is a ring homomorphism
+
+```agda
+hom-ring-raise-ℝ :
+  (l1 l2 : Level) →
+  hom-Commutative-Ring (commutative-ring-ℝ l1) (commutative-ring-ℝ (l1 ⊔ l2))
+hom-ring-raise-ℝ = hom-raise-Large-Commutative-Ring large-commutative-ring-ℝ
+```
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
new file mode 100644
index 00000000000..043a580d269
--- /dev/null
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -0,0 +1,121 @@
+# Powers of real numbers
+
+```agda
+module real-numbers.powers-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.powers-of-elements-commutative-rings
+open import commutative-algebra.powers-of-elements-large-commutative-rings
+
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.powers-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.ring-of-rational-numbers
+
+open import foundation.identity-types
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.large-ring-of-real-numbers
+open import real-numbers.multiplication-real-numbers
+open import real-numbers.rational-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising a real number to a natural number power" Agda=power-ℝ}}
+on the [real numbers](real-numbers.dedekind-real-numbers.md) `n x ↦ xⁿ`, is
+defined by [iteratively](foundation.iterating-functions.md)
+[multiplying](real-numbers.multiplication-real-numbers.md) `x` with itself `n`
+times.
+
+## Definition
+
+```agda
+power-ℝ : {l : Level} → ℕ → ℝ l → ℝ l
+power-ℝ = power-Large-Commutative-Ring large-commutative-ring-ℝ
+```
+
+## Properties
+
+### The canonical embedding of rational numbers preserves powers
+
+```agda
+abstract
+  power-real-ℚ : (n : ℕ) (q : ℚ) → power-ℝ n (real-ℚ q) ＝ real-ℚ (power-ℚ n q)
+  power-real-ℚ n q =
+    inv
+      ( preserves-powers-hom-Commutative-Ring
+        ( commutative-ring-ℚ)
+        ( commutative-ring-ℝ lzero)
+        ( hom-ring-real-ℚ)
+        ( n)
+        ( q))
+```
+
+### `1ⁿ ＝ 1`
+
+```agda
+abstract
+  power-one-ℝ : (n : ℕ) → power-ℝ n one-ℝ ＝ one-ℝ
+  power-one-ℝ = power-one-Large-Commutative-Ring large-commutative-ring-ℝ
+```
+
+### `xⁿ⁺¹ = xⁿx`
+
+```agda
+abstract
+  power-succ-ℝ :
+    {l : Level} (n : ℕ) (x : ℝ l) → power-ℝ (succ-ℕ n) x ＝ power-ℝ n x *ℝ x
+  power-succ-ℝ = power-succ-Large-Commutative-Ring large-commutative-ring-ℝ
+```
+
+### `xⁿ⁺¹ = xxⁿ`
+
+```agda
+abstract
+  power-succ-ℝ' :
+    {l : Level} (n : ℕ) (x : ℝ l) → power-ℝ (succ-ℕ n) x ＝ x *ℝ power-ℝ n x
+  power-succ-ℝ' = power-succ-Large-Commutative-Ring' large-commutative-ring-ℝ
+```
+
+### Powers by sums of natural numbers are products of powers
+
+```agda
+abstract
+  distributive-power-add-ℝ :
+    {l : Level} (m n : ℕ) {x : ℝ l} →
+    power-ℝ (m +ℕ n) x ＝ power-ℝ m x *ℝ power-ℝ n x
+  distributive-power-add-ℝ =
+    distributive-power-add-Large-Commutative-Ring large-commutative-ring-ℝ
+```
+
+### Powers by products of natural numbers are iterated powers
+
+```agda
+abstract
+  power-mul-ℝ :
+    {l : Level} (m n : ℕ) {x : ℝ l} →
+    power-ℝ (m *ℕ n) x ＝ power-ℝ n (power-ℝ m x)
+  power-mul-ℝ =
+    power-mul-Large-Commutative-Ring large-commutative-ring-ℝ
+```
+
+### `(xy)ⁿ = xⁿyⁿ`
+
+```agda
+abstract
+  distributive-power-mul-ℝ :
+    {l1 l2 : Level} (n : ℕ) {x : ℝ l1} {y : ℝ l2} →
+    power-ℝ n (x *ℝ y) ＝ power-ℝ n x *ℝ power-ℝ n y
+  distributive-power-mul-ℝ =
+    distributive-power-mul-Large-Commutative-Ring large-commutative-ring-ℝ
+```
diff --git a/src/ring-theory.lagda.md b/src/ring-theory.lagda.md
index 945dff35bf9..4c47fab0905 100644
--- a/src/ring-theory.lagda.md
+++ b/src/ring-theory.lagda.md
@@ -73,6 +73,7 @@ open import ring-theory.poset-of-cyclic-rings public
 open import ring-theory.poset-of-ideals-rings public
 open import ring-theory.poset-of-left-ideals-rings public
 open import ring-theory.poset-of-right-ideals-rings public
+open import ring-theory.powers-of-elements-large-rings public
 open import ring-theory.powers-of-elements-rings public
 open import ring-theory.powers-of-elements-semirings public
 open import ring-theory.precategory-of-rings public
diff --git a/src/ring-theory/large-rings.lagda.md b/src/ring-theory/large-rings.lagda.md
index 488461591db..4189d3517de 100644
--- a/src/ring-theory/large-rings.lagda.md
+++ b/src/ring-theory/large-rings.lagda.md
@@ -19,6 +19,7 @@ open import group-theory.large-monoids
 open import group-theory.large-semigroups
 open import group-theory.monoids
 
+open import ring-theory.homomorphisms-rings
 open import ring-theory.rings
 ```
 
@@ -289,3 +290,25 @@ module _
       left-distributive-mul-add-Large-Ring R ,
       right-distributive-mul-add-Large-Ring R)
 ```
+
+### The raise operation is a ring homomorphism
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  (l1 l2 : Level)
+  where
+
+  hom-raise-Large-Ring :
+    hom-Ring
+      ( ring-Large-Ring R l1)
+      ( ring-Large-Ring R (l1 ⊔ l2))
+  hom-raise-Large-Ring =
+    ( hom-raise-Large-Ab (large-ab-Large-Ring R) l1 l2 ,
+      inv
+        ( raise-mul-Large-Monoid
+          ( multiplicative-large-monoid-Large-Ring R)
+          ( _)
+          ( _)) ,
+      raise-raise-Large-Ring R _)
+```
diff --git a/src/ring-theory/powers-of-elements-large-rings.lagda.md b/src/ring-theory/powers-of-elements-large-rings.lagda.md
new file mode 100644
index 00000000000..7552fe1a2a9
--- /dev/null
+++ b/src/ring-theory/powers-of-elements-large-rings.lagda.md
@@ -0,0 +1,154 @@
+# Powers of elements in large rings
+
+```agda
+module ring-theory.powers-of-elements-large-rings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.natural-numbers
+
+open import foundation.identity-types
+open import foundation.universe-levels
+
+open import group-theory.large-monoids
+open import group-theory.powers-of-elements-large-monoids
+
+open import ring-theory.large-rings
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising an element of a large ring to a natural number power" Agda=power-Large-Ring}}
+on a [large ring](ring-theory.large-rings.md) is the map `n x ↦ xⁿ`, which is
+defined by [iteratively](foundation.iterating-functions.md) multiplying `x` with
+itself `n` times.
+
+## Definition
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  where
+
+  power-Large-Ring : {l : Level} → ℕ → type-Large-Ring R l → type-Large-Ring R l
+  power-Large-Ring =
+    power-Large-Monoid (multiplicative-large-monoid-Large-Ring R)
+```
+
+## Properties
+
+### The power operation preserves similarity
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  where
+
+  abstract
+    preserves-sim-power-Large-Ring :
+      {l1 l2 : Level} (n : ℕ) →
+      (x : type-Large-Ring R l1) (y : type-Large-Ring R l2) →
+      sim-Large-Ring R x y →
+      sim-Large-Ring R (power-Large-Ring R n x) (power-Large-Ring R n y)
+    preserves-sim-power-Large-Ring =
+      preserves-sim-power-Large-Monoid
+        ( multiplicative-large-monoid-Large-Ring R)
+```
+
+### `1ⁿ = 1`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  where
+
+  abstract
+    raise-power-one-Large-Ring :
+      (l : Level) (n : ℕ) →
+      power-Large-Ring R n (raise-one-Large-Ring R l) ＝
+      raise-one-Large-Ring R l
+    raise-power-one-Large-Ring =
+      raise-power-unit-Large-Monoid
+        ( multiplicative-large-monoid-Large-Ring R)
+
+    power-one-Large-Ring :
+      (n : ℕ) → power-Large-Ring R n (one-Large-Ring R) ＝ one-Large-Ring R
+    power-one-Large-Ring =
+      power-unit-Large-Monoid
+        ( multiplicative-large-monoid-Large-Ring R)
+```
+
+### `xⁿ⁺¹ = xⁿx`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  where
+
+  abstract
+    power-succ-Large-Ring :
+      {l : Level} (n : ℕ) (x : type-Large-Ring R l) →
+      power-Large-Ring R (succ-ℕ n) x ＝
+      mul-Large-Ring R (power-Large-Ring R n x) x
+    power-succ-Large-Ring =
+      power-succ-Large-Monoid
+        ( multiplicative-large-monoid-Large-Ring R)
+```
+
+### `xⁿ⁺¹ = xxⁿ`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  where
+
+  abstract
+    power-succ-Large-Ring' :
+      {l : Level} (n : ℕ) (x : type-Large-Ring R l) →
+      power-Large-Ring R (succ-ℕ n) x ＝
+      mul-Large-Ring R x (power-Large-Ring R n x)
+    power-succ-Large-Ring' =
+      power-succ-Large-Monoid'
+        ( multiplicative-large-monoid-Large-Ring R)
+```
+
+### Powers by sums of natural numbers are products of powers
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  where
+
+  abstract
+    distributive-power-add-Large-Ring :
+      {l : Level} (m n : ℕ) {x : type-Large-Ring R l} →
+      power-Large-Ring R (m +ℕ n) x ＝
+      mul-Large-Ring R (power-Large-Ring R m x) (power-Large-Ring R n x)
+    distributive-power-add-Large-Ring =
+      distributive-power-add-Large-Monoid
+        ( multiplicative-large-monoid-Large-Ring R)
+```
+
+### Powers by products of natural numbers are iterated powers
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level} (R : Large-Ring α β)
+  where
+
+  abstract
+    power-mul-Large-Ring :
+      {l : Level} (m n : ℕ) {x : type-Large-Ring R l} →
+      power-Large-Ring R (m *ℕ n) x ＝
+      power-Large-Ring R n (power-Large-Ring R m x)
+    power-mul-Large-Ring =
+      power-mul-Large-Monoid
+        ( multiplicative-large-monoid-Large-Ring R)
+```
diff --git a/src/ring-theory/powers-of-elements-rings.lagda.md b/src/ring-theory/powers-of-elements-rings.lagda.md
index 3143f9edd75..4e431cf8260 100644
--- a/src/ring-theory/powers-of-elements-rings.lagda.md
+++ b/src/ring-theory/powers-of-elements-rings.lagda.md
@@ -18,7 +18,10 @@ open import foundation.function-types
 open import foundation.identity-types
 open import foundation.universe-levels
 
+open import group-theory.powers-of-elements-monoids
+
 open import ring-theory.central-elements-rings
+open import ring-theory.homomorphisms-rings
 open import ring-theory.powers-of-elements-semirings
 open import ring-theory.rings
 ```
@@ -189,3 +192,22 @@ module _
                   ( ap (neg-Ring R) (inv (power-succ-Ring R n x)))))) ∙
             ( neg-neg-Ring R (power-Ring R (succ-ℕ n) x))))))
 ```
+
+### Ring homomorphisms preserve powers of elements
+
+```agda
+module _
+  {l1 l2 : Level} (R : Ring l1) (S : Ring l2) (f : hom-Ring R S)
+  where
+
+  abstract
+    preserves-powers-hom-Ring :
+      (n : ℕ) (x : type-Ring R) →
+      map-hom-Ring R S f (power-Ring R n x) ＝
+      power-Ring S n (map-hom-Ring R S f x)
+    preserves-powers-hom-Ring =
+      preserves-powers-hom-Monoid
+        ( multiplicative-monoid-Ring R)
+        ( multiplicative-monoid-Ring S)
+        ( hom-multiplicative-monoid-hom-Ring R S f)
+```

From d2bfcb7560c704f8eb036a1fdf5bca86e51f9077 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 18:40:52 -0800
Subject: [PATCH 029/134] Powers of real numbers

---
 .../least-upper-bounds-large-posets.lagda.md  |  27 +++
 src/real-numbers.lagda.md                     |   1 +
 .../absolute-value-real-numbers.lagda.md      |  23 ++
 .../binary-maximum-real-numbers.lagda.md      |  23 ++
 ...ositive-and-negative-real-numbers.lagda.md |  53 +++++
 .../positive-real-numbers.lagda.md            |  20 ++
 src/real-numbers/powers-real-numbers.lagda.md | 204 ++++++++++++++++++
 .../squares-real-numbers.lagda.md             |  30 +++
 8 files changed, 381 insertions(+)
 create mode 100644 src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md

diff --git a/src/order-theory/least-upper-bounds-large-posets.lagda.md b/src/order-theory/least-upper-bounds-large-posets.lagda.md
index be741937000..dfa918f385f 100644
--- a/src/order-theory/least-upper-bounds-large-posets.lagda.md
+++ b/src/order-theory/least-upper-bounds-large-posets.lagda.md
@@ -346,3 +346,30 @@ module _
   is-least-upper-bound-family-of-elements-poset-Large-Poset y is-lub-y z =
     is-lub-y z
 ```
+
+### If `x ≤ y`, `y` is a least upper bound of `x` and `y`
+
+```agda
+module _
+  {α : Level → Level} {β : Level → Level → Level}
+  (P : Large-Poset α β)
+  where
+
+  abstract
+    left-leq-right-least-upper-bound-Large-Poset :
+      {l1 l2 : Level} (x : type-Large-Poset P l1) (y : type-Large-Poset P l2) →
+      leq-Large-Poset P x y →
+      is-least-binary-upper-bound-Large-Poset P x y y
+    pr1 (left-leq-right-least-upper-bound-Large-Poset x y x≤y z) (x≤z , y≤z) =
+      y≤z
+    pr2 (left-leq-right-least-upper-bound-Large-Poset x y x≤y z) y≤z =
+      ( transitive-leq-Large-Poset P x y z y≤z x≤y , y≤z)
+
+    right-leq-left-least-upper-bound-Large-Poset :
+      {l1 l2 : Level} (x : type-Large-Poset P l1) (y : type-Large-Poset P l2) →
+      leq-Large-Poset P y x →
+      is-least-binary-upper-bound-Large-Poset P x y x
+    right-leq-left-least-upper-bound-Large-Poset x y y≤x =
+      is-binary-least-upper-bound-swap-Large-Poset P y x x
+        ( left-leq-right-least-upper-bound-Large-Poset y x y≤x)
+```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 436312c0d49..3e2872c1e15 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -49,6 +49,7 @@ open import real-numbers.minimum-lower-dedekind-real-numbers public
 open import real-numbers.minimum-upper-dedekind-real-numbers public
 open import real-numbers.multiplication-negative-real-numbers public
 open import real-numbers.multiplication-nonnegative-real-numbers public
+open import real-numbers.multiplication-positive-and-negative-real-numbers public
 open import real-numbers.multiplication-positive-real-numbers public
 open import real-numbers.multiplication-real-numbers public
 open import real-numbers.multiplicative-inverses-negative-real-numbers public
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index 8695d96e159..b2b49882e2c 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -34,6 +34,8 @@ open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
+open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.saturation-inequality-real-numbers
 open import real-numbers.similarity-real-numbers
@@ -113,6 +115,27 @@ opaque
     ap (max-ℝ (neg-ℝ x)) (neg-neg-ℝ x) ∙ commutative-max-ℝ _ _
 ```
 
+### The absolute value of a nonnegative real number is itself
+
+```agda
+abstract opaque
+  unfolding abs-ℝ
+
+  abs-real-ℝ⁰⁺ : {l : Level} (x : ℝ⁰⁺ l) → abs-ℝ (real-ℝ⁰⁺ x) ＝ real-ℝ⁰⁺ x
+  abs-real-ℝ⁰⁺ (x , 0≤x) =
+    eq-sim-ℝ
+      ( right-leq-left-max-ℝ
+        ( transitive-leq-ℝ
+          ( neg-ℝ x)
+          ( zero-ℝ)
+          ( x)
+          ( 0≤x)
+          ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ _ _ 0≤x))))
+
+  abs-real-ℝ⁺ : {l : Level} (x : ℝ⁺ l) → abs-ℝ (real-ℝ⁺ x) ＝ real-ℝ⁺ x
+  abs-real-ℝ⁺ x = abs-real-ℝ⁰⁺ (nonnegative-ℝ⁺ x)
+```
+
 ### `x` is between `-|x|` and `|x|`
 
 ```agda
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 7d521f85c1b..4a127ff1164 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -305,6 +305,29 @@ abstract
   preserves-sim-right-max-ℝ a = preserves-sim-max-ℝ a a (refl-sim-ℝ a)
 ```
 
+### If `x ≤ y`, the maximum of `x` and `y` is similar to `y`
+
+```agda
+abstract
+  left-leq-right-max-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → leq-ℝ x y →
+    sim-ℝ (max-ℝ x y) y
+  left-leq-right-max-ℝ {x = x} {y = y} x≤y =
+    sim-sim-leq-ℝ
+      ( sim-is-least-binary-upper-bound-Large-Poset ℝ-Large-Poset x y
+        ( is-least-binary-upper-bound-max-ℝ x y)
+        ( left-leq-right-least-upper-bound-Large-Poset ℝ-Large-Poset x y x≤y))
+
+  right-leq-left-max-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → leq-ℝ y x →
+    sim-ℝ (max-ℝ x y) x
+  right-leq-left-max-ℝ {x = x} {y = y} y≤x =
+    sim-sim-leq-ℝ
+      ( sim-is-least-binary-upper-bound-Large-Poset ℝ-Large-Poset x y
+        ( is-least-binary-upper-bound-max-ℝ x y)
+        ( right-leq-left-least-upper-bound-Large-Poset ℝ-Large-Poset x y y≤x))
+```
+
 ### The binary maximum preserves lower neighborhoods
 
 ```agda
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
new file mode 100644
index 00000000000..7ecc0768e38
--- /dev/null
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -0,0 +1,53 @@
+# Multiplication by positive, negative, and nonnegative real numbers
+
+```agda
+module real-numbers.multiplication-positive-and-negative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.dependent-pair-types
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.multiplication-positive-real-numbers
+open import real-numbers.multiplication-real-numbers
+open import real-numbers.negative-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+When we have information about the sign of the factors of a
+[product](real-numbers.multiplication-real-numbers.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md), we can deduce the sign of
+their product too.
+
+## Lemmas
+
+### The product of a positive and a negative rational number is negative
+
+```agda
+abstract
+  is-negative-mul-positive-negative-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → is-positive-ℝ x → is-negative-ℝ y →
+    is-negative-ℝ (x *ℝ y)
+  is-negative-mul-positive-negative-ℝ {x = x} {y = y} is-pos-x is-neg-y =
+    preserves-le-right-sim-ℝ
+      ( x *ℝ y)
+      ( x *ℝ zero-ℝ)
+      ( zero-ℝ)
+      ( right-zero-law-mul-ℝ x)
+      ( preserves-le-left-mul-ℝ⁺ (x , is-pos-x) y zero-ℝ is-neg-y)
+
+mul-positive-negative-ℝ :
+  {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁻ l2 → ℝ⁻ (l1 ⊔ l2)
+mul-positive-negative-ℝ (x , is-pos-x) (y , is-neg-y) =
+  ( x *ℝ y , is-negative-mul-positive-negative-ℝ is-pos-x is-neg-y)
+```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 1891f071d65..004dbd53454 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -37,6 +37,7 @@ open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
@@ -80,6 +81,17 @@ is-positive-real-ℝ⁺ = pr2
 
 ## Properties
 
+### Positivity is preserved by similarity
+
+```agda
+abstract
+  is-positive-sim-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} →
+    is-positive-ℝ x → sim-ℝ x y → is-positive-ℝ y
+  is-positive-sim-ℝ {x = x} {y = y} 0<x x~y =
+    preserves-le-right-sim-ℝ zero-ℝ x y x~y 0<x
+```
+
 ### Dedekind cuts of positive real numbers
 
 ```agda
@@ -297,3 +309,11 @@ sim-prop-ℝ⁺ (x , _) (y , _) = sim-prop-ℝ x y
 sim-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → UU (l1 ⊔ l2)
 sim-ℝ⁺ (x , _) (y , _) = sim-ℝ x y
 ```
+
+### Raising the universe level of positive real numbers
+
+```agda
+raise-ℝ⁺ : {l1 : Level} (l : Level) → ℝ⁺ l1 → ℝ⁺ (l ⊔ l1)
+raise-ℝ⁺ l (x , 0<x) =
+  ( raise-ℝ l x , preserves-le-right-sim-ℝ zero-ℝ x _ (sim-raise-ℝ _ _) 0<x)
+```
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 043a580d269..aee3f51d45b 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -1,6 +1,8 @@
 # Powers of real numbers
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module real-numbers.powers-real-numbers where
 ```
 
@@ -13,17 +15,36 @@ open import commutative-algebra.powers-of-elements-large-commutative-rings
 open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.multiplication-natural-numbers
 open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.parity-natural-numbers
 open import elementary-number-theory.powers-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.ring-of-rational-numbers
 
+open import foundation.action-on-identifications-functions
+open import foundation.dependent-pair-types
+open import foundation.disjunction
 open import foundation.identity-types
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
+open import order-theory.large-posets
+
+open import real-numbers.absolute-value-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-real-numbers
 open import real-numbers.large-ring-of-real-numbers
+open import real-numbers.multiplication-nonnegative-real-numbers
+open import real-numbers.multiplication-positive-and-negative-real-numbers
+open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.negative-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.squares-real-numbers
 ```
 
 </details>
@@ -107,6 +128,17 @@ abstract
     power-ℝ (m *ℕ n) x ＝ power-ℝ n (power-ℝ m x)
   power-mul-ℝ =
     power-mul-Large-Commutative-Ring large-commutative-ring-ℝ
+
+  power-mul-ℝ' :
+    {l : Level} (m n : ℕ) {x : ℝ l} →
+    power-ℝ (m *ℕ n) x ＝ power-ℝ m (power-ℝ n x)
+  power-mul-ℝ' m n {x = x} =
+    equational-reasoning
+      power-ℝ (m *ℕ n) x
+      ＝ power-ℝ (n *ℕ m) x
+        by ap (λ k → power-ℝ k x) (commutative-mul-ℕ m n)
+      ＝ power-ℝ m (power-ℝ n x)
+        by power-mul-ℝ n m
 ```
 
 ### `(xy)ⁿ = xⁿyⁿ`
@@ -119,3 +151,175 @@ abstract
   distributive-power-mul-ℝ =
     distributive-power-mul-Large-Commutative-Ring large-commutative-ring-ℝ
 ```
+
+### Even powers of real numbers are nonnegative
+
+```agda
+abstract
+  is-nonnegative-even-power-ℝ :
+    {l : Level} (n : ℕ) (x : ℝ l) → is-even-ℕ n → is-nonnegative-ℝ (power-ℝ n x)
+  is-nonnegative-even-power-ℝ _ x (k , refl) =
+    inv-tr
+      ( is-nonnegative-ℝ)
+      ( power-mul-ℝ k 2)
+      ( is-nonnegative-square-ℝ (power-ℝ k x))
+
+nonnegative-even-power-ℝ : {l : Level} (n : ℕ) (x : ℝ l) → is-even-ℕ n → ℝ⁰⁺ l
+nonnegative-even-power-ℝ n x even-n =
+  ( power-ℝ n x , is-nonnegative-even-power-ℝ n x even-n)
+```
+
+### Powers of positive real numbers are positive
+
+```agda
+abstract
+  is-positive-power-ℝ⁺ :
+    {l : Level} (n : ℕ) (x : ℝ⁺ l) → is-positive-ℝ (power-ℝ n (real-ℝ⁺ x))
+  is-positive-power-ℝ⁺ 0 _ =
+    is-positive-sim-ℝ (is-positive-real-ℝ⁺ one-ℝ⁺) (sim-raise-ℝ _ _)
+  is-positive-power-ℝ⁺ 1 (_ , is-pos-x) = is-pos-x
+  is-positive-power-ℝ⁺ (succ-ℕ n@(succ-ℕ _)) x⁺@(x , is-pos-x) =
+    is-positive-mul-ℝ (is-positive-power-ℝ⁺ n x⁺) is-pos-x
+```
+
+### Even powers of negative real numbers are positive
+
+```agda
+abstract
+  is-positive-even-power-ℝ⁻ :
+    {l : Level} (n : ℕ) (x : ℝ⁻ l) → is-even-ℕ n →
+    is-positive-ℝ (power-ℝ n (real-ℝ⁻ x))
+  is-positive-even-power-ℝ⁻ _ x (k , refl) =
+    inv-tr
+      ( is-positive-ℝ)
+      ( power-mul-ℝ' k 2)
+      ( is-positive-power-ℝ⁺ k (square-ℝ⁻ x))
+```
+
+### Odd powers of negative real numbers are negative
+
+```agda
+abstract
+  is-negative-odd-power-ℝ⁻ :
+    {l : Level} (n : ℕ) (x : ℝ⁻ l) → is-odd-ℕ n →
+    is-negative-ℝ (power-ℝ n (real-ℝ⁻ x))
+  is-negative-odd-power-ℝ⁻ n x⁻@(x , is-neg-x) odd-n =
+    let (k , k2+1=n) = has-odd-expansion-is-odd n odd-n
+    in
+      tr
+        ( is-negative-ℝ)
+        ( equational-reasoning
+          power-ℝ (k *ℕ 2) x *ℝ x
+          ＝ power-ℝ (succ-ℕ (k *ℕ 2)) x
+            by inv (power-succ-ℝ _ _)
+          ＝ power-ℝ n x
+            by ap (λ m → power-ℝ m x) k2+1=n)
+        ( is-negative-mul-positive-negative-ℝ
+          ( is-positive-even-power-ℝ⁻ _ x⁻ (k , refl))
+          ( is-neg-x))
+```
+
+### For even `n`, `(-x)ⁿ ＝ xⁿ`
+
+```agda
+abstract
+  even-power-neg-ℝ :
+    {l : Level} (n : ℕ) (x : ℝ l) → is-even-ℕ n →
+    power-ℝ n (neg-ℝ x) ＝ power-ℝ n x
+  even-power-neg-ℝ _ x (k , refl) =
+    equational-reasoning
+      power-ℝ (k *ℕ 2) (neg-ℝ x)
+      ＝ power-ℝ k (square-ℝ (neg-ℝ x))
+        by power-mul-ℝ' k 2
+      ＝ power-ℝ k (square-ℝ x)
+        by ap (power-ℝ k) (square-neg-ℝ x)
+      ＝ power-ℝ (k *ℕ 2) x
+        by inv (power-mul-ℝ' k 2)
+```
+
+### For odd `n`, `(-x)ⁿ = -(xⁿ)`
+
+```agda
+abstract
+  odd-power-neg-ℝ :
+    {l : Level} (n : ℕ) (x : ℝ l) → is-odd-ℕ n →
+    power-ℝ n (neg-ℝ x) ＝ neg-ℝ (power-ℝ n x)
+  odd-power-neg-ℝ n x odd-n =
+    let (k , k2+1=n) = has-odd-expansion-is-odd n odd-n
+    in
+      equational-reasoning
+        power-ℝ n (neg-ℝ x)
+        ＝ power-ℝ (succ-ℕ (k *ℕ 2)) (neg-ℝ x)
+          by ap (λ m → power-ℝ m (neg-ℝ x)) (inv k2+1=n)
+        ＝ power-ℝ (k *ℕ 2) (neg-ℝ x) *ℝ neg-ℝ x
+          by power-succ-ℝ _ _
+        ＝ power-ℝ (k *ℕ 2) x *ℝ neg-ℝ x
+          by ap-mul-ℝ (even-power-neg-ℝ _ x (k , refl)) refl
+        ＝ neg-ℝ (power-ℝ (k *ℕ 2) x *ℝ x)
+          by right-negative-law-mul-ℝ _ _
+        ＝ neg-ℝ (power-ℝ (succ-ℕ (k *ℕ 2)) x)
+          by ap neg-ℝ (inv (power-succ-ℝ _ _))
+        ＝ neg-ℝ (power-ℝ n x)
+          by ap neg-ℝ (ap (λ m → power-ℝ m x) k2+1=n)
+```
+
+### `|x|ⁿ=|xⁿ|`
+
+```agda
+abstract
+  power-abs-ℝ :
+    {l : Level} (n : ℕ) (x : ℝ l) → power-ℝ n (abs-ℝ x) ＝ abs-ℝ (power-ℝ n x)
+  power-abs-ℝ 0 x = inv (abs-real-ℝ⁺ (raise-ℝ⁺ _ one-ℝ⁺))
+  power-abs-ℝ (succ-ℕ n) x =
+    equational-reasoning
+      power-ℝ (succ-ℕ n) (abs-ℝ x)
+      ＝ power-ℝ n (abs-ℝ x) *ℝ abs-ℝ x
+        by power-succ-ℝ _ _
+      ＝ abs-ℝ (power-ℝ n x) *ℝ abs-ℝ x
+        by ap-mul-ℝ (power-abs-ℝ n x) refl
+      ＝ abs-ℝ (power-ℝ n x *ℝ x)
+        by inv (abs-mul-ℝ _ _)
+      ＝ abs-ℝ (power-ℝ (succ-ℕ n) x)
+        by ap abs-ℝ (inv (power-succ-ℝ n x))
+```
+
+### If `|p| ≤ |q|`, `|pⁿ| ≤ |qⁿ|`
+
+```agda
+abstract
+  preserves-leq-abs-power-ℝ :
+    {l1 l2 : Level} (n : ℕ) (x : ℝ l1) (y : ℝ l2) → leq-ℝ (abs-ℝ x) (abs-ℝ y) →
+    leq-ℝ (abs-ℝ (power-ℝ n x)) (abs-ℝ (power-ℝ n y))
+  preserves-leq-abs-power-ℝ 0 _ _ _ =
+    preserves-leq-sim-ℝ _ _ _ _
+      ( inv-tr
+        ( sim-ℝ one-ℝ)
+        ( abs-real-ℝ⁺ (raise-ℝ⁺ _ one-ℝ⁺))
+        ( sim-raise-ℝ _ one-ℝ))
+      ( inv-tr
+        ( sim-ℝ one-ℝ)
+        ( abs-real-ℝ⁺ (raise-ℝ⁺ _ one-ℝ⁺))
+        ( sim-raise-ℝ _ one-ℝ))
+      ( refl-leq-ℝ one-ℝ)
+  preserves-leq-abs-power-ℝ (succ-ℕ n) x y |x|≤|y| =
+    let open inequality-reasoning-Large-Poset ℝ-Large-Poset
+    in chain-of-inequalities
+      abs-ℝ (power-ℝ (succ-ℕ n) x)
+      ≤ abs-ℝ (power-ℝ n x *ℝ x)
+        by leq-eq-ℝ _ _ (ap abs-ℝ (power-succ-ℝ n x))
+      ≤ abs-ℝ (power-ℝ n x) *ℝ abs-ℝ x
+        by leq-eq-ℝ _ _ (abs-mul-ℝ _ _)
+      ≤ abs-ℝ (power-ℝ n y) *ℝ abs-ℝ y
+        by
+          preserves-leq-mul-ℝ⁰⁺
+            ( nonnegative-abs-ℝ (power-ℝ n x))
+            ( nonnegative-abs-ℝ (power-ℝ n y))
+            ( nonnegative-abs-ℝ x)
+            ( nonnegative-abs-ℝ y)
+            ( preserves-leq-abs-power-ℝ n x y |x|≤|y|)
+            ( |x|≤|y|)
+      ≤ abs-ℝ (power-ℝ n y *ℝ y)
+        by leq-eq-ℝ _ _ (inv (abs-mul-ℝ _ _))
+      ≤ abs-ℝ (power-ℝ (succ-ℕ n) y)
+        by leq-eq-ℝ _ _ (ap abs-ℝ (inv (power-succ-ℝ n y)))
+```
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 7c9c42cc2d9..992433dcb1c 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -1,6 +1,8 @@
 # Squares of real numbers
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module real-numbers.squares-real-numbers where
 ```
 
@@ -42,8 +44,10 @@ open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
+open import real-numbers.negative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
+open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
@@ -181,6 +185,32 @@ abstract
   distributive-square-mul-ℝ x y = interchange-law-mul-mul-ℝ x y x y
 ```
 
+### The square of a positive real number is positive
+
+```agda
+abstract
+  is-positive-square-ℝ⁺ :
+    {l : Level} (x : ℝ⁺ l) → is-positive-ℝ (square-ℝ (real-ℝ⁺ x))
+  is-positive-square-ℝ⁺ (x , is-pos-x) =
+    is-positive-mul-ℝ is-pos-x is-pos-x
+```
+
+### The square of a negative real number is positive
+
+```agda
+abstract
+  is-positive-square-ℝ⁻ :
+    {l : Level} (x : ℝ⁻ l) → is-positive-ℝ (square-ℝ (real-ℝ⁻ x))
+  is-positive-square-ℝ⁻ x⁻@(x , _) =
+    tr
+      ( is-positive-ℝ)
+      ( square-neg-ℝ x)
+      ( is-positive-square-ℝ⁺ (neg-ℝ⁻ x⁻))
+
+square-ℝ⁻ : {l : Level} → ℝ⁻ l → ℝ⁺ l
+square-ℝ⁻ x⁻@(x , _) = (square-ℝ x , is-positive-square-ℝ⁻ x⁻)
+```
+
 ### For nonnegative real numbers, squaring preserves strict inequality
 
 ```agda

From a0e010cc4b0cfb1be171f43ddd6db2366348f6ce Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 19:26:52 -0800
Subject: [PATCH 030/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../geometric-sequences-commutative-rings.lagda.md | 14 +++++++-------
 .../geometric-sequences-rational-numbers.lagda.md  |  2 +-
 src/ring-theory/geometric-sequences-rings.lagda.md |  8 ++++----
 3 files changed, 12 insertions(+), 12 deletions(-)

diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
index ee6b868c9e9..1c27ccb4a0b 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -37,17 +37,17 @@ open import lists.sequences
 ## Ideas
 
 A
-{{#concept "geometric sequence" Disambiguation="in a commutative semiring" Agda=geometric-sequence-Commutative-Ring}}
-in a [semiring](ring-theory.semirings.md) is an
+{{#concept "geometric sequence" Disambiguation="in a commutative ring" Agda=geometric-sequence-Commutative-Ring}}
+in a [commutative ring](ring-theory.commutative-rings.md) is an
 [geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
-semiring multiplicative [semigroup](group-theory.semigroups.md).
+ring's multiplicative [semigroup](group-theory.semigroups.md).
 
 These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
-semiring.
+ring.
 
 ## Definitions
 
-### Geometric sequences in semirings
+### Geometric sequences in commutative rings
 
 ```agda
 module _
@@ -112,7 +112,7 @@ module _
       ( u)
 ```
 
-### The standard geometric sequences in a semiring
+### The standard geometric sequences in a commutative ring
 
 The standard geometric sequence with initial term `a` and common factor `r` is
 the sequence `u` defined by:
@@ -180,7 +180,7 @@ module _
 
 ## Properties
 
-### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+### Any geometric sequence is homotopic to a standard geometric sequence
 
 ```agda
 module _
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 747150c04bb..01a86c46296 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -117,7 +117,7 @@ module _
             by right-zero-law-mul-ℚ a)
     compute-sum-standard-geometric-fin-sequence-ℚ (succ-ℕ n) =
       let
-        1/1-r = rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
+        1/⟨1-r⟩ = rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
       in
         equational-reasoning
           sum-standard-geometric-fin-sequence-ℚ a r (succ-ℕ n)
diff --git a/src/ring-theory/geometric-sequences-rings.lagda.md b/src/ring-theory/geometric-sequences-rings.lagda.md
index 68d24fa3921..700adbe9ee0 100644
--- a/src/ring-theory/geometric-sequences-rings.lagda.md
+++ b/src/ring-theory/geometric-sequences-rings.lagda.md
@@ -34,12 +34,12 @@ open import ring-theory.rings
 
 A
 {{#concept "geometric sequence" Disambiguation="in a ring" Agda=geometric-sequence-Ring}}
-in a [ring](ring-theory.semirings.md) is a
+in a [ring](ring-theory.rings.md) is a
 [geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
 underlying [semiring](ring-theory.semirings.md).
 
 These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
-semiring.
+ring.
 
 ## Definitions
 
@@ -86,7 +86,7 @@ module _
       ( u)
 ```
 
-### The standard geometric sequences in a semiring
+### The standard geometric sequences in a ring
 
 The standard geometric sequence with initial term `a` and common factor `r` is
 the sequence `u` defined by:
@@ -138,7 +138,7 @@ module _
 
 ## Properties
 
-### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+### Any geometric sequence is homotopic to a standard geometric sequence
 
 ```agda
 module _

From dc3c1b3213598c4a6046c1426b1f06bc58233cce Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 19:29:02 -0800
Subject: [PATCH 031/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../geometric-sequences-commutative-semirings.lagda.md | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
index 821a30d43c3..5e011a77f93 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
@@ -43,16 +43,16 @@ open import univalent-combinatorics.standard-finite-types
 
 A
 {{#concept "geometric sequence" Disambiguation="in a commutative semiring" Agda=geometric-sequence-Commutative-Semiring}}
-in a [semiring](ring-theory.semirings.md) is an
+in a [commutative semiring](ring-theory.commutative-semirings.md) is an
 [geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
-semiring multiplicative [semigroup](group-theory.semigroups.md).
+semiring's multiplicative [semigroup](group-theory.semigroups.md).
 
 These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
 semiring.
 
 ## Definitions
 
-### Geometric sequences in semirings
+### Geometric sequences in commutative semirings
 
 ```agda
 module _
@@ -116,7 +116,7 @@ module _
       ( u)
 ```
 
-### The standard geometric sequences in a semiring
+### The standard geometric sequences in a commutative semiring
 
 The standard geometric sequence with initial term `a` and common factor `r` is
 the sequence `u` defined by:
@@ -182,7 +182,7 @@ module _
 
 ## Properties
 
-### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+### Any geometric sequence is homotopic to a standard geometric sequence
 
 ```agda
 module _

From b61e819a3da9a0031a0153675176b59e91441bb3 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 19:33:34 -0800
Subject: [PATCH 032/134] Update

---
 ...c-sequences-commutative-semirings.lagda.md | 39 +++++++++++--------
 ...metric-sequences-rational-numbers.lagda.md | 18 ++++-----
 2 files changed, 32 insertions(+), 25 deletions(-)

diff --git a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
index 5e011a77f93..d1beb4c07ee 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
@@ -309,9 +309,18 @@ module _
           ( succ-ℕ n)
         ＝
           add-Commutative-Semiring R
-            ( term-Fin a r (succ-ℕ n) (zero-Fin n))
+            ( standard-geometric-fin-sequence-Commutative-Semiring R
+              ( a)
+              ( r)
+              ( succ-ℕ n)
+              ( zero-Fin n))
             ( sum-fin-sequence-type-Commutative-Semiring R n
-              ( λ i → term-Fin a r (succ-ℕ n) (inr-Fin n i)))
+              ( λ i →
+                standard-geometric-fin-sequence-Commutative-Semiring R
+                  ( a)
+                  ( r)
+                  ( succ-ℕ n)
+                  ( inr-Fin n i)))
           by
             snoc-sum-fin-sequence-type-Commutative-Semiring R n
               ( standard-geometric-fin-sequence-Commutative-Semiring R
@@ -321,16 +330,25 @@ module _
               ( refl)
         ＝
           add-Commutative-Semiring R
-            ( term a r 0)
+            ( seq-standard-geometric-sequence-Commutative-Semiring R a r 0)
             ( sum-fin-sequence-type-Commutative-Semiring R n
-              ( λ i → term a r (succ-ℕ (nat-Fin n i))))
+              ( λ i →
+                seq-standard-geometric-sequence-Commutative-Semiring R
+                  ( a)
+                  ( r)
+                  ( succ-ℕ (nat-Fin n i))))
           by
             ap-add-Commutative-Semiring R
               ( ap
                 ( seq-standard-geometric-sequence-Commutative-Semiring R a r)
                 ( is-zero-nat-zero-Fin {n}))
               ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
-                ( λ i → ap (term a r) (nat-inr-Fin n i)))
+                ( λ i →
+                  ap
+                    ( seq-standard-geometric-sequence-Commutative-Semiring R
+                      ( a)
+                      ( r))
+                    ( nat-inr-Fin n i)))
         ＝
           add-Commutative-Semiring R
             ( a)
@@ -425,15 +443,4 @@ module _
                       ( a)
                       ( r) ∘
                     nat-Fin n)))
-      where
-        term :
-          type-Commutative-Semiring R → type-Commutative-Semiring R →
-          sequence (type-Commutative-Semiring R)
-        term a' r' =
-          seq-standard-geometric-sequence-Commutative-Semiring R a' r'
-        term-Fin :
-          type-Commutative-Semiring R → type-Commutative-Semiring R →
-          (n : ℕ) → fin-sequence (type-Commutative-Semiring R) n
-        term-Fin a' r' =
-          standard-geometric-fin-sequence-Commutative-Semiring R a' r'
 ```
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 01a86c46296..d59c56016e8 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -132,7 +132,7 @@ module _
                 ( refl)
           ＝
             ( a *ℚ
-              ( (one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r)) +ℚ
+              ( (one-ℚ -ℚ power-ℚ n r) *ℚ 1/⟨1-r⟩)) +ℚ
             ( a *ℚ power-ℚ n r)
             by
               ap-add-ℚ
@@ -140,12 +140,12 @@ module _
                 ( compute-standard-geometric-sequence-ℚ a r n)
           ＝
             a *ℚ
-            (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r) +ℚ power-ℚ n r)
+            (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/⟨1-r⟩) +ℚ power-ℚ n r)
             by inv (left-distributive-mul-add-ℚ a _ _)
           ＝
             a *ℚ
-            ( (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r) +ℚ
-              (power-ℚ n r *ℚ (one-ℚ -ℚ r)) *ℚ 1/1-r))
+            ( (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/⟨1-r⟩) +ℚ
+              (power-ℚ n r *ℚ (one-ℚ -ℚ r)) *ℚ 1/⟨1-r⟩))
             by
               ap-mul-ℚ
                 ( refl)
@@ -157,16 +157,16 @@ module _
           ＝
             a *ℚ
             ( ( (one-ℚ -ℚ power-ℚ n r) +ℚ (power-ℚ n r *ℚ (one-ℚ -ℚ r))) *ℚ
-              1/1-r)
+              1/⟨1-r⟩)
             by
               ap-mul-ℚ
                 ( refl)
-                ( inv (right-distributive-mul-add-ℚ _ _ 1/1-r))
+                ( inv (right-distributive-mul-add-ℚ _ _ 1/⟨1-r⟩))
           ＝
             a *ℚ
             ( ( one-ℚ -ℚ power-ℚ n r +ℚ
                 ((power-ℚ n r *ℚ one-ℚ) -ℚ (power-ℚ n r *ℚ r))) *ℚ
-              1/1-r)
+              1/⟨1-r⟩)
             by
               ap-mul-ℚ
                 ( refl)
@@ -177,7 +177,7 @@ module _
             a *ℚ
             ( ( one-ℚ -ℚ power-ℚ n r +ℚ
                 ((power-ℚ n r -ℚ power-ℚ (succ-ℕ n) r))) *ℚ
-              1/1-r)
+              1/⟨1-r⟩)
             by
               ap-mul-ℚ
                 ( refl)
@@ -188,7 +188,7 @@ module _
                       ( right-unit-law-mul-ℚ _)
                       ( inv (power-succ-ℚ n r))))
                   ( refl))
-          ＝ a *ℚ ((one-ℚ -ℚ power-ℚ (succ-ℕ n) r) *ℚ 1/1-r)
+          ＝ a *ℚ ((one-ℚ -ℚ power-ℚ (succ-ℕ n) r) *ℚ 1/⟨1-r⟩)
             by
               ap-mul-ℚ
                 ( refl)

From 66ebb46ac80c481233d55a6ec5672ab28493e89b Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 6 Nov 2025 20:04:22 -0800
Subject: [PATCH 033/134] Fixes

---
 .../geometric-sequences-commutative-rings.lagda.md       | 9 ++++-----
 .../geometric-sequences-commutative-semirings.lagda.md   | 2 +-
 src/ring-theory/geometric-sequences-rings.lagda.md       | 3 +--
 3 files changed, 6 insertions(+), 8 deletions(-)

diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
index 1c27ccb4a0b..9343e05ed82 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -38,12 +38,11 @@ open import lists.sequences
 
 A
 {{#concept "geometric sequence" Disambiguation="in a commutative ring" Agda=geometric-sequence-Commutative-Ring}}
-in a [commutative ring](ring-theory.commutative-rings.md) is an
-[geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
-ring's multiplicative [semigroup](group-theory.semigroups.md).
+in a [commutative ring](commutative-algebra.commutative-rings.md) is an
+[geometric sequence](ring-theory.geometric-sequences-semirings.md) in the ring's
+multiplicative [semigroup](group-theory.semigroups.md).
 
-These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
-ring.
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the ring.
 
 ## Definitions
 
diff --git a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
index d1beb4c07ee..1e4b7ed9f99 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
@@ -43,7 +43,7 @@ open import univalent-combinatorics.standard-finite-types
 
 A
 {{#concept "geometric sequence" Disambiguation="in a commutative semiring" Agda=geometric-sequence-Commutative-Semiring}}
-in a [commutative semiring](ring-theory.commutative-semirings.md) is an
+in a [commutative semiring](commutative-algebra.commutative-semirings.md) is an
 [geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
 semiring's multiplicative [semigroup](group-theory.semigroups.md).
 
diff --git a/src/ring-theory/geometric-sequences-rings.lagda.md b/src/ring-theory/geometric-sequences-rings.lagda.md
index 700adbe9ee0..2d415a0512d 100644
--- a/src/ring-theory/geometric-sequences-rings.lagda.md
+++ b/src/ring-theory/geometric-sequences-rings.lagda.md
@@ -38,8 +38,7 @@ in a [ring](ring-theory.rings.md) is a
 [geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
 underlying [semiring](ring-theory.semirings.md).
 
-These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
-ring.
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the ring.
 
 ## Definitions
 

From 6969f98674aa2e427296f978d7c858711b42d924 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 09:59:22 -0800
Subject: [PATCH 034/134] Merge

---
 .../geometric-sequences-rational-numbers.lagda.md | 15 ---------------
 1 file changed, 15 deletions(-)

diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index c9d8cf9b621..21695e9d15a 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -36,21 +36,6 @@ open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.negated-equality
 open import foundation.transport-along-identifications
-open import commutative-algebra.geometric-sequences-commutative-rings
-open import commutative-algebra.sums-of-finite-sequences-of-elements-commutative-rings
-
-open import elementary-number-theory.addition-rational-numbers
-open import elementary-number-theory.additive-group-of-rational-numbers
-open import elementary-number-theory.difference-rational-numbers
-open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.multiplicative-group-of-rational-numbers
-open import elementary-number-theory.natural-numbers
-open import elementary-number-theory.powers-rational-numbers
-open import elementary-number-theory.rational-numbers
-open import elementary-number-theory.ring-of-rational-numbers
-
-open import foundation.identity-types
-open import foundation.negated-equality
 open import foundation.universe-levels
 
 open import group-theory.groups

From 966d80dbef7f3f8ecd26614e9cfa0953ac136822 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 12:01:55 -0800
Subject: [PATCH 035/134] Progress

---
 .../distance-real-numbers.lagda.md            |  38 +++++-
 .../nonnegative-real-numbers.lagda.md         |  20 ++++
 src/real-numbers/powers-real-numbers.lagda.md |  74 ++++++++++++
 ...real-sequences-approximating-zero.lagda.md | 111 ++++++++++++++++++
 4 files changed, 242 insertions(+), 1 deletion(-)
 create mode 100644 src/real-numbers/real-sequences-approximating-zero.lagda.md

diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 446e7c220ae..6d6e4c05bd2 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -303,7 +303,7 @@ abstract
         by sim-eq-ℝ (ap abs-ℝ (right-unit-law-add-ℝ (x -ℝ y)))
 ```
 
-## Distributivity laws
+### Distributivity laws
 
 ```agda
 module _
@@ -331,3 +331,39 @@ module _
         ＝ dist-ℝ (x *ℝ z) (y *ℝ z)
           by ap abs-ℝ (right-distributive-mul-diff-ℝ x y z)
 ```
+
+### Zero laws
+
+```agda
+abstract
+  right-zero-law-dist-ℝ : {l : Level} (x : ℝ l) → dist-ℝ x zero-ℝ ＝ abs-ℝ x
+  right-zero-law-dist-ℝ x = ap abs-ℝ (right-unit-law-diff-ℝ x)
+
+  left-zero-law-dist-ℝ : {l : Level} (x : ℝ l) → dist-ℝ zero-ℝ x ＝ abs-ℝ x
+  left-zero-law-dist-ℝ x = commutative-dist-ℝ zero-ℝ x ∙ right-zero-law-dist-ℝ x
+```
+
+### Distance is preserved by similarity
+
+```agda
+abstract
+  preserves-dist-left-sim-ℝ :
+    {l1 l2 l3 : Level} {z : ℝ l1} {x : ℝ l2} {y : ℝ l3} → sim-ℝ x y →
+    sim-ℝ (dist-ℝ x z) (dist-ℝ y z)
+  preserves-dist-left-sim-ℝ {z = z} {x = x} {y = y} x~y =
+    preserves-sim-abs-ℝ (preserves-sim-right-add-ℝ (neg-ℝ z) x y x~y)
+
+  preserves-dist-right-sim-ℝ :
+    {l1 l2 l3 : Level} {z : ℝ l1} {x : ℝ l2} {y : ℝ l3} → sim-ℝ x y →
+    sim-ℝ (dist-ℝ z x) (dist-ℝ z y)
+  preserves-dist-right-sim-ℝ {z = z} x~y =
+    preserves-sim-abs-ℝ (preserves-sim-diff-ℝ (refl-sim-ℝ z) x~y)
+
+  preserves-dist-sim-ℝ :
+    {l1 l2 l3 l4 : Level} {x : ℝ l1} {x' : ℝ l2} {y : ℝ l3} {y' : ℝ l4} →
+    sim-ℝ x x' → sim-ℝ y y' → sim-ℝ (dist-ℝ x y) (dist-ℝ x' y')
+  preserves-dist-sim-ℝ x~x' y~y' =
+    transitive-sim-ℝ _ _ _
+      ( preserves-dist-right-sim-ℝ y~y')
+      ( preserves-dist-left-sim-ℝ x~x')
+```
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index bf51c3d32b5..1cf848c3742 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -41,6 +41,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.saturation-inequality-real-numbers
 open import real-numbers.similarity-real-numbers
@@ -577,3 +578,22 @@ abstract
     is-nonnegative-ℝ y
   is-nonnegative-leq-ℝ⁰⁺ (x , 0≤x) y x≤y = transitive-leq-ℝ zero-ℝ x y x≤y 0≤x
 ```
+
+### Nonnegativity is preserved by similarity
+
+```agda
+abstract
+  is-nonnegative-sim-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → is-nonnegative-ℝ x → sim-ℝ x y →
+    is-nonnegative-ℝ y
+  is-nonnegative-sim-ℝ {x = x} {y = y} 0≤x x~y =
+    preserves-leq-right-sim-ℝ zero-ℝ x y x~y 0≤x
+```
+
+### Raising the universe levels of nonnegative real numbers
+
+```agda
+raise-ℝ⁰⁺ : {l1 : Level} (l : Level) → ℝ⁰⁺ l1 → ℝ⁰⁺ (l ⊔ l1)
+raise-ℝ⁰⁺ l (x , is-nonneg-x) =
+  (raise-ℝ l x , is-nonnegative-sim-ℝ is-nonneg-x (sim-raise-ℝ l x))
+```
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index aee3f51d45b..97d316963b8 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -26,6 +26,7 @@ open import foundation.disjunction
 open import foundation.identity-types
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
+open import foundation.propositional-truncations
 
 open import order-theory.large-posets
 
@@ -43,8 +44,10 @@ open import real-numbers.nonnegative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.squares-real-numbers
+open import real-numbers.real-sequences-approximating-zero
 ```
 
 </details>
@@ -182,6 +185,24 @@ abstract
     is-positive-mul-ℝ (is-positive-power-ℝ⁺ n x⁺) is-pos-x
 ```
 
+### Powers of nonnegative real numbers are nonnegative
+
+```agda
+abstract
+  is-nonnegative-power-ℝ⁰⁺ :
+    {l : Level} (n : ℕ) (x : ℝ⁰⁺ l) → is-nonnegative-ℝ (power-ℝ n (real-ℝ⁰⁺ x))
+  is-nonnegative-power-ℝ⁰⁺ {l} 0 _ =
+    is-nonnegative-real-ℝ⁰⁺ (raise-ℝ⁰⁺ l one-ℝ⁰⁺)
+  is-nonnegative-power-ℝ⁰⁺ 1 (x , is-nonneg-x) = is-nonneg-x
+  is-nonnegative-power-ℝ⁰⁺ (succ-ℕ n@(succ-ℕ _)) x⁰⁺@(x , is-nonneg-x) =
+    is-nonnegative-mul-ℝ
+      ( is-nonnegative-power-ℝ⁰⁺ n x⁰⁺)
+      ( is-nonneg-x)
+
+power-ℝ⁰⁺ : {l : Level} → ℕ → ℝ⁰⁺ l → ℝ⁰⁺ l
+power-ℝ⁰⁺ n x⁰⁺@(x , _) = (power-ℝ n x , is-nonnegative-power-ℝ⁰⁺ n x⁰⁺)
+```
+
 ### Even powers of negative real numbers are positive
 
 ```agda
@@ -323,3 +344,56 @@ abstract
       ≤ abs-ℝ (power-ℝ (succ-ℕ n) y)
         by leq-eq-ℝ _ _ (ap abs-ℝ (inv (power-succ-ℝ n y)))
 ```
+
+### If `x` and `y` are nonnegative, and `x ≤ y`, `xⁿ ≤ yⁿ`
+
+```agda
+abstract
+  preserves-leq-power-real-ℝ⁰⁺ :
+    {l1 l2 : Level} (n : ℕ) (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) → leq-ℝ⁰⁺ x y →
+    leq-ℝ (power-ℝ n (real-ℝ⁰⁺ x)) (power-ℝ n (real-ℝ⁰⁺ y))
+  preserves-leq-power-real-ℝ⁰⁺ {l1} {l2} 0 _ _ _ =
+    leq-sim-ℝ _ _
+      ( transitive-sim-ℝ _ one-ℝ _
+        ( sim-raise-ℝ l2 one-ℝ)
+        ( symmetric-sim-ℝ (sim-raise-ℝ l1 one-ℝ)))
+  preserves-leq-power-real-ℝ⁰⁺ (succ-ℕ n) x⁰⁺@(x , _) y⁰⁺@(y , _) x≤y =
+    let
+      open inequality-reasoning-Large-Poset ℝ-Large-Poset
+    in
+      chain-of-inequalities
+      power-ℝ (succ-ℕ n) x
+      ≤ power-ℝ n x *ℝ x
+        by leq-eq-ℝ _ _ (power-succ-ℝ n x)
+      ≤ power-ℝ n y *ℝ y
+        by
+          preserves-leq-mul-ℝ⁰⁺
+            ( power-ℝ⁰⁺ n x⁰⁺)
+            ( power-ℝ⁰⁺ n y⁰⁺)
+            ( x⁰⁺)
+            ( y⁰⁺)
+            ( preserves-leq-power-real-ℝ⁰⁺ n x⁰⁺ y⁰⁺ x≤y)
+            ( x≤y)
+      ≤ power-ℝ (succ-ℕ n) y
+        by leq-eq-ℝ _ _ (inv (power-succ-ℝ n y))
+```
+
+### If `|r| < 1`, `|r|ⁿ` approaches 0
+
+```agda
+abstract
+  is-zero-lim-power-le-one-abs-ℝ :
+    {l : Level} (r : ℝ l) → le-ℝ (abs-ℝ r) one-ℝ →
+    is-zero-limit-sequence-ℝ (λ n → power-ℝ n r)
+  is-zero-lim-power-le-one-abs-ℝ r |r|<1 =
+    let
+      open
+        do-syntax-trunc-Prop (is-zero-limit-prop-sequence-ℝ (λ n → power-ℝ n r))
+    in do
+      (ε , |r|<ε , ε<1ℝ) ← dense-rational-le-ℝ _ _ |r|<1
+      let ε<1 = reflects-le-real-ℚ ε one-ℚ ε<1ℝ
+      is-zero-limit-sequence-leq-abs-rational-zero-limit-sequence-ℝ
+        ( λ n → power-ℝ n r)
+        ( {! λ n → power-ℚ n ε  !} , {!   !})
+        {!   !}
+```
diff --git a/src/real-numbers/real-sequences-approximating-zero.lagda.md b/src/real-numbers/real-sequences-approximating-zero.lagda.md
new file mode 100644
index 00000000000..8ca717450f9
--- /dev/null
+++ b/src/real-numbers/real-sequences-approximating-zero.lagda.md
@@ -0,0 +1,111 @@
+# Real sequences approximating zero
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.real-sequences-approximating-zero where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import metric-spaces.limits-of-sequences-metric-spaces
+open import lists.sequences
+open import foundation.identity-types
+open import real-numbers.absolute-value-real-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import foundation.action-on-identifications-functions
+open import metric-spaces.metric-space-of-rational-numbers
+open import elementary-number-theory.absolute-value-rational-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.similarity-real-numbers
+open import foundation.function-types
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.distance-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import logic.functoriality-existential-quantification
+open import order-theory.large-posets
+open import foundation.existential-quantification
+open import real-numbers.rational-real-numbers
+open import real-numbers.distance-real-numbers
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+open import real-numbers.distance-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import foundation.propositions
+open import metric-spaces.rational-sequences-approximating-zero
+open import real-numbers.metric-space-of-real-numbers
+```
+
+</details>
+
+## Idea
+
+A [sequence](lists.sequences.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md) is an
+{{#concept "approximation of zero" Disambiguation="sequence of real numbers" Agda=zero-limit-sequence-ℝ}}
+if it [converges](metric-spaces.limits-of-sequences-metric-spaces.md) to 0 in
+the
+[standard metric space of real numbers](real-numbers.metric-space-of-real-numbers.md).
+
+## Definition
+
+```agda
+is-zero-limit-prop-sequence-ℝ : {l : Level} → sequence (ℝ l) → Prop l
+is-zero-limit-prop-sequence-ℝ {l} σ =
+  is-limit-prop-sequence-Metric-Space
+    ( metric-space-ℝ l)
+    ( σ)
+    ( raise-ℝ l zero-ℝ)
+
+is-zero-limit-sequence-ℝ : {l : Level} → sequence (ℝ l) → UU l
+is-zero-limit-sequence-ℝ σ = type-Prop (is-zero-limit-prop-sequence-ℝ σ)
+```
+
+## Properties
+
+### If the absolute value of a sequence of reals is bounded by a rational approximation of zero, the sequence of reals is an approximation of zero
+
+```agda
+abstract
+  is-zero-limit-sequence-leq-abs-rational-zero-limit-sequence-ℝ :
+    {l : Level} (a : sequence (ℝ l)) (b : zero-limit-sequence-ℚ) →
+    ((n : ℕ) → leq-ℝ (abs-ℝ (a n)) (real-ℚ (seq-zero-limit-sequence-ℚ b n))) →
+    is-zero-limit-sequence-ℝ a
+  is-zero-limit-sequence-leq-abs-rational-zero-limit-sequence-ℝ
+    {l} a (b , lim-b=0) H =
+    let
+      open inequality-reasoning-Large-Poset ℝ-Large-Poset
+    in
+      map-tot-exists
+        ( λ μ is-mod-μ ε n με≤n →
+          neighborhood-dist-ℝ
+            ( ε)
+            ( a n)
+            ( raise-ℝ l zero-ℝ)
+            ( chain-of-inequalities
+              dist-ℝ (a n) (raise-ℝ l zero-ℝ)
+              ≤ dist-ℝ (a n) zero-ℝ
+                by
+                  leq-sim-ℝ _ _
+                    ( preserves-dist-right-sim-ℝ
+                      ( symmetric-sim-ℝ (sim-raise-ℝ l zero-ℝ)))
+              ≤ abs-ℝ (a n)
+                by leq-eq-ℝ _ _ (right-zero-law-dist-ℝ (a n))
+              ≤ real-ℚ (b n)
+                by H n
+              ≤ real-ℚ (rational-abs-ℚ (b n))
+                by preserves-leq-real-ℚ _ _ (leq-abs-ℚ (b n))
+              ≤ real-ℚ (rational-dist-ℚ (b n) zero-ℚ)
+                by
+                  leq-eq-ℝ _ _
+                    ( ap
+                      ( real-ℚ ∘ rational-ℚ⁰⁺)
+                      ( inv (right-zero-law-dist-ℚ (b n))))
+              ≤ real-ℚ⁺ ε
+                by
+                  preserves-leq-real-ℚ _ _
+                    ( leq-dist-neighborhood-ℚ ε _ _ (is-mod-μ ε n με≤n))))
+      ( lim-b=0)
+```

From 2882532a1cda04c5e4bfa60840fe2634d891bdd7 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 12:25:07 -0800
Subject: [PATCH 036/134] Progress

---
 src/real-numbers.lagda.md                     |  1 +
 .../nonnegative-real-numbers.lagda.md         | 11 +++++
 ...ositive-and-negative-real-numbers.lagda.md |  6 ---
 .../positive-real-numbers.lagda.md            | 14 ++++--
 src/real-numbers/powers-real-numbers.lagda.md | 45 ++++++++++++++++---
 ...real-sequences-approximating-zero.lagda.md | 45 ++++++++++---------
 6 files changed, 85 insertions(+), 37 deletions(-)

diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 3e2872c1e15..ec376e07eff 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -69,6 +69,7 @@ open import real-numbers.rational-real-numbers public
 open import real-numbers.rational-upper-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-lower-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
+open import real-numbers.real-sequences-approximating-zero public
 open import real-numbers.saturation-inequality-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.square-roots-nonnegative-real-numbers public
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 1cf848c3742..96d515c30c5 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -130,6 +130,11 @@ abstract
   is-nonnegative-real-ℚ⁰⁺ (q , nonneg-q) =
     preserves-leq-real-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ nonneg-q)
 
+  reflects-is-nonnegative-real-ℚ :
+    (q : ℚ) → is-nonnegative-ℝ (real-ℚ q) → is-nonnegative-ℚ q
+  reflects-is-nonnegative-real-ℚ q 0≤qℝ =
+    is-nonnegative-leq-zero-ℚ (reflects-leq-real-ℚ zero-ℚ q 0≤qℝ)
+
 nonnegative-real-ℚ⁰⁺ : ℚ⁰⁺ → ℝ⁰⁺ lzero
 nonnegative-real-ℚ⁰⁺ q = (real-ℚ⁰⁺ q , is-nonnegative-real-ℚ⁰⁺ q)
 
@@ -577,6 +582,12 @@ abstract
     {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) → leq-ℝ (real-ℝ⁰⁺ x) y →
     is-nonnegative-ℝ y
   is-nonnegative-leq-ℝ⁰⁺ (x , 0≤x) y x≤y = transitive-leq-ℝ zero-ℝ x y x≤y 0≤x
+
+  is-nonnegative-le-ℝ⁰⁺ :
+    {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) → le-ℝ (real-ℝ⁰⁺ x) y →
+    is-nonnegative-ℝ y
+  is-nonnegative-le-ℝ⁰⁺ x y x<y =
+    is-nonnegative-leq-ℝ⁰⁺ x y (leq-le-ℝ (real-ℝ⁰⁺ x) y x<y)
 ```
 
 ### Nonnegativity is preserved by similarity
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index a3349458452..9ae4439776a 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -89,10 +89,4 @@ abstract
     {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) → le-ℝ (real-ℝ⁰⁺ x) y →
     is-positive-ℝ y
   is-positive-le-ℝ⁰⁺ (x , 0≤x) y = concatenate-leq-le-ℝ zero-ℝ x y 0≤x
-
-  is-nonnegative-le-ℝ⁰⁺ :
-    {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) → le-ℝ (real-ℝ⁰⁺ x) y →
-    is-nonnegative-ℝ y
-  is-nonnegative-le-ℝ⁰⁺ x y x<y =
-    is-nonnegative-is-positive-ℝ (is-positive-le-ℝ⁰⁺ x y x<y)
 ```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 004dbd53454..fadf5b6430b 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -266,10 +266,16 @@ abstract
 ### The canonical embedding of rational numbers preserves positivity
 
 ```agda
-preserves-is-positive-real-ℚ :
-  (q : ℚ) → is-positive-ℚ q → is-positive-ℝ (real-ℚ q)
-preserves-is-positive-real-ℚ q pos-q =
-  preserves-le-real-ℚ zero-ℚ q (le-zero-is-positive-ℚ pos-q)
+abstract
+  preserves-is-positive-real-ℚ :
+    (q : ℚ) → is-positive-ℚ q → is-positive-ℝ (real-ℚ q)
+  preserves-is-positive-real-ℚ q pos-q =
+    preserves-le-real-ℚ zero-ℚ q (le-zero-is-positive-ℚ pos-q)
+
+  reflects-is-positive-real-ℚ :
+    (q : ℚ) → is-positive-ℝ (real-ℚ q) → is-positive-ℚ q
+  reflects-is-positive-real-ℚ q 0<qℝ =
+    is-positive-le-zero-ℚ (reflects-le-real-ℚ zero-ℚ q 0<qℝ)
 
 opaque
   unfolding le-ℝ real-ℚ
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 97d316963b8..d78b0dbdfb7 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -16,6 +16,8 @@ open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.multiplication-natural-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.parity-natural-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.powers-positive-rational-numbers
 open import elementary-number-theory.powers-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.ring-of-rational-numbers
@@ -24,9 +26,9 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.identity-types
+open import foundation.propositional-truncations
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
-open import foundation.propositional-truncations
 
 open import order-theory.large-posets
 
@@ -41,13 +43,14 @@ open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.real-sequences-approximating-zero
 open import real-numbers.similarity-real-numbers
 open import real-numbers.squares-real-numbers
-open import real-numbers.real-sequences-approximating-zero
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
@@ -183,6 +186,9 @@ abstract
   is-positive-power-ℝ⁺ 1 (_ , is-pos-x) = is-pos-x
   is-positive-power-ℝ⁺ (succ-ℕ n@(succ-ℕ _)) x⁺@(x , is-pos-x) =
     is-positive-mul-ℝ (is-positive-power-ℝ⁺ n x⁺) is-pos-x
+
+power-ℝ⁺ : {l : Level} → ℕ → ℝ⁺ l → ℝ⁺ l
+power-ℝ⁺ n x⁺@(x , _) = (power-ℝ n x , is-positive-power-ℝ⁺ n x⁺)
 ```
 
 ### Powers of nonnegative real numbers are nonnegative
@@ -378,7 +384,7 @@ abstract
         by leq-eq-ℝ _ _ (inv (power-succ-ℝ n y))
 ```
 
-### If `|r| < 1`, `|r|ⁿ` approaches 0
+### If `|r| < 1`, `rⁿ` approaches 0
 
 ```agda
 abstract
@@ -389,11 +395,36 @@ abstract
     let
       open
         do-syntax-trunc-Prop (is-zero-limit-prop-sequence-ℝ (λ n → power-ℝ n r))
+      open inequality-reasoning-Large-Poset ℝ-Large-Poset
     in do
       (ε , |r|<ε , ε<1ℝ) ← dense-rational-le-ℝ _ _ |r|<1
-      let ε<1 = reflects-le-real-ℚ ε one-ℚ ε<1ℝ
+      let
+        is-pos-ε =
+          reflects-is-positive-real-ℚ
+            ( ε)
+            ( is-positive-le-ℝ⁰⁺ (nonnegative-abs-ℝ r) (real-ℚ ε) |r|<ε)
+        ε⁺ = (ε , is-pos-ε)
       is-zero-limit-sequence-leq-abs-rational-zero-limit-sequence-ℝ
         ( λ n → power-ℝ n r)
-        ( {! λ n → power-ℚ n ε  !} , {!   !})
-        {!   !}
+        ( (λ n → rational-ℚ⁺ (power-ℚ⁺ n ε⁺)) ,
+          is-zero-limit-power-le-one-ℚ⁺ ε⁺ (reflects-le-real-ℚ ε one-ℚ ε<1ℝ))
+        ( λ n →
+          chain-of-inequalities
+            abs-ℝ (power-ℝ n r)
+            ≤ abs-ℝ (power-ℝ n (real-ℚ ε))
+              by
+                preserves-leq-abs-power-ℝ
+                  ( n)
+                  ( r)
+                  ( real-ℚ ε)
+                  ( inv-tr
+                    ( leq-ℝ (abs-ℝ r))
+                    ( abs-real-ℝ⁺ (positive-real-ℚ⁺ ε⁺))
+                    ( leq-le-ℝ _ _ |r|<ε))
+            ≤ power-ℝ n (real-ℚ ε)
+              by leq-eq-ℝ _ _ (abs-real-ℝ⁺ (power-ℝ⁺ n (positive-real-ℚ⁺ ε⁺)))
+            ≤ real-ℚ (power-ℚ n ε)
+              by leq-eq-ℝ _ _ (power-real-ℚ n ε)
+            ≤ real-ℚ⁺ (power-ℚ⁺ n ε⁺)
+              by leq-eq-ℝ _ _ (ap real-ℚ (power-rational-ℚ⁺ n ε⁺)))
 ```
diff --git a/src/real-numbers/real-sequences-approximating-zero.lagda.md b/src/real-numbers/real-sequences-approximating-zero.lagda.md
index 8ca717450f9..4f50b606de9 100644
--- a/src/real-numbers/real-sequences-approximating-zero.lagda.md
+++ b/src/real-numbers/real-sequences-approximating-zero.lagda.md
@@ -9,33 +9,38 @@ module real-numbers.real-sequences-approximating-zero where
 <details><summary>Imports</summary>
 
 ```agda
-open import metric-spaces.limits-of-sequences-metric-spaces
-open import lists.sequences
-open import foundation.identity-types
-open import real-numbers.absolute-value-real-numbers
-open import elementary-number-theory.nonnegative-rational-numbers
-open import foundation.action-on-identifications-functions
-open import metric-spaces.metric-space-of-rational-numbers
 open import elementary-number-theory.absolute-value-rational-numbers
-open import real-numbers.inequality-real-numbers
-open import real-numbers.similarity-real-numbers
-open import foundation.function-types
-open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.distance-rational-numbers
 open import elementary-number-theory.natural-numbers
-open import logic.functoriality-existential-quantification
-open import order-theory.large-posets
-open import foundation.existential-quantification
-open import real-numbers.rational-real-numbers
-open import real-numbers.distance-real-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.rational-numbers
+
+open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
-open import foundation.universe-levels
-open import real-numbers.distance-real-numbers
-open import real-numbers.raising-universe-levels-real-numbers
-open import real-numbers.dedekind-real-numbers
+open import foundation.existential-quantification
+open import foundation.function-types
+open import foundation.identity-types
 open import foundation.propositions
+open import foundation.universe-levels
+
+open import lists.sequences
+
+open import logic.functoriality-existential-quantification
+
+open import metric-spaces.limits-of-sequences-metric-spaces
+open import metric-spaces.metric-space-of-rational-numbers
 open import metric-spaces.rational-sequences-approximating-zero
+
+open import order-theory.large-posets
+
+open import real-numbers.absolute-value-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.distance-real-numbers
+open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
 ```
 
 </details>

From 380382bfd9c357f6017d92126b8533f7b2065606 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 14:41:01 -0800
Subject: [PATCH 037/134] Progress

---
 .../absolute-value-rational-numbers.lagda.md  |   2 +-
 ...ddition-positive-rational-numbers.lagda.md |  13 +-
 .../distance-rational-numbers.lagda.md        |   5 +-
 .../inequality-rational-numbers.lagda.md      |  15 +-
 ...closed-intervals-rational-numbers.lagda.md |  16 +-
 ...ication-positive-rational-numbers.lagda.md |   2 +-
 .../negative-rational-numbers.lagda.md        |   4 +-
 .../positive-rational-numbers.lagda.md        |   2 +-
 ...e-roots-positive-rational-numbers.lagda.md |  22 +-
 ...trict-inequality-rational-numbers.lagda.md |  33 +-
 .../metrics-of-metric-spaces.lagda.md         |   8 +-
 src/metric-spaces/metrics.lagda.md            |  12 +-
 src/real-numbers.lagda.md                     |   7 +
 ...lue-closed-intervals-real-numbers.lagda.md |   2 +-
 .../absolute-value-real-numbers.lagda.md      |  26 +-
 ...ition-lower-dedekind-real-numbers.lagda.md |   2 +-
 ...addition-nonnegative-real-numbers.lagda.md | 136 ++++++++
 ...ition-upper-dedekind-real-numbers.lagda.md |   2 +-
 ...thmetically-located-dedekind-cuts.lagda.md |  20 +-
 .../binary-maximum-real-numbers.lagda.md      |  23 +-
 .../binary-minimum-real-numbers.lagda.md      |  13 +-
 ...ompleteness-dedekind-real-numbers.lagda.md |  13 +-
 .../closed-intervals-real-numbers.lagda.md    |   6 +-
 .../dedekind-real-numbers.lagda.md            |   8 +-
 .../distance-real-numbers.lagda.md            |   6 +-
 ...nequalities-addition-real-numbers.lagda.md | 234 +++++++++++++
 ...equality-nonnegative-real-numbers.lagda.md | 159 +++++++++
 .../inequality-real-numbers.lagda.md          | 267 ++-------------
 .../infima-families-real-numbers.lagda.md     |  17 +-
 ...ally-bounded-subsets-real-numbers.lagda.md |  11 +-
 .../isometry-addition-real-numbers.lagda.md   |   1 +
 .../isometry-negation-real-numbers.lagda.md   |  29 ++
 ...nuity-multiplication-real-numbers.lagda.md |  27 +-
 ...imum-finite-families-real-numbers.lagda.md |   1 +
 ...space-of-nonnegative-real-numbers.lagda.md |  34 ++
 .../metric-space-of-real-numbers.lagda.md     |  21 +-
 ...tiplication-negative-real-numbers.lagda.md |   4 +-
 ...lication-nonnegative-real-numbers.lagda.md |   5 +-
 .../multiplication-real-numbers.lagda.md      |  12 +-
 ...ve-inverses-positive-real-numbers.lagda.md |   3 +-
 .../negation-real-numbers.lagda.md            |   2 +-
 .../nonnegative-real-numbers.lagda.md         | 319 +----------------
 ...ositive-and-negative-real-numbers.lagda.md |   6 +-
 .../positive-real-numbers.lagda.md            |  39 +--
 ...sing-universe-levels-real-numbers.lagda.md |   2 +-
 .../rational-real-numbers.lagda.md            |  14 +-
 ...equality-nonnegative-real-numbers.lagda.md |  71 ++++
 ...aturation-inequality-real-numbers.lagda.md |  29 +-
 ...milarity-nonnegative-real-numbers.lagda.md |  70 ++++
 ...re-roots-nonnegative-real-numbers.lagda.md |   9 +-
 .../squares-real-numbers.lagda.md             |  10 +-
 ...nequalities-addition-real-numbers.lagda.md | 289 ++++++++++++++++
 ...equality-nonnegative-real-numbers.lagda.md |  42 ++-
 .../strict-inequality-real-numbers.lagda.md   | 323 ++----------------
 .../suprema-families-real-numbers.lagda.md    |   5 +-
 ...action-cuts-dedekind-real-numbers.lagda.md |  13 +-
 56 files changed, 1340 insertions(+), 1126 deletions(-)
 create mode 100644 src/real-numbers/addition-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/inequalities-addition-real-numbers.lagda.md
 create mode 100644 src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md

diff --git a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
index 4b0ad01d885..031a23160f4 100644
--- a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
@@ -166,7 +166,7 @@ abstract
         ( q)
         ( rational-abs-ℚ q)
         ( zero-ℚ)
-        ( leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ abs=0))
+        ( leq-eq-ℚ (ap rational-ℚ⁰⁺ abs=0))
         ( leq-abs-ℚ q))
       ( binary-tr
         ( leq-ℚ)
diff --git a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
index 85668704afe..7e1cf13e5b6 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -169,7 +169,7 @@ module _
   where
 
   le-diff-ℚ⁺ : ℚ⁺
-  le-diff-ℚ⁺ = positive-diff-le-ℚ (rational-ℚ⁺ x) (rational-ℚ⁺ y) H
+  le-diff-ℚ⁺ = positive-diff-le-ℚ H
 
   left-diff-law-add-ℚ⁺ : le-diff-ℚ⁺ +ℚ⁺ x ＝ y
   left-diff-law-add-ℚ⁺ =
@@ -342,15 +342,8 @@ module _
             ( le-right-mediant-ℚ y x I)
             ( tr
               ( leq-ℚ x)
-              ( right-law-positive-diff-le-ℚ
-                ( y)
-                ( mediant-ℚ y x)
-                ( le-left-mediant-ℚ y x I))
-              ( H
-                ( positive-diff-le-ℚ
-                  ( y)
-                  ( mediant-ℚ y x)
-                  ( le-left-mediant-ℚ y x I))))))
+              ( right-law-positive-diff-le-ℚ (le-left-mediant-ℚ y x I))
+              ( H (positive-diff-le-ℚ (le-left-mediant-ℚ y x I))))))
       ( id)
       ( decide-le-leq-ℚ y x)
 
diff --git a/src/elementary-number-theory/distance-rational-numbers.lagda.md b/src/elementary-number-theory/distance-rational-numbers.lagda.md
index 6a9bcc3d5d9..3926d248dc4 100644
--- a/src/elementary-number-theory/distance-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-rational-numbers.lagda.md
@@ -191,10 +191,7 @@ abstract
       ( rational-dist-ℚ p q)
       ( (rational-abs-ℚ p) +ℚ (rational-abs-ℚ (neg-ℚ q)))
       ( rational-abs-ℚ p +ℚ rational-abs-ℚ q)
-      ( leq-eq-ℚ
-        ( (rational-abs-ℚ p) +ℚ (rational-abs-ℚ (neg-ℚ q)))
-        ( rational-abs-ℚ p +ℚ rational-abs-ℚ q)
-        ( ap (add-ℚ (rational-abs-ℚ p) ∘ rational-ℚ⁰⁺) (abs-neg-ℚ q)))
+      ( leq-eq-ℚ (ap (add-ℚ (rational-abs-ℚ p) ∘ rational-ℚ⁰⁺) (abs-neg-ℚ q)))
       ( triangle-inequality-abs-ℚ p (neg-ℚ q))
 ```
 
diff --git a/src/elementary-number-theory/inequality-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
index 14d1a9398e4..29d6fb57a98 100644
--- a/src/elementary-number-theory/inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
@@ -126,8 +126,8 @@ opaque
     refl-leq-ℤ (numerator-ℚ x *ℤ denominator-ℚ x)
 
 abstract
-  leq-eq-ℚ : (x y : ℚ) → x ＝ y → leq-ℚ x y
-  leq-eq-ℚ x y x=y = tr (leq-ℚ x) x=y (refl-leq-ℚ x)
+  leq-eq-ℚ : {x y : ℚ} → x ＝ y → leq-ℚ x y
+  leq-eq-ℚ {x} refl = refl-leq-ℚ x
 ```
 
 ### Inequality on the rational numbers is antisymmetric
@@ -294,8 +294,8 @@ opaque
   unfolding leq-ℚ-Prop
 
   preserves-leq-rational-ℤ :
-    (x y : ℤ) → leq-ℤ x y → leq-ℚ (rational-ℤ x) (rational-ℤ y)
-  preserves-leq-rational-ℤ x y =
+    {x y : ℤ} → leq-ℤ x y → leq-ℚ (rational-ℤ x) (rational-ℤ y)
+  preserves-leq-rational-ℤ {x} {y} =
     binary-tr leq-ℤ
       ( inv (right-unit-law-mul-ℤ x))
       ( inv (right-unit-law-mul-ℤ y))
@@ -309,7 +309,7 @@ opaque
 
   iff-leq-rational-ℤ :
     (x y : ℤ) → leq-ℤ x y ↔ leq-ℚ (rational-ℤ x) (rational-ℤ y)
-  pr1 (iff-leq-rational-ℤ x y) = preserves-leq-rational-ℤ x y
+  pr1 (iff-leq-rational-ℤ x y) = preserves-leq-rational-ℤ
   pr2 (iff-leq-rational-ℤ x y) = reflects-leq-rational-ℤ x y
 ```
 
@@ -324,8 +324,9 @@ abstract
     iff-leq-int-ℕ x y
 
   preserves-leq-rational-ℕ :
-    (x y : ℕ) → leq-ℕ x y → leq-ℚ (rational-ℕ x) (rational-ℕ y)
-  preserves-leq-rational-ℕ x y = forward-implication (iff-leq-rational-ℕ x y)
+    {x y : ℕ} → leq-ℕ x y → leq-ℚ (rational-ℕ x) (rational-ℕ y)
+  preserves-leq-rational-ℕ {x} {y} =
+    forward-implication (iff-leq-rational-ℕ x y)
 
   reflects-leq-rational-ℕ :
     (x y : ℕ) → leq-ℚ (rational-ℕ x) (rational-ℕ y) → leq-ℕ x y
diff --git a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
index 8ec9ae546c5..698a8adb827 100644
--- a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
@@ -924,7 +924,7 @@ abstract
           rational-dist-ℚ (a *ℚ q) (b *ℚ q)
           ≤ rational-dist-ℚ a b *ℚ rational-abs-ℚ q
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ( ap
                     ( rational-abs-ℚ)
                     ( inv (right-distributive-mul-diff-ℚ _ _ _))) ∙
@@ -939,7 +939,7 @@ abstract
           ≤ width-closed-interval-ℚ [a,b] *ℚ
             rational-max-abs-closed-interval-ℚ [c,d]
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap-mul-ℚ
                   ( eq-width-dist-lower-upper-bounds-closed-interval-ℚ [a,b])
                   ( refl))
@@ -951,7 +951,7 @@ abstract
           rational-dist-ℚ (p *ℚ c) (p *ℚ d)
           ≤ rational-dist-ℚ c d *ℚ rational-abs-ℚ p
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ( ap
                     ( rational-abs-ℚ)
                     ( inv (left-distributive-mul-diff-ℚ _ _ _))) ∙
@@ -966,7 +966,7 @@ abstract
                 ( leq-max-abs-is-in-closed-interval-ℚ [a,b] p p∈[a,b])
           ≤ _
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap-mul-ℚ
                   ( eq-width-dist-lower-upper-bounds-closed-interval-ℚ [c,d])
                   ( refl))
@@ -983,7 +983,7 @@ abstract
           rational-dist-ℚ (a *ℚ c) (b *ℚ d)
           ≤ rational-abs-ℚ ((a *ℚ c -ℚ b *ℚ c) +ℚ (b *ℚ c -ℚ b *ℚ d))
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap
                   ( rational-abs-ℚ)
                   ( inv ( mul-right-div-Group group-add-ℚ _ _ _)))
@@ -996,7 +996,7 @@ abstract
           rational-dist-ℚ (a *ℚ d) (b *ℚ c)
           ≤ rational-abs-ℚ ((a *ℚ d -ℚ b *ℚ d) +ℚ (b *ℚ d -ℚ b *ℚ c))
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap
                   ( rational-abs-ℚ)
                   ( inv ( mul-right-div-Group group-add-ℚ _ _ _)))
@@ -1006,7 +1006,7 @@ abstract
           ≤ rational-dist-ℚ (a *ℚ d) (b *ℚ d) +ℚ
             rational-dist-ℚ (b *ℚ c) (b *ℚ d)
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap-add-ℚ refl (ap rational-ℚ⁰⁺ (commutative-dist-ℚ _ _)))
           ≤ <b-a><max|c||d|> +ℚ <d-c><max|a||b|>
             by preserves-leq-add-ℚ |ad-bd|≤<b-a>max|c||d| |bc-bd|≤<d-c>max|a||b|
@@ -1042,7 +1042,7 @@ abstract
         chain-of-inequalities
           rational-dist-ℚ q q
           ≤ zero-ℚ
-            by leq-eq-ℚ _ _ (rational-dist-self-ℚ q)
+            by leq-eq-ℚ (rational-dist-self-ℚ q)
           ≤ <b-a><max|c||d|> +ℚ <d-c><max|a||b|>
             by
               leq-zero-rational-ℚ⁰⁺ (<b-a><max|c||d|>⁰⁺ +ℚ⁰⁺ <d-c><max|a||b|>⁰⁺)
diff --git a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
index e0fb4cbbe41..492703832cb 100644
--- a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
@@ -357,6 +357,6 @@ abstract
   leq-left-mul-leq-one-ℚ⁺ p⁺@(p , _) p≤1 q⁺@(q , _) =
     trichotomy-le-ℚ p one-ℚ
       ( λ p<1 → leq-le-ℚ (le-left-mul-less-than-one-ℚ⁺ p⁺ p<1 q⁺))
-      ( λ p=1 → leq-eq-ℚ _ _ (ap-mul-ℚ p=1 refl ∙ left-unit-law-mul-ℚ q))
+      ( λ p=1 → leq-eq-ℚ (ap-mul-ℚ p=1 refl ∙ left-unit-law-mul-ℚ q))
       ( λ 1<p → ex-falso (not-leq-le-ℚ one-ℚ p 1<p p≤1))
 ```
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 20a8ca5963f..f5754c5d866 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -198,7 +198,7 @@ abstract
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (H : le-ℚ x y)
   where
 
   opaque
@@ -209,7 +209,7 @@ module _
       inv-tr
         ( is-positive-ℚ)
         ( distributive-neg-diff-ℚ x y)
-        ( is-positive-diff-le-ℚ x y H)
+        ( is-positive-diff-le-ℚ H)
 
   negative-diff-le-ℚ : ℚ⁻
   negative-diff-le-ℚ = x -ℚ y , is-negative-diff-le-ℚ
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index 4eed2615132..dec1ae2814b 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -235,7 +235,7 @@ abstract
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (H : le-ℚ x y)
   where
 
   abstract
diff --git a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
index efd3983670e..7687c6d3d4b 100644
--- a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
@@ -102,7 +102,7 @@ module _
               ( le-ℚ)
               ( inv q=1)
               ( inv (mul-rational-ℕ 2 2))
-              ( preserves-le-rational-ℕ 1 4 _)))
+              ( preserves-le-rational-ℕ {1} {4} _)))
         ( λ 1<q →
           ( q , le-left-mul-greater-than-one-ℚ⁺ q 1<q q))
 
@@ -120,7 +120,7 @@ abstract
   bound-rounded-below-square-le-ℚ⁺ p⁺@(p , _) q⁺@(q , _) p²<q =
     let
       (δ₁⁺@(δ₁ , _) , δ₁+δ₁<⟨q-p²⟩/p) =
-        bound-double-le-ℚ⁺ (positive-diff-le-ℚ (p *ℚ p) q p²<q *ℚ⁺ inv-ℚ⁺ p⁺)
+        bound-double-le-ℚ⁺ (positive-diff-le-ℚ p²<q *ℚ⁺ inv-ℚ⁺ p⁺)
       (δ₂⁺@(δ₂ , _) , δ₂+δ₂<δ₁) = bound-double-le-ℚ⁺ δ₁⁺
       δ₃⁺@(δ₃ , _) = min-ℚ⁺ δ₂⁺ p⁺
       δ₂<δ₁ : le-ℚ δ₂ δ₁
@@ -141,22 +141,22 @@ abstract
           ( chain-of-inequalities
             (p +ℚ δ₃) *ℚ (p +ℚ δ₃)
             ≤ (p +ℚ δ₃) *ℚ p +ℚ (p +ℚ δ₃) *ℚ δ₃
-              by leq-eq-ℚ _ _ (left-distributive-mul-add-ℚ (p +ℚ δ₃) p δ₃)
+              by leq-eq-ℚ (left-distributive-mul-add-ℚ (p +ℚ δ₃) p δ₃)
             ≤ (p +ℚ δ₃) *ℚ p +ℚ ((p *ℚ δ₃) +ℚ (δ₃ *ℚ δ₃))
-              by leq-eq-ℚ _ _
+              by leq-eq-ℚ
                 ( ap-add-ℚ refl (right-distributive-mul-add-ℚ _ _ _))
             ≤ (p +ℚ δ₃) *ℚ p +ℚ ((δ₃ *ℚ p) +ℚ (δ₃ *ℚ p))
               by
                 preserves-leq-right-add-ℚ _ _ _
                   ( preserves-leq-add-ℚ
-                    ( leq-eq-ℚ _ _ (commutative-mul-ℚ _ _))
+                    ( leq-eq-ℚ (commutative-mul-ℚ _ _))
                     ( preserves-leq-left-mul-ℚ⁺
                       ( δ₃⁺)
                       ( δ₃)
                       ( p)
                       ( leq-right-min-ℚ⁺ _ _)))
             ≤ (p +ℚ (δ₃ +ℚ δ₃ +ℚ δ₃)) *ℚ p
-              by leq-eq-ℚ _ _
+              by leq-eq-ℚ
                 ( equational-reasoning
                   (p +ℚ δ₃) *ℚ p +ℚ (δ₃ *ℚ p +ℚ δ₃ *ℚ p)
                   ＝ (p +ℚ δ₃) *ℚ p +ℚ (δ₃ +ℚ δ₃) *ℚ p
@@ -185,7 +185,7 @@ abstract
                       ( leq-le-ℚ δ₂+δ₂<δ₁)
                       ( leq-le-ℚ δ₂<δ₁)))
             ≤ p *ℚ p +ℚ (δ₁ +ℚ δ₁) *ℚ p
-              by leq-eq-ℚ _ _ (right-distributive-mul-add-ℚ _ _ _))
+              by leq-eq-ℚ (right-distributive-mul-add-ℚ _ _ _))
           ( tr
             ( le-ℚ (p *ℚ p +ℚ (δ₁ +ℚ δ₁) *ℚ p))
             ( equational-reasoning
@@ -200,7 +200,7 @@ abstract
                       ( is-section-right-div-Group
                         ( group-mul-ℚ⁺)
                         ( p⁺)
-                        ( positive-diff-le-ℚ _ _ p²<q)))
+                        ( positive-diff-le-ℚ p²<q)))
               ＝ q
                 by is-identity-right-conjugation-add-ℚ (p *ℚ p) q)
             ( preserves-le-right-add-ℚ (p *ℚ p) _ _
@@ -223,7 +223,7 @@ abstract
   bound-rounded-above-square-le-ℚ⁺ p⁺@(p , _) q⁺@(q , _) q<p² =
     let
       (δ⁺@(δ , _) , δ+δ<⟨p²-q⟩/p) =
-        bound-double-le-ℚ⁺ ( positive-diff-le-ℚ _ _ q<p² *ℚ⁺ inv-ℚ⁺ p⁺)
+        bound-double-le-ℚ⁺ ( positive-diff-le-ℚ q<p² *ℚ⁺ inv-ℚ⁺ p⁺)
       δ<p =
         transitive-le-ℚ δ (δ +ℚ δ) p
           ( transitive-le-ℚ
@@ -238,7 +238,7 @@ abstract
             ( δ+δ<⟨p²-q⟩/p))
           ( le-right-add-rational-ℚ⁺ δ δ⁺)
     in
-      ( positive-diff-le-ℚ _ _ δ<p ,
+      ( positive-diff-le-ℚ δ<p ,
         le-diff-rational-ℚ⁺ p δ⁺ ,
         transitive-le-ℚ
           ( q)
@@ -271,7 +271,7 @@ abstract
                     ( is-section-right-div-Group
                       ( group-mul-ℚ⁺)
                       ( p⁺)
-                      ( positive-diff-le-ℚ _ _ q<p²)))
+                      ( positive-diff-le-ℚ q<p²)))
                   ( preserves-le-right-mul-ℚ⁺ p⁺ _ _ δ+δ<⟨p²-q⟩/p))))))
 
   rounded-above-square-le-ℚ⁺ :
diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
index fad1efcdb1e..db95edd8b78 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -148,19 +148,15 @@ irreflexive-le-ℚ =
 ### Strict inequality on the rationals is transitive
 
 ```agda
-module _
-  (x y z : ℚ)
-  where
-
-  opaque
-    unfolding le-ℚ-Prop
+abstract opaque
+  unfolding le-ℚ-Prop
 
-    transitive-le-ℚ : le-ℚ y z → le-ℚ x y → le-ℚ x z
-    transitive-le-ℚ =
-      transitive-le-fraction-ℤ
-        ( fraction-ℚ x)
-        ( fraction-ℚ y)
-        ( fraction-ℚ z)
+  transitive-le-ℚ : (x y z : ℚ) → le-ℚ y z → le-ℚ x y → le-ℚ x z
+  transitive-le-ℚ x y z =
+    transitive-le-fraction-ℤ
+      ( fraction-ℚ x)
+      ( fraction-ℚ y)
+      ( fraction-ℚ z)
 ```
 
 ### Concatenation rules for inequality and strict inequality on the rational numbers
@@ -297,7 +293,7 @@ module _
 
 ```agda
 module _
-  (x y : ℤ)
+  {x y : ℤ}
   where
 
   opaque
@@ -322,12 +318,12 @@ module _
 
 ```agda
 module _
-  (m n : ℕ)
+  {m n : ℕ}
   where
 
   abstract
     iff-le-rational-ℕ : le-ℕ m n ↔ le-ℚ (rational-ℕ m) (rational-ℕ n)
-    iff-le-rational-ℕ = iff-le-rational-ℤ _ _ ∘iff iff-le-int-ℕ m n
+    iff-le-rational-ℕ = iff-le-rational-ℤ ∘iff iff-le-int-ℕ m n
 
     preserves-le-rational-ℕ : le-ℕ m n → le-ℚ (rational-ℕ m) (rational-ℕ n)
     preserves-le-rational-ℕ = forward-implication iff-le-rational-ℕ
@@ -344,9 +340,7 @@ module _
   where
 
   opaque
-    unfolding add-ℚ
-    unfolding le-ℚ-Prop
-    unfolding neg-ℚ
+    unfolding add-ℚ le-ℚ-Prop neg-ℚ
 
     iff-translate-diff-le-zero-ℚ : le-ℚ zero-ℚ (y -ℚ x) ↔ le-ℚ x y
     iff-translate-diff-le-zero-ℚ =
@@ -478,8 +472,7 @@ module _
 
 ```agda
 opaque
-  unfolding le-ℚ-Prop
-  unfolding leq-ℚ-Prop
+  unfolding le-ℚ-Prop leq-ℚ-Prop
 
   decide-le-leq-ℚ : (x y : ℚ) → le-ℚ x y + leq-ℚ y x
   decide-le-leq-ℚ x y =
diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index 641b00f2ba4..7a8cea0f775 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -23,7 +23,11 @@ open import metric-spaces.equality-of-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.metrics
 
+open import real-numbers.addition-nonnegative-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.saturation-inequality-nonnegative-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -154,10 +158,10 @@ module _
                     ( r⁺)
                     ( backward-implication
                       ( is-metric-M-ρ r⁺ y z)
-                      ( leq-le-ℝ _ _ ρyz<r))
+                      ( leq-le-ℝ ρyz<r))
                     ( backward-implication
                       ( is-metric-M-ρ q⁺ x y)
-                      ( leq-le-ℝ _ _ ρxy<q)))))
+                      ( leq-le-ℝ ρxy<q)))))
               ( d'<d))
 
     is-extensional-is-metric-of-Metric-Space :
diff --git a/src/metric-spaces/metrics.lagda.md b/src/metric-spaces/metrics.lagda.md
index 66155d05b2d..6ed22735b3d 100644
--- a/src/metric-spaces/metrics.lagda.md
+++ b/src/metric-spaces/metrics.lagda.md
@@ -1,6 +1,8 @@
 # Metrics
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module metric-spaces.metrics where
 ```
 
@@ -36,9 +38,13 @@ open import metric-spaces.saturated-rational-neighborhood-relations
 open import metric-spaces.symmetric-rational-neighborhood-relations
 open import metric-spaces.triangular-rational-neighborhood-relations
 
+open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.saturation-inequality-nonnegative-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -338,9 +344,7 @@ module _
         ( not-leq-le-ℝ
           ( real-ℚ⁺ δ)
           ( real-ℝ⁰⁺ (dist-Metric X μ x y)))
-        ( leq-le-ℝ
-          ( real-ℝ⁰⁺ (dist-Metric X μ x y))
-          ( real-ℚ⁺ ε))
+        ( leq-le-ℝ)
         ( is-located-le-ℝ
           ( real-ℝ⁰⁺ (dist-Metric X μ x y))
           ( rational-ℚ⁺ δ)
@@ -365,6 +369,6 @@ module _
       bounded-dist-Metric-Space (metric-space-Metric X μ) x y
     bounded-dist-metric-space-Metric x y =
       map-tot-exists
-        ( λ ε → leq-le-ℝ (real-ℝ⁰⁺ (dist-Metric X μ x y)) (real-ℚ⁺ ε))
+        ( λ ε → leq-le-ℝ)
         ( le-some-positive-rational-ℝ⁰⁺ (dist-Metric X μ x y))
 ```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index dec3db12f13..1a10077ffcd 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -8,6 +8,7 @@ module real-numbers where
 open import real-numbers.absolute-value-closed-intervals-real-numbers public
 open import real-numbers.absolute-value-real-numbers public
 open import real-numbers.addition-lower-dedekind-real-numbers public
+open import real-numbers.addition-nonnegative-real-numbers public
 open import real-numbers.addition-real-numbers public
 open import real-numbers.addition-upper-dedekind-real-numbers public
 open import real-numbers.apartness-real-numbers public
@@ -22,7 +23,9 @@ open import real-numbers.difference-real-numbers public
 open import real-numbers.distance-real-numbers public
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers public
 open import real-numbers.finitely-enumerable-subsets-real-numbers public
+open import real-numbers.inequalities-addition-real-numbers public
 open import real-numbers.inequality-lower-dedekind-real-numbers public
+open import real-numbers.inequality-nonnegative-real-numbers public
 open import real-numbers.inequality-positive-real-numbers public
 open import real-numbers.inequality-real-numbers public
 open import real-numbers.inequality-upper-dedekind-real-numbers public
@@ -42,6 +45,7 @@ open import real-numbers.maximum-finite-families-real-numbers public
 open import real-numbers.maximum-inhabited-finitely-enumerable-subsets-real-numbers public
 open import real-numbers.maximum-lower-dedekind-real-numbers public
 open import real-numbers.maximum-upper-dedekind-real-numbers public
+open import real-numbers.metric-space-of-nonnegative-real-numbers public
 open import real-numbers.metric-space-of-real-numbers public
 open import real-numbers.minimum-finite-families-real-numbers public
 open import real-numbers.minimum-inhabited-finitely-enumerable-subsets-real-numbers public
@@ -67,10 +71,13 @@ open import real-numbers.rational-real-numbers public
 open import real-numbers.rational-upper-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-lower-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
+open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
+open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.square-roots-nonnegative-real-numbers public
 open import real-numbers.squares-real-numbers public
+open import real-numbers.strict-inequalities-addition-real-numbers public
 open import real-numbers.strict-inequality-nonnegative-real-numbers public
 open import real-numbers.strict-inequality-positive-real-numbers public
 open import real-numbers.strict-inequality-real-numbers public
diff --git a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
index 194cae093fc..f040689b7c2 100644
--- a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
@@ -52,7 +52,7 @@ abstract
       ( z)
       ( max-ℝ (neg-ℝ x) y)
       ( transitive-leq-ℝ _ _ _ (leq-right-max-ℝ _ _) z≤y)
-      ( transitive-leq-ℝ _ _ _ (leq-left-max-ℝ _ _) (neg-leq-ℝ _ _ x≤z))
+      ( transitive-leq-ℝ _ _ _ (leq-left-max-ℝ _ _) (neg-leq-ℝ x≤z))
 ```
 
 ### The maximum absolute value of an interval is nonnegative
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index 8695d96e159..d31b66d8b89 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -28,7 +28,9 @@ open import metric-spaces.short-functions-metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.isometry-negation-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
@@ -96,7 +98,7 @@ opaque
       ( lower-cut-ℝ (abs-ℝ x) q)
       ( id)
       ( λ (x<0 , 0<x) → ex-falso (is-disjoint-cut-ℝ x zero-ℚ (0<x , x<0)))
-      ( is-located-lower-upper-cut-ℝ (abs-ℝ x) q zero-ℚ q<0)
+      ( is-located-lower-upper-cut-ℝ (abs-ℝ x) q<0)
 
 nonnegative-abs-ℝ : {l : Level} → ℝ l → ℝ⁰⁺ l
 nonnegative-abs-ℝ x = (abs-ℝ x , is-nonnegative-abs-ℝ x)
@@ -134,10 +136,10 @@ module _
       tr
         ( leq-ℝ (neg-ℝ (abs-ℝ x)))
         ( neg-neg-ℝ x)
-        ( neg-leq-ℝ (neg-ℝ x) (abs-ℝ x) neg-leq-abs-ℝ)
+        ( neg-leq-ℝ neg-leq-abs-ℝ)
 
     neg-leq-neg-abs-ℝ : leq-ℝ (neg-ℝ (abs-ℝ x)) (neg-ℝ x)
-    neg-leq-neg-abs-ℝ = neg-leq-ℝ x (abs-ℝ x) leq-abs-ℝ
+    neg-leq-neg-abs-ℝ = neg-leq-ℝ leq-abs-ℝ
 ```
 
 ### If `|x| ＝ 0` then `x ＝ 0`
@@ -231,23 +233,11 @@ module _
       leq-abs-leq-leq-neg-ℝ
         ( x +ℝ y)
         ( abs-ℝ x +ℝ abs-ℝ y)
-        ( preserves-leq-add-ℝ
-          ( x)
-          ( abs-ℝ x)
-          ( y)
-          ( abs-ℝ y)
-          ( leq-abs-ℝ x)
-          ( leq-abs-ℝ y))
+        ( preserves-leq-add-ℝ (leq-abs-ℝ x) (leq-abs-ℝ y))
         ( inv-tr
           ( λ z → leq-ℝ z (abs-ℝ x +ℝ abs-ℝ y))
           ( distributive-neg-add-ℝ x y)
-          ( preserves-leq-add-ℝ
-            ( neg-ℝ x)
-            ( abs-ℝ x)
-            ( neg-ℝ y)
-            ( abs-ℝ y)
-            ( neg-leq-abs-ℝ x)
-            ( neg-leq-abs-ℝ y)))
+          ( preserves-leq-add-ℝ (neg-leq-abs-ℝ x) (neg-leq-abs-ℝ y)))
 ```
 
 ### The absolute value is a short function
@@ -354,7 +344,7 @@ module _
     leq-abs-sqrt-square-ℝ' :
       leq-ℝ' (real-sqrt-ℝ⁰⁺ (nonnegative-square-ℝ x)) (abs-ℝ x)
     leq-abs-sqrt-square-ℝ' q |x|<q@(x<q , -x<q) =
-      ( is-positive-is-in-upper-cut-ℝ⁰⁺ (nonnegative-abs-ℝ x) q (x<q , -x<q) ,
+      ( is-positive-is-in-upper-cut-ℝ⁰⁺ (nonnegative-abs-ℝ x) (x<q , -x<q) ,
         is-in-upper-cut-square-pos-neg-ℝ x q x<q -x<q)
 
     eq-abs-sqrt-square-ℝ : abs-ℝ x ＝ real-sqrt-ℝ⁰⁺ (nonnegative-square-ℝ x)
diff --git a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
index d970a20b789..bd1e58abfe1 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -114,7 +114,7 @@ module _
         (r , q<r , r<x+y) ← ∃r
         ((rx , ry) , (rx<x , ry<y , r=rx+ry)) ← r<x+y
         let
-          r-q⁺ = positive-diff-le-ℚ q r q<r
+          r-q⁺ = positive-diff-le-ℚ q<r
           ε⁺@(ε , _) = mediant-zero-ℚ⁺ r-q⁺
         intro-exists
           ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
diff --git a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
new file mode 100644
index 00000000000..0735a723925
--- /dev/null
+++ b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,136 @@
+# Addition of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.addition-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-nonnegative-rational-numbers
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.identity-types
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequality-nonnegative-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+The [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are closed under
+[addition](real-numbers.addition-real-numbers.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  real-add-ℝ⁰⁺ : ℝ (l1 ⊔ l2)
+  real-add-ℝ⁰⁺ = real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y
+
+  abstract
+    is-nonnegative-real-add-ℝ⁰⁺ : is-nonnegative-ℝ real-add-ℝ⁰⁺
+    is-nonnegative-real-add-ℝ⁰⁺ =
+      tr
+        ( λ z → leq-ℝ z (real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y))
+        ( left-unit-law-add-ℝ zero-ℝ)
+        ( preserves-leq-add-ℝ
+          ( is-nonnegative-real-ℝ⁰⁺ x)
+          ( is-nonnegative-real-ℝ⁰⁺ y))
+
+  add-ℝ⁰⁺ : ℝ⁰⁺ (l1 ⊔ l2)
+  add-ℝ⁰⁺ = (real-add-ℝ⁰⁺ , is-nonnegative-real-add-ℝ⁰⁺)
+
+infixl 35 _+ℝ⁰⁺_
+
+_+ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → ℝ⁰⁺ (l1 ⊔ l2)
+_+ℝ⁰⁺_ = add-ℝ⁰⁺
+```
+
+## Properties
+
+### Unit laws for addition
+
+```agda
+module _
+  {l : Level} (x : ℝ⁰⁺ l)
+  where
+
+  abstract
+    left-unit-law-add-ℝ⁰⁺ : zero-ℝ⁰⁺ +ℝ⁰⁺ x ＝ x
+    left-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (left-unit-law-add-ℝ _)
+
+    right-unit-law-add-ℝ⁰⁺ : x +ℝ⁰⁺ zero-ℝ⁰⁺ ＝ x
+    right-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (right-unit-law-add-ℝ _)
+```
+
+### Addition preserves inequality
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
+  where
+
+  abstract
+    preserves-leq-add-ℝ⁰⁺ :
+      leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ z w → leq-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
+    preserves-leq-add-ℝ⁰⁺ = preserves-leq-add-ℝ
+```
+
+### The canonical embedding of nonnegative rational numbers to nonnegative real numbers preserves addition
+
+```agda
+abstract
+  add-nonnegative-real-ℚ⁰⁺ :
+    (p q : ℚ⁰⁺) →
+    nonnegative-real-ℚ⁰⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁰⁺ q ＝
+    nonnegative-real-ℚ⁰⁺ (p +ℚ⁰⁺ q)
+  add-nonnegative-real-ℚ⁰⁺ p q =
+    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q))
+```
+
+### The canonical embedding of positive rational numbers to nonnegative real numbers preserves addition
+
+```agda
+abstract
+  add-nonnegative-real-ℚ⁺ :
+    (p q : ℚ⁺) →
+    nonnegative-real-ℚ⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁺ q ＝
+    nonnegative-real-ℚ⁺ (p +ℚ⁺ q)
+  add-nonnegative-real-ℚ⁺ p q =
+    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁺ p) (rational-ℚ⁺ q))
+```
+
+### Addition preserves strict inequality
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
+  where
+
+  abstract
+    preserves-le-add-ℝ⁰⁺ :
+      le-ℝ⁰⁺ x y → le-ℝ⁰⁺ z w → le-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
+    preserves-le-add-ℝ⁰⁺ = preserves-le-add-ℝ
+```
diff --git a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
index 44bd7f51aee..616124b9515 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -111,7 +111,7 @@ module _
         (p , p<q , x+y<p) ← ∃p
         ((px , py) , (x<px , y<py , p=px+py)) ← x+y<p
         let
-          q-p⁺ = positive-diff-le-ℚ p q p<q
+          q-p⁺ = positive-diff-le-ℚ p<q
           ε⁺@(ε , _) = mediant-zero-ℚ⁺ q-p⁺
         intro-exists
           ( px +ℚ ε , q -ℚ (px +ℚ ε))
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index af401404875..ffca8483f91 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -204,7 +204,7 @@ module _
                       ( preserves-leq-left-add-ℚ (q -ℚ p) p' p p'≤p)))
                   ( q'∈U)))
             ( decide-le-leq-ℚ p p'))
-        ( arithmetically-located (positive-diff-le-ℚ p q p<q))
+        ( arithmetically-located (positive-diff-le-ℚ p<q))
 
     is-located-is-weakly-arithmetically-located-lower-upper-ℝ :
       is-weakly-arithmetically-located-lower-upper-ℝ x y →
@@ -270,11 +270,7 @@ module _
                 ( is-in-upper-cut-ℝ x)
                 ( associative-add-ℚ p ε ε)
                 ( x<p+2ε)))
-        ( is-located-lower-upper-cut-ℝ
-          ( x)
-          ( p +ℚ ε)
-          ( (p +ℚ ε) +ℚ ε)
-          ( le-right-add-rational-ℚ⁺ (p +ℚ ε) ε⁺))
+        ( is-located-lower-upper-cut-ℝ x (le-right-add-rational-ℚ⁺ (p +ℚ ε) ε⁺))
 
     is-arithmetically-located-ℝ :
       is-arithmetically-located-lower-upper-ℝ (lower-real-ℝ x) (upper-real-ℝ x)
@@ -376,7 +372,7 @@ abstract
                     ≤ rational-ℕ m +ℚ rational-ℕ n
                       by preserves-leq-right-add-ℚ _ _ _ (leq-le-ℚ max|p||q|<n)
                     ≤ rational-ℕ (m +ℕ n)
-                      by leq-eq-ℚ _ _ (add-rational-ℕ _ _))
+                      by leq-eq-ℚ (add-rational-ℕ _ _))
                 ( chain-of-inequalities
                     neg-ℚ a
                     ≤ neg-ℚ (b -ℚ ε)
@@ -384,7 +380,7 @@ abstract
                         neg-leq-ℚ
                           ( leq-transpose-right-add-ℚ _ _ _ (leq-le-ℚ b<a+ε))
                     ≤ ε -ℚ b
-                      by leq-eq-ℚ _ _ (distributive-neg-diff-ℚ _ _)
+                      by leq-eq-ℚ (distributive-neg-diff-ℚ _ _)
                     ≤ ε -ℚ p
                       by
                         preserves-leq-right-add-ℚ ε
@@ -399,7 +395,7 @@ abstract
                     ≤ ε +ℚ max-ℚ (rational-abs-ℚ p) (rational-abs-ℚ q)
                       by
                         preserves-leq-add-ℚ
-                          ( leq-eq-ℚ _ _ (rational-abs-rational-ℚ⁺ ε⁺))
+                          ( leq-eq-ℚ (rational-abs-rational-ℚ⁺ ε⁺))
                           ( leq-left-max-ℚ _ _)
                     ≤ rational-ℕ m +ℚ rational-ℕ n
                       by
@@ -407,7 +403,7 @@ abstract
                           ( leq-le-ℚ ε<m)
                           ( leq-le-ℚ max|p||q|<n)
                     ≤ rational-ℕ (m +ℕ n)
-                      by leq-eq-ℚ _ _ (add-rational-ℕ _ _))
+                      by leq-eq-ℚ (add-rational-ℕ _ _))
             |b|≤|a|+ε =
               leq-abs-leq-leq-neg-ℚ b (rational-abs-ℚ a +ℚ ε)
                 ( chain-of-inequalities
@@ -419,9 +415,7 @@ abstract
                     ≤ rational-abs-ℚ a +ℚ rational-abs-ℚ ε
                       by triangle-inequality-abs-ℚ _ _
                     ≤ rational-abs-ℚ a +ℚ ε
-                      by
-                        leq-eq-ℚ _ _
-                          ( ap-add-ℚ refl (rational-abs-rational-ℚ⁺ ε⁺)))
+                      by leq-eq-ℚ (ap-add-ℚ refl (rational-abs-rational-ℚ⁺ ε⁺)))
                 ( chain-of-inequalities
                     neg-ℚ b
                     ≤ neg-ℚ a
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 7d521f85c1b..2482c2f8552 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -32,6 +32,7 @@ open import order-theory.least-upper-bounds-large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
@@ -96,8 +97,8 @@ module _
           map-disjunction
             ( λ p<y → inr-disjunction p<y)
             ( x<q ,_)
-            ( is-located-lower-upper-cut-ℝ y p q p<q))
-        ( is-located-lower-upper-cut-ℝ x p q p<q)
+            ( is-located-lower-upper-cut-ℝ y p<q))
+        ( is-located-lower-upper-cut-ℝ x p<q)
       where
         claim : Prop (l1 ⊔ l2)
         claim =
@@ -331,8 +332,6 @@ module _
             ( max-ℝ x z)
             ( max-ℝ x z +ℝ real-ℚ⁺ d)
             ( leq-le-ℝ
-              ( max-ℝ x z)
-              ( max-ℝ x z +ℝ real-ℚ⁺ d)
               ( le-left-add-real-ℝ⁺
                 ( max-ℝ x z)
                 ( positive-real-ℚ⁺ d)))
@@ -453,14 +452,14 @@ module _
             ( max-ℝ x y)
             ( le-diff-real-ℝ⁺ (max-ℝ x y) (positive-real-ℚ⁺ ε⁺))
         (r , q-<ℝ-r , r<max) ← dense-rational-le-ℝ (real-ℚ q) (max-ℝ x y) q<max
-        let q<r = reflects-le-real-ℚ q r q-<ℝ-r
+        let q<r = reflects-le-real-ℚ q-<ℝ-r
         map-disjunction
           ( λ q<x →
             transitive-le-ℝ
               ( max-ℝ x y -ℝ real-ℚ ε)
               ( real-ℚ q)
               ( x)
-              ( le-real-is-in-lower-cut-ℚ q x q<x)
+              ( le-real-is-in-lower-cut-ℚ x q<x)
               ( max-ε<q))
           ( λ x<r →
             elim-disjunction
@@ -470,7 +469,7 @@ module _
                   ( max-ℝ x y -ℝ real-ℚ ε)
                   ( real-ℚ q)
                   ( y)
-                  ( le-real-is-in-lower-cut-ℚ q y q<y)
+                  ( le-real-is-in-lower-cut-ℚ y q<y)
                   ( max-ε<q))
               ( λ y<r →
                 ex-falso
@@ -478,11 +477,9 @@ module _
                     ( max-ℝ x y)
                     ( concatenate-leq-le-ℝ (max-ℝ x y) (real-ℚ r) (max-ℝ x y)
                       ( leq-max-leq-leq-ℝ x y (real-ℚ r)
-                        ( leq-le-ℝ x (real-ℚ r)
-                          ( le-real-is-in-upper-cut-ℚ r x x<r))
-                        ( leq-le-ℝ y (real-ℚ r)
-                          ( le-real-is-in-upper-cut-ℚ r y y<r)))
+                        ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ x x<r))
+                        ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ y y<r)))
                       ( r<max))))
-              ( is-located-lower-upper-cut-ℝ y q r q<r))
-          ( is-located-lower-upper-cut-ℝ x q r q<r)
+              ( is-located-lower-upper-cut-ℝ y q<r))
+          ( is-located-lower-upper-cut-ℝ x q<r)
 ```
diff --git a/src/real-numbers/binary-minimum-real-numbers.lagda.md b/src/real-numbers/binary-minimum-real-numbers.lagda.md
index f156bb46744..121bf5d42b9 100644
--- a/src/real-numbers/binary-minimum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-minimum-real-numbers.lagda.md
@@ -79,10 +79,8 @@ module _
       tr
         ( λ w → leq-ℝ w (min-ℝ x y))
         ( neg-neg-ℝ z)
-        ( neg-leq-ℝ (max-ℝ (neg-ℝ x) (neg-ℝ y)) (neg-ℝ z)
-          ( leq-max-leq-leq-ℝ _ _ (neg-ℝ z)
-            ( neg-leq-ℝ z x z≤x)
-            ( neg-leq-ℝ z y z≤y)))
+        ( neg-leq-ℝ
+          ( leq-max-leq-leq-ℝ _ _ (neg-ℝ z) (neg-leq-ℝ z≤x) (neg-leq-ℝ z≤y)))
     pr2 (is-greatest-binary-lower-bound-min-ℝ z) z≤min =
       let
         case :
@@ -93,7 +91,7 @@ module _
             ( tr
               ( leq-ℝ (min-ℝ x y))
               ( neg-neg-ℝ _)
-              ( neg-leq-ℝ (neg-ℝ w) (max-ℝ (neg-ℝ x) (neg-ℝ y)) -w≤max))
+              ( neg-leq-ℝ -w≤max))
             ( z≤min)
       in (case x (leq-left-max-ℝ _ _) , case y (leq-right-max-ℝ _ _))
 
@@ -172,10 +170,11 @@ module _
           le-ℝ (max-ℝ (neg-ℝ x) (neg-ℝ y) -ℝ real-ℚ⁺ ε) (neg-ℝ w) →
           le-ℝ w (min-ℝ x y +ℝ real-ℚ⁺ ε)
         case w max-ε<-w =
-          binary-tr le-ℝ
+          binary-tr
+            ( le-ℝ)
             ( neg-neg-ℝ w)
             ( distributive-neg-diff-ℝ _ _ ∙ commutative-add-ℝ _ _)
-            ( neg-le-ℝ _ (neg-ℝ w) max-ε<-w)
+            ( neg-le-ℝ max-ε<-w)
       in
         elim-disjunction
           ( (le-prop-ℝ x (min-ℝ x y +ℝ real-ℚ⁺ ε)) ∨
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index c80918effe6..2a3e5294511 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -45,6 +45,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.distance-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.metric-space-of-real-numbers
@@ -369,10 +370,8 @@ module _
         ( upper-real-lim-cauchy-approximation-ℝ)
     is-located-lower-upper-cut-lim-cauchy-approximation-ℝ p q p<q =
       let
-        ε'⁺@(ε' , _) , 2ε'⁺<q-p =
-          bound-double-le-ℚ⁺ (positive-diff-le-ℚ p q p<q)
-        ε⁺@(ε , _) , 2ε⁺<ε'⁺ =
-          bound-double-le-ℚ⁺ ε'⁺
+        ε'⁺@(ε' , _) , 2ε'⁺<q-p = bound-double-le-ℚ⁺ (positive-diff-le-ℚ p<q)
+        ε⁺@(ε , _) , 2ε⁺<ε'⁺ = bound-double-le-ℚ⁺ ε'⁺
         2ε' = ε' +ℚ ε'
         2ε = ε +ℚ ε
         4ε = 2ε +ℚ 2ε
@@ -383,8 +382,6 @@ module _
           ( intro-exists (ε⁺ , ε⁺))
           ( is-located-lower-upper-cut-ℝ
             ( map-cauchy-approximation-ℝ x ε⁺)
-            ( p +ℚ 2ε)
-            ( q -ℚ 2ε)
             ( le-transpose-left-add-ℚ
               ( p +ℚ 2ε)
               ( 2ε)
@@ -479,8 +476,6 @@ module _
               ( ε+θ)
               ( lim)
               ( leq-le-ℝ
-                ( xε -ℝ ε+θ)
-                ( lim)
                 ( intro-exists
                   ( q)
                   ( xε-ε-θ<q ,
@@ -500,8 +495,6 @@ module _
                 ( xε)
                 ( ε+θ)
                 ( leq-le-ℝ
-                  ( lim)
-                  ( xε +ℝ ε+θ)
                   ( intro-exists
                     ( r)
                     ( intro-exists
diff --git a/src/real-numbers/closed-intervals-real-numbers.lagda.md b/src/real-numbers/closed-intervals-real-numbers.lagda.md
index 2369f9cf543..846087aa483 100644
--- a/src/real-numbers/closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/closed-intervals-real-numbers.lagda.md
@@ -84,7 +84,7 @@ upper-bound-closed-interval-ℝ =
 ```agda
 unit-closed-interval-ℝ : closed-interval-ℝ lzero lzero
 unit-closed-interval-ℝ =
-  ((zero-ℝ , one-ℝ) , preserves-leq-real-ℚ zero-ℚ one-ℚ leq-zero-one-ℚ)
+  ((zero-ℝ , one-ℝ) , preserves-leq-real-ℚ leq-zero-one-ℚ)
 ```
 
 ### Closed intervals in the real numbers are closed in the metric space of real numbers
@@ -103,7 +103,7 @@ opaque
         let open do-syntax-trunc-Prop (lower-cut-ℝ x q)
         in do
           (r , q<r , r<a) ← forward-implication (is-rounded-lower-cut-ℝ a q) q<a
-          let ε⁺@(ε , _) = positive-diff-le-ℚ q r q<r
+          let ε⁺@(ε , _) = positive-diff-le-ℚ q<r
           (y , Nεxy , a≤y , _) ← H ε⁺
           pr1 Nεxy
             ( q)
@@ -117,7 +117,7 @@ opaque
         let open do-syntax-trunc-Prop (lower-cut-ℝ b q)
         in do
           (r , q<r , r<x) ← forward-implication (is-rounded-lower-cut-ℝ x q) q<x
-          let ε⁺@(ε , _) = positive-diff-le-ℚ q r q<r
+          let ε⁺@(ε , _) = positive-diff-le-ℚ q<r
           (y , Nεxy , _ , y≤b) ← H ε⁺
           y≤b
             ( q)
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 3ce56e8f4b3..a6183b2bc47 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -179,9 +179,9 @@ module _
   is-disjoint-cut-ℝ = pr1 (pr2 (pr2 x))
 
   is-located-lower-upper-cut-ℝ :
-    (q r : ℚ) → le-ℚ q r →
+    {q r : ℚ} → le-ℚ q r →
     type-disjunction-Prop (lower-cut-ℝ q) (upper-cut-ℝ r)
-  is-located-lower-upper-cut-ℝ = pr2 (pr2 (pr2 x))
+  is-located-lower-upper-cut-ℝ {q} {r} = pr2 (pr2 (pr2 x)) q r
 
   cut-ℝ : subtype l ℚ
   cut-ℝ q =
@@ -315,7 +315,7 @@ module _
           ( lower-cut-ℝ x p)
           ( upper-cut-ℝ x q)
           ( pr2 I)
-          ( is-located-lower-upper-cut-ℝ x p q ( pr1 I)))
+          ( is-located-lower-upper-cut-ℝ x ( pr1 I)))
 
   subset-lower-complement-upper-cut-lower-cut-ℝ :
     lower-cut-ℝ x ⊆ lower-complement-upper-cut-ℝ x
@@ -382,7 +382,7 @@ module _
           ( lower-cut-ℝ x p)
           ( upper-cut-ℝ x q)
           ( pr2 I)
-          ( is-located-lower-upper-cut-ℝ x p q (pr1 I)))
+          ( is-located-lower-upper-cut-ℝ x (pr1 I)))
 
   subset-upper-complement-lower-cut-upper-cut-ℝ :
     upper-cut-ℝ x ⊆ upper-complement-lower-cut-ℝ x
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 446e7c220ae..df3a005ee36 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -28,6 +28,7 @@ open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.multiplication-real-numbers
@@ -138,14 +139,12 @@ abstract
     lower-neighborhood-ℝ d y x
   lower-neighborhood-diff-ℝ d⁺@(d , _) x y x-y≤d q q+d<x =
     is-in-lower-cut-le-real-ℚ
-      ( q)
       ( y)
       ( concatenate-le-leq-ℝ
         ( real-ℚ q)
         ( x -ℝ real-ℚ d)
         ( y)
         ( le-real-is-in-lower-cut-ℚ
-          ( q)
           ( x -ℝ real-ℚ d)
           ( transpose-add-is-in-lower-cut-ℝ x q d q+d<x))
         ( swap-right-diff-leq-ℝ x y (real-ℚ d) x-y≤d))
@@ -213,9 +212,6 @@ abstract
     dist-ℝ x z ≤-ℝ dist-ℝ x y +ℝ dist-ℝ y z
   triangle-inequality-dist-ℝ x y z =
     preserves-leq-left-sim-ℝ
-      ( dist-ℝ x y +ℝ dist-ℝ y z)
-      ( abs-ℝ (x -ℝ y +ℝ y -ℝ z))
-      ( abs-ℝ (x -ℝ z))
       ( preserves-sim-abs-ℝ
         ( similarity-reasoning-ℝ
           x -ℝ y +ℝ y -ℝ z
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
new file mode 100644
index 00000000000..654f922bc10
--- /dev/null
+++ b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
@@ -0,0 +1,234 @@
+# Inequalities about addition and subtraction on the real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.inequalities-addition-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.function-types
+open import foundation.functoriality-cartesian-product-types
+open import foundation.logical-equivalences
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import logic.functoriality-existential-quantification
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.rational-real-numbers
+```
+
+</details>
+
+## Idea
+
+This file describes lemmas about
+[inequalities](real-numbers.inequality-real-numbers.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md) related to
+[addition](real-numbers.addition-real-numbers.md) and
+[subtraction](real-numbers.difference-real-numbers.md).
+
+## Lemmas
+
+### Inequality on the real numbers is translation invariant
+
+```agda
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  opaque
+    unfolding add-ℝ leq-ℝ
+
+    preserves-leq-right-add-ℝ : leq-ℝ x y → leq-ℝ (x +ℝ z) (y +ℝ z)
+    preserves-leq-right-add-ℝ x≤y _ =
+      map-tot-exists (λ (qx , _) → map-product (x≤y qx) id)
+
+    preserves-leq-left-add-ℝ : leq-ℝ x y → leq-ℝ (z +ℝ x) (z +ℝ y)
+    preserves-leq-left-add-ℝ x≤y _ =
+      map-tot-exists (λ (_ , qx) → map-product id (map-product (x≤y qx) id))
+
+abstract
+  preserves-leq-diff-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    leq-ℝ x y → leq-ℝ (x -ℝ z) (y -ℝ z)
+  preserves-leq-diff-ℝ z = preserves-leq-right-add-ℝ (neg-ℝ z)
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  abstract
+    reflects-leq-right-add-ℝ : leq-ℝ (x +ℝ z) (y +ℝ z) → leq-ℝ x y
+    reflects-leq-right-add-ℝ x+z≤y+z =
+      preserves-leq-sim-ℝ
+        ( (x +ℝ z) -ℝ z)
+        ( x)
+        ( (y +ℝ z) -ℝ z)
+        ( y)
+        ( cancel-right-add-diff-ℝ x z)
+        ( cancel-right-add-diff-ℝ y z)
+        ( preserves-leq-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z≤y+z)
+
+    reflects-leq-left-add-ℝ : leq-ℝ (z +ℝ x) (z +ℝ y) → leq-ℝ x y
+    reflects-leq-left-add-ℝ z+x≤z+y =
+      reflects-leq-right-add-ℝ
+        ( binary-tr
+          ( leq-ℝ)
+          ( commutative-add-ℝ z x)
+          ( commutative-add-ℝ z y)
+          ( z+x≤z+y))
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  iff-translate-right-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (x +ℝ z) (y +ℝ z)
+  pr1 iff-translate-right-leq-ℝ = preserves-leq-right-add-ℝ z x y
+  pr2 iff-translate-right-leq-ℝ = reflects-leq-right-add-ℝ z x y
+
+  iff-translate-left-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (z +ℝ x) (z +ℝ y)
+  pr1 iff-translate-left-leq-ℝ = preserves-leq-left-add-ℝ z x y
+  pr2 iff-translate-left-leq-ℝ = reflects-leq-left-add-ℝ z x y
+
+abstract
+  preserves-leq-add-ℝ :
+    {l1 l2 l3 l4 : Level} {a : ℝ l1} {b : ℝ l2} {c : ℝ l3} {d : ℝ l4} →
+    leq-ℝ a b → leq-ℝ c d → leq-ℝ (a +ℝ c) (b +ℝ d)
+  preserves-leq-add-ℝ {a = a} {b = b} {c = c} {d = d} a≤b c≤d =
+    transitive-leq-ℝ
+      ( a +ℝ c)
+      ( a +ℝ d)
+      ( b +ℝ d)
+      ( preserves-leq-right-add-ℝ d a b a≤b)
+      ( preserves-leq-left-add-ℝ a c d c≤d)
+```
+
+### Transposition laws
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    leq-transpose-left-diff-ℝ : leq-ℝ (x -ℝ y) z → leq-ℝ x (z +ℝ y)
+    leq-transpose-left-diff-ℝ x-y≤z =
+      preserves-leq-left-sim-ℝ
+        ( cancel-right-diff-add-ℝ x y)
+        ( preserves-leq-right-add-ℝ y (x -ℝ y) z x-y≤z)
+
+    leq-transpose-left-add-ℝ : leq-ℝ (x +ℝ y) z → leq-ℝ x (z -ℝ y)
+    leq-transpose-left-add-ℝ x+y≤z =
+      preserves-leq-left-sim-ℝ
+        ( cancel-right-add-diff-ℝ x y)
+        ( preserves-leq-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y≤z)
+
+    leq-transpose-right-add-ℝ : leq-ℝ x (y +ℝ z) → leq-ℝ (x -ℝ z) y
+    leq-transpose-right-add-ℝ x≤y+z =
+      preserves-leq-right-sim-ℝ
+        ( cancel-right-add-diff-ℝ y z)
+        ( preserves-leq-right-add-ℝ (neg-ℝ z) x (y +ℝ z) x≤y+z)
+
+    leq-transpose-right-diff-ℝ : leq-ℝ x (y -ℝ z) → leq-ℝ (x +ℝ z) y
+    leq-transpose-right-diff-ℝ x≤y-z =
+      preserves-leq-right-sim-ℝ
+        ( cancel-right-diff-add-ℝ y z)
+        ( preserves-leq-right-add-ℝ z x (y -ℝ z) x≤y-z)
+```
+
+### Swapping laws
+
+```agda
+abstract
+  swap-right-diff-leq-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    leq-ℝ (x -ℝ y) z → leq-ℝ (x -ℝ z) y
+  swap-right-diff-leq-ℝ x y z x-y≤z =
+    leq-transpose-right-add-ℝ
+      ( x)
+      ( y)
+      ( z)
+      ( tr
+        ( leq-ℝ x)
+        ( commutative-add-ℝ _ _)
+        ( leq-transpose-left-diff-ℝ x y z x-y≤z))
+```
+
+### Addition of real numbers preserves lower neighborhoods
+
+```agda
+module _
+  {l1 l2 l3 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    preserves-lower-neighborhood-leq-left-add-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d =
+      inv-tr
+        ( leq-ℝ (x +ℝ y))
+        ( associative-add-ℝ x z (real-ℚ⁺ d))
+        ( preserves-leq-left-add-ℝ
+          ( x)
+          ( y)
+          ( z +ℝ real-ℚ⁺ d)
+          ( z≤y+d))
+
+    preserves-lower-neighborhood-leq-right-add-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-right-add-ℝ z≤y+d =
+      binary-tr
+        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
+        ( commutative-add-ℝ x y)
+        ( commutative-add-ℝ x z)
+        ( preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d)
+```
+
+### Addition of real numbers reflects lower neighborhoods
+
+```agda
+module _
+  {l1 l2 l3 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    reflects-lower-neighborhood-leq-left-add-ℝ :
+      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d) →
+      leq-ℝ y (z +ℝ real-ℚ⁺ d)
+    reflects-lower-neighborhood-leq-left-add-ℝ x+y≤x+z+d =
+      reflects-leq-left-add-ℝ
+        ( x)
+        ( y)
+        ( z +ℝ real-ℚ⁺ d)
+        ( tr
+          ( leq-ℝ (x +ℝ y))
+          ( associative-add-ℝ x z (real-ℚ⁺ d))
+          ( x+y≤x+z+d))
+
+    reflects-lower-neighborhood-leq-right-add-ℝ :
+      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d) →
+      leq-ℝ y (z +ℝ real-ℚ⁺ d)
+    reflects-lower-neighborhood-leq-right-add-ℝ y+x≤z+y+d =
+      reflects-lower-neighborhood-leq-left-add-ℝ
+        ( binary-tr
+          ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
+          ( commutative-add-ℝ y x)
+          ( commutative-add-ℝ z x)
+          ( y+x≤z+y+d))
+```
diff --git a/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
new file mode 100644
index 00000000000..fa71aea44a1
--- /dev/null
+++ b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,159 @@
+# Inequality of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.inequality-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.coproduct-types
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.function-types
+open import foundation.logical-equivalences
+open import foundation.propositional-truncations
+open import foundation.propositions
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "standard ordering" Disambiguation="on the nonnegative real numbers" Agda=leq-ℝ⁰⁺}}
+on the [nonnegative real numbers](real-numbers.nonnegative-real-numbers.md) is
+inherited from the [standard ordering](real-numbers.inequality-real-numbers.md)
+on [real numbers](real-numbers.dedekind-real-numbers.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  leq-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
+  leq-prop-ℝ⁰⁺ = leq-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+
+  leq-ℝ⁰⁺ : UU (l1 ⊔ l2)
+  leq-ℝ⁰⁺ = type-Prop leq-prop-ℝ⁰⁺
+```
+
+## Properties
+
+### Zero is less than or equal to every nonnegative real number
+
+```agda
+leq-zero-ℝ⁰⁺ : {l : Level} (x : ℝ⁰⁺ l) → leq-ℝ⁰⁺ zero-ℝ⁰⁺ x
+leq-zero-ℝ⁰⁺ = is-nonnegative-real-ℝ⁰⁺
+```
+
+### Similarity preserves inequality
+
+```agda
+module _
+  {l1 l2 l3 : Level} (z : ℝ⁰⁺ l1) (x : ℝ⁰⁺ l2) (y : ℝ⁰⁺ l3) (x~y : sim-ℝ⁰⁺ x y)
+  where
+
+  abstract
+    preserves-leq-left-sim-ℝ⁰⁺ : leq-ℝ⁰⁺ x z → leq-ℝ⁰⁺ y z
+    preserves-leq-left-sim-ℝ⁰⁺ = preserves-leq-left-sim-ℝ x~y
+```
+
+### Inequality is transitive
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3)
+  where
+
+  transitive-leq-ℝ⁰⁺ : leq-ℝ⁰⁺ y z → leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ x z
+  transitive-leq-ℝ⁰⁺ = transitive-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z)
+```
+
+### If `x` is less than all the positive rational numbers `y` is less than, then `x ≤ y`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    leq-le-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) → le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
+        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
+      leq-ℝ⁰⁺ x y
+    leq-le-positive-rational-ℝ⁰⁺ H =
+      leq-le-rational-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+        ( λ q y<q →
+          rec-coproduct
+            ( λ 0<q →
+              let open do-syntax-trunc-Prop (le-prop-ℝ (real-ℝ⁰⁺ x) (real-ℚ q))
+              in do
+                (p , y<p , p<q) ← dense-rational-le-ℝ _ _ y<q
+                transitive-le-ℝ _ (real-ℚ p) _
+                  ( p<q)
+                  ( H
+                    ( p , is-positive-le-nonnegative-real-ℚ y p y<p)
+                    ( y<p)))
+            ( λ q≤0 → ex-falso (not-le-leq-zero-rational-ℝ⁰⁺ y q q≤0 y<q))
+            ( decide-le-leq-ℚ zero-ℚ q))
+```
+
+### If `x` is less than or equal to all the positive rational numbers `y` is less than or equal to, then `x ≤ y`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    leq-leq-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) → leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
+        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
+      leq-ℝ⁰⁺ x y
+    leq-leq-positive-rational-ℝ⁰⁺ H =
+      leq-le-positive-rational-ℝ⁰⁺ x y
+        ( λ q y<q →
+          let open do-syntax-trunc-Prop (le-prop-ℝ _ _)
+          in do
+            (r , y<r , r<q) ← dense-rational-le-ℝ _ _ y<q
+            concatenate-leq-le-ℝ _ _ _
+              ( H
+                ( r , is-positive-le-nonnegative-real-ℚ y r y<r)
+                ( leq-le-ℝ y<r))
+              ( r<q))
+```
+
+### If `x` is less than or equal to the same positive rational numbers `y` is less than or equal to, then `x` and `y` are similar
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    sim-leq-same-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) →
+        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
+      sim-ℝ⁰⁺ x y
+    sim-leq-same-positive-rational-ℝ⁰⁺ H =
+      sim-sim-leq-ℝ
+        ( leq-leq-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
+          leq-leq-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
+```
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index d441631cb99..d9b18435f16 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -193,14 +193,14 @@ opaque
   refl-leq-ℝ : {l : Level} (x : ℝ l) → leq-ℝ x x
   refl-leq-ℝ x = refl-leq-Large-Preorder lower-ℝ-Large-Preorder (lower-real-ℝ x)
 
-  leq-eq-ℝ : {l : Level} (x y : ℝ l) → x ＝ y → leq-ℝ x y
-  leq-eq-ℝ x y x=y = tr (leq-ℝ x) x=y (refl-leq-ℝ x)
+  leq-eq-ℝ : {l : Level} {x y : ℝ l} → x ＝ y → leq-ℝ x y
+  leq-eq-ℝ {x = x} refl = refl-leq-ℝ x
 
 opaque
   unfolding leq-ℝ sim-ℝ
 
-  leq-sim-ℝ : {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ x y → leq-ℝ x y
-  leq-sim-ℝ _ _ = pr1
+  leq-sim-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → sim-ℝ x y → leq-ℝ x y
+  leq-sim-ℝ = pr1
 ```
 
 ### Inequality on the real numbers is antisymmetric
@@ -288,24 +288,28 @@ opaque
 ### The canonical map from rational numbers to the reals preserves and reflects inequality
 
 ```agda
-opaque
-  unfolding leq-ℝ real-ℚ
+module _
+  {x y : ℚ}
+  where
+
+  opaque
+    unfolding leq-ℝ real-ℚ
 
-  preserves-leq-real-ℚ : (x y : ℚ) → leq-ℚ x y → leq-ℝ (real-ℚ x) (real-ℚ y)
-  preserves-leq-real-ℚ = preserves-leq-lower-real-ℚ
+    preserves-leq-real-ℚ : leq-ℚ x y → leq-ℝ (real-ℚ x) (real-ℚ y)
+    preserves-leq-real-ℚ = preserves-leq-lower-real-ℚ x y
 
-  reflects-leq-real-ℚ : (x y : ℚ) → leq-ℝ (real-ℚ x) (real-ℚ y) → leq-ℚ x y
-  reflects-leq-real-ℚ = reflects-leq-lower-real-ℚ
+    reflects-leq-real-ℚ : leq-ℝ (real-ℚ x) (real-ℚ y) → leq-ℚ x y
+    reflects-leq-real-ℚ = reflects-leq-lower-real-ℚ x y
 
-  iff-leq-real-ℚ : (x y : ℚ) → leq-ℚ x y ↔ leq-ℝ (real-ℚ x) (real-ℚ y)
-  iff-leq-real-ℚ = iff-leq-lower-real-ℚ
+    iff-leq-real-ℚ : leq-ℚ x y ↔ leq-ℝ (real-ℚ x) (real-ℚ y)
+    iff-leq-real-ℚ = iff-leq-lower-real-ℚ x y
 ```
 
 ### Negation reverses inequality on the real numbers
 
 ```agda
 module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
   opaque
@@ -319,7 +323,7 @@ module _
 
 ```agda
 module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) (x~y : sim-ℝ x y)
+  {l1 l2 l3 : Level} {z : ℝ l1} {x : ℝ l2} {y : ℝ l3} (x~y : sim-ℝ x y)
   where
 
   opaque
@@ -340,241 +344,8 @@ module _
   preserves-leq-sim-ℝ : leq-ℝ x1 y1 → leq-ℝ x2 y2
   preserves-leq-sim-ℝ x1≤y1 =
     preserves-leq-left-sim-ℝ
-      ( y2)
-      ( x1)
-      ( x2)
       ( x1~x2)
-      ( preserves-leq-right-sim-ℝ x1 y1 y2 y1~y2 x1≤y1)
-```
-
-### Inequality on the real numbers is translation invariant
-
-```agda
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  opaque
-    unfolding add-ℝ leq-ℝ
-
-    preserves-leq-right-add-ℝ : leq-ℝ x y → leq-ℝ (x +ℝ z) (y +ℝ z)
-    preserves-leq-right-add-ℝ x≤y _ =
-      map-tot-exists (λ (qx , _) → map-product (x≤y qx) id)
-
-    preserves-leq-left-add-ℝ : leq-ℝ x y → leq-ℝ (z +ℝ x) (z +ℝ y)
-    preserves-leq-left-add-ℝ x≤y _ =
-      map-tot-exists (λ (_ , qx) → map-product id (map-product (x≤y qx) id))
-
-abstract
-  preserves-leq-diff-ℝ :
-    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
-    leq-ℝ x y → leq-ℝ (x -ℝ z) (y -ℝ z)
-  preserves-leq-diff-ℝ z = preserves-leq-right-add-ℝ (neg-ℝ z)
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  abstract
-    reflects-leq-right-add-ℝ : leq-ℝ (x +ℝ z) (y +ℝ z) → leq-ℝ x y
-    reflects-leq-right-add-ℝ x+z≤y+z =
-      preserves-leq-sim-ℝ
-        ( (x +ℝ z) -ℝ z)
-        ( x)
-        ( (y +ℝ z) -ℝ z)
-        ( y)
-        ( cancel-right-add-diff-ℝ x z)
-        ( cancel-right-add-diff-ℝ y z)
-        ( preserves-leq-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z≤y+z)
-
-    reflects-leq-left-add-ℝ : leq-ℝ (z +ℝ x) (z +ℝ y) → leq-ℝ x y
-    reflects-leq-left-add-ℝ z+x≤z+y =
-      reflects-leq-right-add-ℝ
-        ( binary-tr
-          ( leq-ℝ)
-          ( commutative-add-ℝ z x)
-          ( commutative-add-ℝ z y)
-          ( z+x≤z+y))
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  iff-translate-right-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (x +ℝ z) (y +ℝ z)
-  pr1 iff-translate-right-leq-ℝ = preserves-leq-right-add-ℝ z x y
-  pr2 iff-translate-right-leq-ℝ = reflects-leq-right-add-ℝ z x y
-
-  iff-translate-left-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (z +ℝ x) (z +ℝ y)
-  pr1 iff-translate-left-leq-ℝ = preserves-leq-left-add-ℝ z x y
-  pr2 iff-translate-left-leq-ℝ = reflects-leq-left-add-ℝ z x y
-
-abstract
-  preserves-leq-add-ℝ :
-    {l1 l2 l3 l4 : Level} (a : ℝ l1) (b : ℝ l2) (c : ℝ l3) (d : ℝ l4) →
-    leq-ℝ a b → leq-ℝ c d → leq-ℝ (a +ℝ c) (b +ℝ d)
-  preserves-leq-add-ℝ a b c d a≤b c≤d =
-    transitive-leq-ℝ
-      ( a +ℝ c)
-      ( a +ℝ d)
-      ( b +ℝ d)
-      ( preserves-leq-right-add-ℝ d a b a≤b)
-      ( preserves-leq-left-add-ℝ a c d c≤d)
-```
-
-### Transposition laws
-
-```agda
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    leq-transpose-left-diff-ℝ : leq-ℝ (x -ℝ y) z → leq-ℝ x (z +ℝ y)
-    leq-transpose-left-diff-ℝ x-y≤z =
-      preserves-leq-left-sim-ℝ
-        ( z +ℝ y)
-        ( (x -ℝ y) +ℝ y)
-        ( x)
-        ( cancel-right-diff-add-ℝ x y)
-        ( preserves-leq-right-add-ℝ y (x -ℝ y) z x-y≤z)
-
-    leq-transpose-left-add-ℝ : leq-ℝ (x +ℝ y) z → leq-ℝ x (z -ℝ y)
-    leq-transpose-left-add-ℝ x+y≤z =
-      preserves-leq-left-sim-ℝ
-        (z -ℝ y)
-        ( (x +ℝ y) -ℝ y)
-        ( x)
-        ( cancel-right-add-diff-ℝ x y)
-        ( preserves-leq-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y≤z)
-
-    leq-transpose-right-add-ℝ : leq-ℝ x (y +ℝ z) → leq-ℝ (x -ℝ z) y
-    leq-transpose-right-add-ℝ x≤y+z =
-      preserves-leq-right-sim-ℝ
-        ( x -ℝ z)
-        ( (y +ℝ z) -ℝ z)
-        ( y)
-        ( cancel-right-add-diff-ℝ y z)
-        ( preserves-leq-right-add-ℝ (neg-ℝ z) x (y +ℝ z) x≤y+z)
-
-    leq-transpose-right-diff-ℝ : leq-ℝ x (y -ℝ z) → leq-ℝ (x +ℝ z) y
-    leq-transpose-right-diff-ℝ x≤y-z =
-      preserves-leq-right-sim-ℝ
-        ( x +ℝ z)
-        ( (y -ℝ z) +ℝ z)
-        ( y)
-        ( cancel-right-diff-add-ℝ y z)
-        ( preserves-leq-right-add-ℝ z x (y -ℝ z) x≤y-z)
-```
-
-### Swapping laws
-
-```agda
-abstract
-  swap-right-diff-leq-ℝ :
-    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
-    leq-ℝ (x -ℝ y) z → leq-ℝ (x -ℝ z) y
-  swap-right-diff-leq-ℝ x y z x-y≤z =
-    leq-transpose-right-add-ℝ
-      ( x)
-      ( y)
-      ( z)
-      ( tr
-        ( leq-ℝ x)
-        ( commutative-add-ℝ _ _)
-        ( leq-transpose-left-diff-ℝ x y z x-y≤z))
-```
-
-### Addition of real numbers preserves lower neighborhoods
-
-```agda
-module _
-  {l1 l2 l3 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    preserves-lower-neighborhood-leq-left-add-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d =
-      inv-tr
-        ( leq-ℝ (x +ℝ y))
-        ( associative-add-ℝ x z (real-ℚ⁺ d))
-        ( preserves-leq-left-add-ℝ
-          ( x)
-          ( y)
-          ( z +ℝ real-ℚ⁺ d)
-          ( z≤y+d))
-
-    preserves-lower-neighborhood-leq-right-add-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-right-add-ℝ z≤y+d =
-      binary-tr
-        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
-        ( commutative-add-ℝ x y)
-        ( commutative-add-ℝ x z)
-        ( preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d)
-```
-
-### Addition of real numbers reflects lower neighborhoods
-
-```agda
-module _
-  {l1 l2 l3 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    reflects-lower-neighborhood-leq-left-add-ℝ :
-      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d) →
-      leq-ℝ y (z +ℝ real-ℚ⁺ d)
-    reflects-lower-neighborhood-leq-left-add-ℝ x+y≤x+z+d =
-      reflects-leq-left-add-ℝ
-        ( x)
-        ( y)
-        ( z +ℝ real-ℚ⁺ d)
-        ( tr
-          ( leq-ℝ (x +ℝ y))
-          ( associative-add-ℝ x z (real-ℚ⁺ d))
-          ( x+y≤x+z+d))
-
-    reflects-lower-neighborhood-leq-right-add-ℝ :
-      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d) →
-      leq-ℝ y (z +ℝ real-ℚ⁺ d)
-    reflects-lower-neighborhood-leq-right-add-ℝ y+x≤z+y+d =
-      reflects-lower-neighborhood-leq-left-add-ℝ
-        ( binary-tr
-          ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
-          ( commutative-add-ℝ y x)
-          ( commutative-add-ℝ z x)
-          ( y+x≤z+y+d))
-```
-
-### Negation of real numbers reverses lower neighborhoods
-
-```agda
-module _
-  {l1 l2 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2)
-  where
-
-  reverses-lower-neighborhood-neg-ℝ :
-    leq-ℝ x (y +ℝ real-ℚ⁺ d) →
-    leq-ℝ (neg-ℝ y) (neg-ℝ x +ℝ real-ℚ⁺ d)
-  reverses-lower-neighborhood-neg-ℝ x≤y+d =
-    tr
-      ( leq-ℝ (neg-ℝ y))
-      ( ( distributive-neg-add-ℝ x ((neg-ℝ ∘ real-ℚ ∘ rational-ℚ⁺) d)) ∙
-        ( ap (add-ℝ (neg-ℝ x)) (neg-neg-ℝ (real-ℚ⁺ d))))
-      ( neg-leq-ℝ
-        ( x -ℝ real-ℚ⁺ d)
-        ( y)
-        ( leq-transpose-right-add-ℝ
-          ( x)
-          ( y)
-          ( real-ℚ⁺ d)
-          ( x≤y+d)))
+      ( preserves-leq-right-sim-ℝ y1~y2 x1≤y1)
 ```
 
 ### `x ≤ q` for a rational `q` if and only if `q ∉ lower-cut-ℝ x`
diff --git a/src/real-numbers/infima-families-real-numbers.lagda.md b/src/real-numbers/infima-families-real-numbers.lagda.md
index 6b3be4283aa..add144950d6 100644
--- a/src/real-numbers/infima-families-real-numbers.lagda.md
+++ b/src/real-numbers/infima-families-real-numbers.lagda.md
@@ -41,6 +41,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
@@ -149,7 +150,7 @@ module _
             not-leq-le-ℝ (y i) z
               ( transitive-le-ℝ (y i) (x +ℝ real-ℚ ε) z
                 ( le-transpose-right-diff-ℝ' _ z _
-                  ( le-real-is-in-lower-cut-ℚ ε (z -ℝ x) ε<z-x))
+                  ( le-real-is-in-lower-cut-ℚ (z -ℝ x) ε<z-x))
                 ( yᵢ<x+ε))
               ( z≤yᵢ i))
     pr2 (is-greatest-lower-bound-is-infimum-family-ℝ z) z≤x i =
@@ -257,7 +258,7 @@ module _
         ( i)
         ( transitive-le-ℝ (y i) (x +ℝ real-ℚ ε) z
           ( le-transpose-right-diff-ℝ' _ z _
-            ( le-real-is-in-lower-cut-ℚ ε (z -ℝ x) ε<z-x))
+            ( le-real-is-in-lower-cut-ℚ (z -ℝ x) ε<z-x))
           ( yᵢ<x+ε))
 
   le-infimum-iff-le-element-family-ℝ :
@@ -284,7 +285,7 @@ module _
       tr
         ( leq-ℝ (neg-ℝ x))
         ( neg-neg-ℝ (y i))
-        ( neg-leq-ℝ _ _
+        ( neg-leq-ℝ
           ( is-upper-bound-is-supremum-family-ℝ
             ( neg-ℝ ∘ y)
             ( x)
@@ -299,7 +300,7 @@ module _
           binary-tr le-ℝ
             ( neg-neg-ℝ _)
             ( distributive-neg-diff-ℝ _ _ ∙ commutative-add-ℝ _ _)
-            ( neg-le-ℝ (x -ℝ real-ℚ⁺ ε) (neg-ℝ (y i)) x-ε<-yᵢ))
+            ( neg-le-ℝ x-ε<-yᵢ))
         ( is-approximated-below-is-supremum-family-ℝ
           ( neg-ℝ ∘ y)
           ( x)
@@ -327,7 +328,7 @@ module _
       tr
         ( λ w → leq-ℝ w (neg-ℝ x))
         ( neg-neg-ℝ (y i))
-        ( neg-leq-ℝ _ _
+        ( neg-leq-ℝ
           ( is-lower-bound-is-infimum-family-ℝ
             ( neg-ℝ ∘ y)
             ( x)
@@ -342,7 +343,7 @@ module _
           binary-tr le-ℝ
             ( distributive-neg-add-ℝ _ _)
             ( neg-neg-ℝ (y i))
-            ( neg-le-ℝ (neg-ℝ (y i)) (x +ℝ real-ℚ⁺ ε) -yᵢ<x+ε))
+            ( neg-le-ℝ -yᵢ<x+ε))
         ( is-approximated-above-is-infimum-family-ℝ
           ( neg-ℝ ∘ y)
           ( x)
@@ -373,7 +374,7 @@ module _
       tr
         ( leq-ℝ (neg-ℝ x))
         ( neg-neg-ℝ s)
-        ( neg-leq-ℝ _ _
+        ( neg-leq-ℝ
           ( is-upper-bound-is-supremum-family-ℝ
             ( inclusion-subset-ℝ (neg-subset-ℝ S))
             ( x)
@@ -397,7 +398,7 @@ module _
             ( le-ℝ)
             ( refl)
             ( distributive-neg-diff-ℝ x (real-ℚ⁺ ε) ∙ commutative-add-ℝ _ _)
-            ( neg-le-ℝ _ _ x-ε<s))
+            ( neg-le-ℝ x-ε<s))
 
     is-infimum-neg-supremum-neg-subset-ℝ : is-infimum-subset-ℝ S (neg-ℝ x)
     is-infimum-neg-supremum-neg-subset-ℝ =
diff --git a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
index 0549a213163..21124540b7c 100644
--- a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
@@ -44,6 +44,8 @@ open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.closed-intervals-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.infima-and-suprema-families-real-numbers
 open import real-numbers.infima-families-real-numbers
@@ -53,6 +55,7 @@ open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
@@ -220,14 +223,14 @@ module _
                 ( max-net η +ℝ real-ℚ⁺ η)
                 ( max-net η +ℝ real-ℚ⁺ (ε +ℚ⁺ η))
                 ( preserves-leq-left-add-ℝ (max-net η) _ _
-                  ( preserves-leq-real-ℚ _ _ (leq-left-add-rational-ℚ⁺ _ ε)))
+                  ( preserves-leq-real-ℚ (leq-left-add-rational-ℚ⁺ _ ε)))
                 ( bound ε η))
               ( transitive-leq-ℝ
                 ( max-net η)
                 ( max-net ε +ℝ real-ℚ⁺ ε)
                 ( max-net ε +ℝ real-ℚ⁺ (ε +ℚ⁺ η))
                 ( preserves-leq-left-add-ℝ (max-net ε) _ _
-                  ( preserves-leq-real-ℚ _ _ (leq-right-add-rational-ℚ⁺ _ η)))
+                  ( preserves-leq-real-ℚ (leq-right-add-rational-ℚ⁺ _ η)))
                 ( bound η ε)))
 
     sup-modulated-totally-bounded-subset-ℝ : ℝ l2
@@ -281,7 +284,7 @@ module _
                               ( pr1 ∘ pr1)
                               ( (y , y∈S) , y∈net-ε'))))))
                     ( preserves-le-left-add-ℝ sup _ _
-                      ( preserves-le-real-ℚ _ _ ε'+ε'<ε))))))
+                      ( preserves-le-real-ℚ ε'+ε'<ε))))))
 
     is-approximated-below-sup-modulated-totally-bounded-subset-ℝ :
       is-approximated-below-family-ℝ
@@ -315,7 +318,7 @@ module _
               ( sup -ℝ real-ℚ⁺ ε)
               ( sup -ℝ real-ℚ⁺ (ε' +ℚ⁺ ε'))
               ( max-net ε' -ℝ real-ℚ⁺ ε')
-              ( reverses-le-diff-ℝ sup _ _ (preserves-le-real-ℚ _ _ ε'+ε'<ε))
+              ( reverses-le-diff-ℝ sup _ _ (preserves-le-real-ℚ ε'+ε'<ε))
               ( tr
                 ( λ y → leq-ℝ y (max-net ε' -ℝ real-ℚ⁺ ε'))
                 ( associative-add-ℝ _ _ _ ∙
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index e46f33719ef..a7d3686d03b 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -24,6 +24,7 @@ open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index 8854080f8ea..7e5f7562a20 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -11,19 +11,24 @@ module real-numbers.isometry-negation-real-numbers where
 ```agda
 open import elementary-number-theory.positive-rational-numbers
 
+open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-spaces
 
+open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
+open import real-numbers.rational-real-numbers
 ```
 
 </details>
@@ -36,6 +41,30 @@ open import real-numbers.negation-real-numbers
 
 ## Definitions
 
+### Negation of real numbers reverses lower neighborhoods
+
+```agda
+module _
+  {l1 l2 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2)
+  where
+
+  reverses-lower-neighborhood-neg-ℝ :
+    leq-ℝ x (y +ℝ real-ℚ⁺ d) →
+    leq-ℝ (neg-ℝ y) (neg-ℝ x +ℝ real-ℚ⁺ d)
+  reverses-lower-neighborhood-neg-ℝ x≤y+d =
+    tr
+      ( leq-ℝ (neg-ℝ y))
+      ( ( distributive-neg-add-ℝ x ((neg-ℝ ∘ real-ℚ ∘ rational-ℚ⁺) d)) ∙
+        ( ap (add-ℝ (neg-ℝ x)) (neg-neg-ℝ (real-ℚ⁺ d))))
+      ( neg-leq-ℝ
+        ( leq-transpose-right-add-ℝ
+          ( x)
+          ( y)
+          ( real-ℚ⁺ d)
+          ( x≤y+d)))
+```
+
 ### Negation of a real number preserves neighborhoods
 
 ```agda
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index 9e1a1f4786a..7b3d8f8c669 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -30,11 +30,13 @@ open import order-theory.large-posets
 
 open import real-numbers.absolute-value-closed-intervals-real-numbers
 open import real-numbers.absolute-value-real-numbers
+open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.inhabited-totally-bounded-subsets-real-numbers
 open import real-numbers.metric-space-of-real-numbers
@@ -90,7 +92,7 @@ module _
               ( chain-of-inequalities
                 dist-ℝ (c *ℝ x) (c *ℝ y)
                 ≤ abs-ℝ c *ℝ dist-ℝ x y
-                  by leq-eq-ℝ _ _ (inv (left-distributive-abs-mul-dist-ℝ _ _ _))
+                  by leq-eq-ℝ (inv (left-distributive-abs-mul-dist-ℝ _ _ _))
                 ≤ real-ℚ⁺ q *ℝ real-ℚ⁺ ε
                   by
                     preserves-leq-mul-ℝ⁰⁺
@@ -98,14 +100,10 @@ module _
                       ( nonnegative-real-ℚ⁺ q)
                       ( nonnegative-dist-ℝ x y)
                       ( nonnegative-real-ℚ⁺ ε)
-                      ( leq-le-ℝ _ _
-                        ( le-real-is-in-upper-cut-ℚ
-                          ( rational-ℚ⁺ q)
-                          ( abs-ℝ c)
-                          ( |c|<q)))
+                      ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ (abs-ℝ c) |c|<q))
                       ( leq-dist-neighborhood-ℝ ε x y Nεxy)
                 ≤ real-ℚ⁺ (q *ℚ⁺ ε)
-                  by leq-eq-ℝ _ _ (mul-real-ℚ _ _)))
+                  by leq-eq-ℝ (mul-real-ℚ _ _)))
 
     is-lipschitz-left-mul-ℝ :
       is-lipschitz-function-Metric-Space
@@ -228,14 +226,14 @@ module _
                   by triangle-inequality-dist-ℝ _ _ _
                 ≤ dist-ℝ x₁ x₂ *ℝ abs-ℝ y₁ +ℝ abs-ℝ x₂ *ℝ dist-ℝ y₁ y₂
                   by
-                    leq-eq-ℝ _ _
+                    leq-eq-ℝ
                       ( inv
                         ( ap-add-ℝ
                           ( right-distributive-abs-mul-dist-ℝ x₁ x₂ y₁)
                           ( left-distributive-abs-mul-dist-ℝ x₂ y₁ y₂)))
                 ≤ real-ℚ⁺ ε *ℝ my +ℝ mx *ℝ real-ℚ⁺ ε
                   by
-                    preserves-leq-add-ℝ _ _ _ _
+                    preserves-leq-add-ℝ
                       ( preserves-leq-mul-ℝ⁰⁺
                         ( nonnegative-dist-ℝ x₁ x₂)
                         ( nonnegative-real-ℚ⁺ ε)
@@ -251,12 +249,10 @@ module _
                         ( is-max-mx (x₂ , x₂∈X))
                         ( leq-dist-neighborhood-ℝ ε y₁ y₂ Nεy₁y₂))
                 ≤ my *ℝ real-ℚ⁺ ε +ℝ mx *ℝ real-ℚ⁺ ε
-                  by
-                    leq-eq-ℝ _ _
-                      ( ap-add-ℝ (commutative-mul-ℝ _ _) refl)
+                  by leq-eq-ℝ (ap-add-ℝ (commutative-mul-ℝ _ _) refl)
                 ≤ (my +ℝ mx) *ℝ real-ℚ⁺ ε
                   by
-                    leq-eq-ℝ _ _
+                    leq-eq-ℝ
                       ( inv (right-distributive-mul-add-ℝ my mx (real-ℚ⁺ ε)))
                 ≤ real-ℚ q *ℝ real-ℚ⁺ ε
                   by
@@ -264,10 +260,9 @@ module _
                       ( nonnegative-real-ℚ⁺ ε)
                       ( my +ℝ mx)
                       ( real-ℚ q)
-                      ( leq-le-ℝ _ _
-                        ( le-real-is-in-upper-cut-ℚ q (my +ℝ mx) my+mx<q))
+                      ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ (my +ℝ mx) my+mx<q))
                 ≤ real-ℚ⁺ (q⁺ *ℚ⁺ ε)
-                  by leq-eq-ℝ _ _ (mul-real-ℚ q (rational-ℚ⁺ ε))))
+                  by leq-eq-ℝ (mul-real-ℚ q (rational-ℚ⁺ ε))))
 
   lipschitz-mul-inhabited-totally-bounded-subset-ℝ :
     lipschitz-function-Metric-Space
diff --git a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
index 9486dd60244..5e757a72e78 100644
--- a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
@@ -43,6 +43,7 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.suprema-families-real-numbers
 
diff --git a/src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md b/src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md
new file mode 100644
index 00000000000..6628c69c5c4
--- /dev/null
+++ b/src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,34 @@
+# The metric space of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.metric-space-of-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.universe-levels
+
+open import metric-spaces.metric-spaces
+
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.subsets-real-numbers
+```
+
+</details>
+
+## Idea
+
+The [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) form a
+[subspace](metric-spaces.subspaces-metric-spaces.md) of the
+[metric space of real numbers](real-numbers.metric-space-of-real-numbers.md).
+
+## Definition
+
+```agda
+metric-space-ℝ⁰⁺ : (l : Level) → Metric-Space (lsuc l) l
+metric-space-ℝ⁰⁺ l = metric-space-subset-ℝ (is-nonnegative-prop-ℝ {l})
+```
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index 9c80701de01..5afe6b7b568 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -44,10 +44,12 @@ open import metric-spaces.triangular-rational-neighborhood-relations
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.transposition-addition-subtraction-cuts-dedekind-real-numbers
 ```
@@ -170,15 +172,11 @@ module _
         elim-exists
           ( lower-cut-ℝ x r)
           ( λ r' (K , I') →
-            H ( positive-diff-le-ℚ (r +ℚ rational-ℚ⁺ ε) r' K)
+            H ( positive-diff-le-ℚ K)
               ( r)
               ( tr
                 ( is-in-lower-cut-ℝ y)
-                ( ( inv
-                    ( right-law-positive-diff-le-ℚ
-                      ( r +ℚ rational-ℚ⁺ ε)
-                      ( r')
-                      ( K))) ∙
+                ( ( inv (right-law-positive-diff-le-ℚ K)) ∙
                   ( associative-add-ℚ
                     ( r)
                     ( rational-ℚ⁺ ε)
@@ -221,11 +219,11 @@ module _
               ( lower-cut-ℝ y r)
               ( λ s (r<s , Lxs) →
                 pr2
-                  ( H (positive-diff-le-ℚ r s r<s))
+                  ( H (positive-diff-le-ℚ r<s))
                   ( r)
                   ( inv-tr
                     ( λ u → is-in-lower-cut-ℝ x u)
-                    ( right-law-positive-diff-le-ℚ r s r<s)
+                    ( right-law-positive-diff-le-ℚ r<s)
                     ( Lxs)))
               ( forward-implication (is-rounded-lower-cut-ℝ x r) Lxr))
           ( λ r Lyr →
@@ -233,11 +231,11 @@ module _
               ( lower-cut-ℝ x r)
               ( λ s (r<s , Lys) →
                 pr1
-                  ( H (positive-diff-le-ℚ r s r<s))
+                  ( H (positive-diff-le-ℚ r<s))
                   ( r)
                   ( inv-tr
                     ( λ u → is-in-lower-cut-ℝ y u)
-                    ( right-law-positive-diff-le-ℚ r s r<s)
+                    ( right-law-positive-diff-le-ℚ r<s)
                     ( Lys)))
               ( forward-implication (is-rounded-lower-cut-ℝ y r) Lyr)))
 
@@ -284,7 +282,6 @@ module _
       lower-neighborhood-ℝ d x y
     lower-neighborhood-real-bound-leq-ℝ (d , _) x y y≤x+d q q+d<y =
       is-in-lower-cut-le-real-ℚ
-        ( q)
         ( x)
         ( concatenate-le-leq-ℝ
           ( real-ℚ q)
@@ -297,7 +294,7 @@ module _
             ( inv-tr
               ( λ z → le-ℝ z y)
               ( add-real-ℚ q d)
-              ( le-real-is-in-lower-cut-ℚ (q +ℚ d) y q+d<y)))
+              ( le-real-is-in-lower-cut-ℚ y q+d<y)))
           ( leq-transpose-right-add-ℝ y x (real-ℚ d) y≤x+d))
 
   opaque
diff --git a/src/real-numbers/multiplication-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
index af8ae9cbfaa..e13cdd5cc28 100644
--- a/src/real-numbers/multiplication-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
@@ -76,7 +76,7 @@ abstract
       ( le-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-le-ℝ _ _ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-le-ℝ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
 ```
 
 ### Multiplication by a negative real number reverses inequality
@@ -91,5 +91,5 @@ abstract
       ( leq-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-leq-ℝ _ _ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-leq-ℝ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
 ```
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index 706360afa75..dd18d3c8e50 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -27,6 +27,7 @@ open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.nonnegative-real-numbers
@@ -62,8 +63,8 @@ opaque
               ([c,d]@((c , d) , _) , c<y , y<d)) ,
             [a,b][c,d]<q) ← xy<q
           let
-            b⁺ = (b , is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) b x<b)
-            d⁺ = (d , is-positive-is-in-upper-cut-ℝ⁰⁺ (y , 0≤y) d y<d)
+            b⁺ = (b , is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) x<b)
+            d⁺ = (d , is-positive-is-in-upper-cut-ℝ⁰⁺ (y , 0≤y) y<d)
           is-positive-le-ℚ⁺
             ( b⁺ *ℚ⁺ d⁺)
             ( q)
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index ea789697859..e26e0fb2186 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -477,9 +477,9 @@ module _
                         r<y ,
                         y<s))
               ≤ rational-ℕ N
-                by preserves-leq-rational-ℕ _ _ (right-leq-max-ℕ _ _)
+                by preserves-leq-rational-ℕ (right-leq-max-ℕ _ _)
               ≤ rational-ℕ (succ-ℕ N)
-                by preserves-leq-rational-ℕ _ _ (succ-leq-ℕ N)
+                by preserves-leq-rational-ℕ (succ-leq-ℕ N)
           max|p||q|≤sN =
             chain-of-inequalities
               max-ℚ (rational-abs-ℚ p) (rational-abs-ℚ q)
@@ -495,9 +495,9 @@ module _
                         p<x ,
                         x<q))
               ≤ rational-ℕ N
-                by preserves-leq-rational-ℕ _ _ (left-leq-max-ℕ _ _)
+                by preserves-leq-rational-ℕ (left-leq-max-ℕ _ _)
               ≤ rational-ℕ (succ-ℕ N)
-                by preserves-leq-rational-ℕ _ _ (succ-leq-ℕ N)
+                by preserves-leq-rational-ℕ (succ-leq-ℕ N)
           [p,q] = ((p , q) , p≤q)
           [r,s] = ((r , s) , r≤s)
           a = lower-bound-mul-closed-interval-ℚ [p,q] [r,s]
@@ -531,7 +531,7 @@ module _
               ≤ ( rational-ℚ⁺ εx +ℚ rational-ℚ⁺ εy) *ℚ
                 rational-ℕ (succ-ℕ N)
                 by
-                  leq-eq-ℚ _ _
+                  leq-eq-ℚ
                     ( inv (right-distributive-mul-add-ℚ _ _ _))
               ≤ rational-ℚ⁺
                   ( ( εx₀ *ℚ⁺ positive-reciprocal-rational-succ-ℕ N) +ℚ⁺
@@ -551,7 +551,7 @@ module _
                         ( εy₀ *ℚ⁺ positive-reciprocal-rational-succ-ℕ N)))
               ≤ rational-ℚ⁺ ε
                 by
-                  leq-eq-ℚ _ _
+                  leq-eq-ℚ
                     ( ap-mul-ℚ
                       ( inv (right-distributive-mul-add-ℚ _ _ _))
                       ( refl) ∙
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 73cd01dce63..d17a3591165 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -58,6 +58,7 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-positive-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-positive-real-numbers
 open import real-numbers.strict-inequality-real-numbers
@@ -439,8 +440,6 @@ module _
                         ( x<p⁻¹))))
               ( is-located-lower-upper-cut-ℝ
                 ( real-ℝ⁺ x)
-                ( rational-ℚ⁺ (inv-ℚ⁺ q⁺))
-                ( rational-ℚ⁺ (inv-ℚ⁺ p⁺))
                 ( inv-le-ℚ⁺ p⁺ q⁺ p<q)))
         ( λ is-nonpos-p →
           inl-disjunction
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 832384eb289..183ddbcd090 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -83,7 +83,7 @@ module _
       ( disjunction-Prop (lower-cut-neg-ℝ q) (upper-cut-neg-ℝ r))
       ( inr-disjunction)
       ( inl-disjunction)
-      ( is-located-lower-upper-cut-ℝ x (neg-ℚ r) (neg-ℚ q) (neg-le-ℚ q<r))
+      ( is-located-lower-upper-cut-ℝ x (neg-le-ℚ q<r))
 
   opaque
     neg-ℝ : ℝ l
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index bf51c3d32b5..6b5a11da376 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -40,6 +40,7 @@ open import metric-spaces.metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.saturation-inequality-real-numbers
@@ -125,12 +126,14 @@ eq-ℝ⁰⁺ _ _ = eq-type-subtype is-nonnegative-prop-ℝ
 
 ```agda
 abstract
-  is-nonnegative-real-ℚ⁰⁺ : (q : ℚ⁰⁺) → is-nonnegative-ℝ (real-ℚ⁰⁺ q)
-  is-nonnegative-real-ℚ⁰⁺ (q , nonneg-q) =
-    preserves-leq-real-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ nonneg-q)
+  preserves-is-nonnegative-real-ℚ :
+    {q : ℚ} → is-nonnegative-ℚ q → is-nonnegative-ℝ (real-ℚ q)
+  preserves-is-nonnegative-real-ℚ is-nonneg-q =
+    preserves-leq-real-ℚ (leq-zero-is-nonnegative-ℚ is-nonneg-q)
 
 nonnegative-real-ℚ⁰⁺ : ℚ⁰⁺ → ℝ⁰⁺ lzero
-nonnegative-real-ℚ⁰⁺ q = (real-ℚ⁰⁺ q , is-nonnegative-real-ℚ⁰⁺ q)
+nonnegative-real-ℚ⁰⁺ (q , is-nonneg-q) =
+  ( real-ℚ q , preserves-is-nonnegative-real-ℚ is-nonneg-q)
 
 nonnegative-real-ℚ⁺ : ℚ⁺ → ℝ⁰⁺ lzero
 nonnegative-real-ℚ⁺ q = nonnegative-real-ℚ⁰⁺ (nonnegative-ℚ⁺ q)
@@ -146,102 +149,19 @@ one-ℝ⁰⁺ : ℝ⁰⁺ lzero
 one-ℝ⁰⁺ = nonnegative-real-ℚ⁰⁺ one-ℚ⁰⁺
 ```
 
-### Similarity of nonnegative real numbers
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  sim-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
-  sim-prop-ℝ⁰⁺ = sim-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-
-  sim-ℝ⁰⁺ : UU (l1 ⊔ l2)
-  sim-ℝ⁰⁺ = sim-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-
-infix 6 _~ℝ⁰⁺_
-_~ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → UU (l1 ⊔ l2)
-_~ℝ⁰⁺_ = sim-ℝ⁰⁺
-
-sim-zero-prop-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → Prop l
-sim-zero-prop-ℝ⁰⁺ = sim-prop-ℝ⁰⁺ zero-ℝ⁰⁺
-
-sim-zero-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → UU l
-sim-zero-ℝ⁰⁺ = sim-ℝ⁰⁺ zero-ℝ⁰⁺
-
-eq-sim-ℝ⁰⁺ : {l : Level} (x y : ℝ⁰⁺ l) → sim-ℝ⁰⁺ x y → x ＝ y
-eq-sim-ℝ⁰⁺ x y x~y = eq-ℝ⁰⁺ x y (eq-sim-ℝ {x = real-ℝ⁰⁺ x} {y = real-ℝ⁰⁺ y} x~y)
-```
-
-#### Similarity is symmetric
-
-```agda
-abstract
-  symmetric-sim-ℝ⁰⁺ :
-    {l1 l2 : Level} → (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) → x ~ℝ⁰⁺ y → y ~ℝ⁰⁺ x
-  symmetric-sim-ℝ⁰⁺ _ _ = symmetric-sim-ℝ
-```
-
-### Inequality on nonnegative real numbers
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  leq-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
-  leq-prop-ℝ⁰⁺ = leq-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-
-  leq-ℝ⁰⁺ : UU (l1 ⊔ l2)
-  leq-ℝ⁰⁺ = type-Prop leq-prop-ℝ⁰⁺
-```
-
-### Zero is less than or equal to every nonnegative real number
-
-```agda
-leq-zero-ℝ⁰⁺ : {l : Level} (x : ℝ⁰⁺ l) → leq-ℝ⁰⁺ zero-ℝ⁰⁺ x
-leq-zero-ℝ⁰⁺ = is-nonnegative-real-ℝ⁰⁺
-```
-
-### Similarity preserves inequality
-
-```agda
-module _
-  {l1 l2 l3 : Level} (z : ℝ⁰⁺ l1) (x : ℝ⁰⁺ l2) (y : ℝ⁰⁺ l3) (x~y : sim-ℝ⁰⁺ x y)
-  where
-
-  abstract
-    preserves-leq-left-sim-ℝ⁰⁺ : leq-ℝ⁰⁺ x z → leq-ℝ⁰⁺ y z
-    preserves-leq-left-sim-ℝ⁰⁺ =
-      preserves-leq-left-sim-ℝ (real-ℝ⁰⁺ z) _ _ x~y
-```
-
-### Inequality is transitive
-
-```agda
-module _
-  {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3)
-  where
-
-  transitive-leq-ℝ⁰⁺ : leq-ℝ⁰⁺ y z → leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ x z
-  transitive-leq-ℝ⁰⁺ = transitive-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z)
-```
-
 ### A real number is nonnegative if and only if every element of its upper cut is positive
 
 ```agda
 abstract
   is-positive-is-in-upper-cut-ℝ⁰⁺ :
-    {l : Level} → (x : ℝ⁰⁺ l) (q : ℚ) → is-in-upper-cut-ℝ⁰⁺ x q →
+    {l : Level} → (x : ℝ⁰⁺ l) {q : ℚ} → is-in-upper-cut-ℝ⁰⁺ x q →
     is-positive-ℚ q
-  is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) q x<q =
+  is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) x<q =
     is-positive-le-zero-ℚ
       ( reflects-le-real-ℚ
-        ( zero-ℚ)
-        ( q)
         ( concatenate-leq-le-ℝ zero-ℝ x _
           ( 0≤x)
-          ( le-real-is-in-upper-cut-ℚ q x x<q)))
+          ( le-real-is-in-upper-cut-ℚ x x<q)))
 
 opaque
   unfolding leq-ℝ leq-ℝ' real-ℚ
@@ -283,123 +203,7 @@ abstract
     let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → upper-cut-ℝ⁰⁺ x (rational-ℚ⁺ q)))
     in do
       (q , x<q) ← is-inhabited-upper-cut-ℝ (real-ℝ⁰⁺ x)
-      intro-exists (q , is-positive-is-in-upper-cut-ℝ⁰⁺ x q x<q) x<q
-```
-
-### Addition on nonnegative real numbers
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  real-add-ℝ⁰⁺ : ℝ (l1 ⊔ l2)
-  real-add-ℝ⁰⁺ = real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y
-
-  abstract
-    is-nonnegative-real-add-ℝ⁰⁺ : is-nonnegative-ℝ real-add-ℝ⁰⁺
-    is-nonnegative-real-add-ℝ⁰⁺ =
-      tr
-        ( λ z → leq-ℝ z (real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y))
-        ( left-unit-law-add-ℝ zero-ℝ)
-        ( preserves-leq-add-ℝ
-          ( zero-ℝ)
-          ( real-ℝ⁰⁺ x)
-          ( zero-ℝ)
-          ( real-ℝ⁰⁺ y)
-          ( is-nonnegative-real-ℝ⁰⁺ x)
-          ( is-nonnegative-real-ℝ⁰⁺ y))
-
-  add-ℝ⁰⁺ : ℝ⁰⁺ (l1 ⊔ l2)
-  add-ℝ⁰⁺ = (real-add-ℝ⁰⁺ , is-nonnegative-real-add-ℝ⁰⁺)
-
-infixl 35 _+ℝ⁰⁺_
-
-_+ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → ℝ⁰⁺ (l1 ⊔ l2)
-_+ℝ⁰⁺_ = add-ℝ⁰⁺
-```
-
-### Unit laws for addition
-
-```agda
-module _
-  {l : Level} (x : ℝ⁰⁺ l)
-  where
-
-  abstract
-    left-unit-law-add-ℝ⁰⁺ : zero-ℝ⁰⁺ +ℝ⁰⁺ x ＝ x
-    left-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (left-unit-law-add-ℝ _)
-
-    right-unit-law-add-ℝ⁰⁺ : x +ℝ⁰⁺ zero-ℝ⁰⁺ ＝ x
-    right-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (right-unit-law-add-ℝ _)
-```
-
-### Addition preserves inequality
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
-  where
-
-  abstract
-    preserves-leq-add-ℝ⁰⁺ :
-      leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ z w → leq-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
-    preserves-leq-add-ℝ⁰⁺ =
-      preserves-leq-add-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z) (real-ℝ⁰⁺ w)
-```
-
-### If `x ≤ y + ε` for every `ε : ℚ⁺`, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    saturated-leq-ℝ⁰⁺ :
-      ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (y +ℝ⁰⁺ nonnegative-real-ℚ⁺ ε)) →
-      leq-ℝ⁰⁺ x y
-    saturated-leq-ℝ⁰⁺ = saturated-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-```
-
-### If a nonnegative real number is less than or equal to all positive rational numbers, it is similar to zero
-
-```agda
-sim-zero-le-positive-rational-ℝ⁰⁺ :
-  {l : Level} (x : ℝ⁰⁺ l) →
-  ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (nonnegative-real-ℚ⁺ ε)) →
-  sim-zero-ℝ⁰⁺ x
-sim-zero-le-positive-rational-ℝ⁰⁺ x H =
-  sim-sim-leq-ℝ
-    ( leq-zero-ℝ⁰⁺ x ,
-      saturated-leq-ℝ⁰⁺
-        ( x)
-        ( zero-ℝ⁰⁺)
-        ( λ ε → inv-tr (leq-ℝ⁰⁺ x) (left-unit-law-add-ℝ⁰⁺ _) (H ε)))
-```
-
-### The canonical embedding of nonnegative rational numbers to nonnegative real numbers preserves addition
-
-```agda
-abstract
-  add-nonnegative-real-ℚ⁰⁺ :
-    (p q : ℚ⁰⁺) →
-    nonnegative-real-ℚ⁰⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁰⁺ q ＝
-    nonnegative-real-ℚ⁰⁺ (p +ℚ⁰⁺ q)
-  add-nonnegative-real-ℚ⁰⁺ p q =
-    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q))
-```
-
-### The canonical embedding of positive rational numbers to nonnegative real numbers preserves addition
-
-```agda
-abstract
-  add-nonnegative-real-ℚ⁺ :
-    (p q : ℚ⁺) →
-    nonnegative-real-ℚ⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁺ q ＝
-    nonnegative-real-ℚ⁺ (p +ℚ⁺ q)
-  add-nonnegative-real-ℚ⁺ p q =
-    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁺ p) (rational-ℚ⁺ q))
+      intro-exists (q , is-positive-is-in-upper-cut-ℝ⁰⁺ x x<q) x<q
 ```
 
 ### Nonpositive rational numbers are not less than or equal to nonnegative real numbers
@@ -419,7 +223,7 @@ module _
           ( x<q)
           ( transitive-leq-ℝ (real-ℚ q) zero-ℝ (real-ℝ⁰⁺ x)
             ( is-nonnegative-real-ℝ⁰⁺ x)
-            ( preserves-leq-real-ℚ _ _ q≤0)))
+            ( preserves-leq-real-ℚ q≤0)))
 ```
 
 ### If a nonnegative real number is less than a rational number, the rational number is positive
@@ -434,107 +238,10 @@ module _
       le-ℝ (real-ℝ⁰⁺ x) (real-ℚ q) → is-positive-ℚ q
     is-positive-le-nonnegative-real-ℚ x<q =
       is-positive-le-zero-ℚ
-        ( reflects-le-real-ℚ _ _
+        ( reflects-le-real-ℚ
           ( concatenate-leq-le-ℝ _ _ _ (is-nonnegative-real-ℝ⁰⁺ x) x<q))
 ```
 
-### If `x` is less than all the positive rational numbers `y` is less than, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    leq-le-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) → le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
-        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
-      leq-ℝ⁰⁺ x y
-    leq-le-positive-rational-ℝ⁰⁺ H =
-      leq-le-rational-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-        ( λ q y<q →
-          rec-coproduct
-            ( λ 0<q →
-              let open do-syntax-trunc-Prop (le-prop-ℝ (real-ℝ⁰⁺ x) (real-ℚ q))
-              in do
-                (p , y<p , p<q) ← dense-rational-le-ℝ _ _ y<q
-                transitive-le-ℝ _ (real-ℚ p) _
-                  ( p<q)
-                  ( H
-                    ( p , is-positive-le-nonnegative-real-ℚ y p y<p)
-                    ( y<p)))
-            ( λ q≤0 → ex-falso (not-le-leq-zero-rational-ℝ⁰⁺ y q q≤0 y<q))
-            ( decide-le-leq-ℚ zero-ℚ q))
-```
-
-### If `x` is less than or equal to all the positive rational numbers `y` is less than or equal to, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    leq-leq-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) → leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
-        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
-      leq-ℝ⁰⁺ x y
-    leq-leq-positive-rational-ℝ⁰⁺ H =
-      leq-le-positive-rational-ℝ⁰⁺ x y
-        ( λ q y<q →
-          let open do-syntax-trunc-Prop (le-prop-ℝ _ _)
-          in do
-            (r , y<r , r<q) ← dense-rational-le-ℝ _ _ y<q
-            concatenate-leq-le-ℝ _ _ _
-              ( H
-                ( r , is-positive-le-nonnegative-real-ℚ y r y<r)
-                ( leq-le-ℝ _ _ y<r))
-              ( r<q))
-```
-
-### If `x` is less than or equal to the same positive rational numbers `y` is less than or equal to, then `x` and `y` are similar
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    sim-leq-same-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) →
-        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
-      sim-ℝ⁰⁺ x y
-    sim-leq-same-positive-rational-ℝ⁰⁺ H =
-      sim-sim-leq-ℝ
-        ( leq-leq-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
-          leq-leq-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
-```
-
-### If `x` is less than the same positive rational numbers `y` is less than, then `x` and `y` are similar
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    sim-le-same-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) →
-        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
-      sim-ℝ⁰⁺ x y
-    sim-le-same-positive-rational-ℝ⁰⁺ H =
-      sim-sim-leq-ℝ
-        ( leq-le-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
-          leq-le-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
-```
-
-### The metric space of nonnegative real numbers
-
-```agda
-metric-space-ℝ⁰⁺ : (l : Level) → Metric-Space (lsuc l) l
-metric-space-ℝ⁰⁺ l = metric-space-subset-ℝ (is-nonnegative-prop-ℝ {l})
-```
-
 ### `x ≤ y` if and only if `y - x` is nonnegative
 
 ```agda
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index a3349458452..850496dd467 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -43,7 +43,7 @@ On this page, we outline basic relations between
 abstract
   is-nonnegative-is-positive-ℝ :
     {l : Level} {x : ℝ l} → is-positive-ℝ x → is-nonnegative-ℝ x
-  is-nonnegative-is-positive-ℝ = leq-le-ℝ _ _
+  is-nonnegative-is-positive-ℝ = leq-le-ℝ
 
 nonnegative-ℝ⁺ : {l : Level} → ℝ⁺ l → ℝ⁰⁺ l
 nonnegative-ℝ⁺ (x , is-pos-x) = (x , is-nonnegative-is-positive-ℝ is-pos-x)
@@ -59,7 +59,7 @@ abstract
     tr
       ( le-ℝ (neg-ℝ x))
       ( neg-zero-ℝ)
-      ( neg-le-ℝ zero-ℝ x 0<x)
+      ( neg-le-ℝ 0<x)
 
 neg-ℝ⁺ : {l : Level} → ℝ⁺ l → ℝ⁻ l
 neg-ℝ⁺ (x , is-pos-x) = (neg-ℝ x , neg-is-positive-ℝ x is-pos-x)
@@ -75,7 +75,7 @@ abstract
     tr
       ( λ z → le-ℝ z (neg-ℝ x))
       ( neg-zero-ℝ)
-      ( neg-le-ℝ x zero-ℝ x<0)
+      ( neg-le-ℝ x<0)
 
 neg-ℝ⁻ : {l : Level} → ℝ⁻ l → ℝ⁺ l
 neg-ℝ⁻ (x , is-neg-x) = (neg-ℝ x , neg-is-negative-ℝ x is-neg-x)
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 1891f071d65..3e3881f93cf 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -39,6 +39,7 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -132,8 +133,7 @@ module _
   abstract
     is-positive-iff-zero-in-lower-cut-ℝ :
       is-positive-ℝ x ↔ is-in-lower-cut-ℝ x zero-ℚ
-    is-positive-iff-zero-in-lower-cut-ℝ =
-      inv-iff (le-real-iff-lower-cut-ℚ zero-ℚ x)
+    is-positive-iff-zero-in-lower-cut-ℝ = inv-iff (le-real-iff-lower-cut-ℚ x)
 
     is-positive-zero-in-lower-cut-ℝ :
       is-in-lower-cut-ℝ x zero-ℚ → is-positive-ℝ x
@@ -254,24 +254,19 @@ abstract
 ### The canonical embedding of rational numbers preserves positivity
 
 ```agda
-preserves-is-positive-real-ℚ :
-  (q : ℚ) → is-positive-ℚ q → is-positive-ℝ (real-ℚ q)
-preserves-is-positive-real-ℚ q pos-q =
-  preserves-le-real-ℚ zero-ℚ q (le-zero-is-positive-ℚ pos-q)
-
-opaque
-  unfolding le-ℝ real-ℚ
+abstract
+  preserves-is-positive-real-ℚ :
+    {q : ℚ} → is-positive-ℚ q → is-positive-ℝ (real-ℚ q)
+  preserves-is-positive-real-ℚ pos-q =
+    preserves-le-real-ℚ (le-zero-is-positive-ℚ pos-q)
 
-  is-positive-rational-is-positive-real-ℚ :
-    (q : ℚ) → is-positive-ℝ (real-ℚ q) → is-positive-ℚ q
-  is-positive-rational-is-positive-real-ℚ q =
-    elim-exists
-      ( is-positive-prop-ℚ q)
-      ( λ r (0<r , r<q) →
-        is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ r q r<q 0<r))
+  reflects-is-positive-real-ℚ :
+    {q : ℚ} → is-positive-ℝ (real-ℚ q) → is-positive-ℚ q
+  reflects-is-positive-real-ℚ {q} 0<qℝ =
+    is-positive-le-zero-ℚ (reflects-le-real-ℚ 0<qℝ)
 
 positive-real-ℚ⁺ : ℚ⁺ → ℝ⁺ lzero
-positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , preserves-is-positive-real-ℚ q pos-q)
+positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , preserves-is-positive-real-ℚ pos-q)
 
 one-ℝ⁺ : ℝ⁺ lzero
 one-ℝ⁺ = positive-real-ℚ⁺ one-ℚ⁺
@@ -287,13 +282,3 @@ abstract
     is-positive-le-zero-ℚ
       ( le-lower-upper-cut-ℝ x zero-ℚ q (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
 ```
-
-### Similarity of positive real numbers
-
-```agda
-sim-prop-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → Prop (l1 ⊔ l2)
-sim-prop-ℝ⁺ (x , _) (y , _) = sim-prop-ℝ x y
-
-sim-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → UU (l1 ⊔ l2)
-sim-ℝ⁺ (x , _) (y , _) = sim-ℝ x y
-```
diff --git a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
index 8af1317ffab..2782e63dab5 100644
--- a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
+++ b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
@@ -154,7 +154,7 @@ module _
       map-disjunction
         ( map-raise)
         ( map-raise)
-        ( is-located-lower-upper-cut-ℝ x p q p<q)
+        ( is-located-lower-upper-cut-ℝ x p<q)
 
   raise-ℝ : ℝ (l0 ⊔ l)
   raise-ℝ =
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index 63ec308b130..c6398376914 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -66,12 +66,10 @@ the [image](foundation.images.md) of this embedding
 ```agda
 is-dedekind-lower-upper-real-ℚ :
   (x : ℚ) →
-  is-dedekind-lower-upper-ℝ
-    ( lower-real-ℚ x)
-    ( upper-real-ℚ x)
+  is-dedekind-lower-upper-ℝ (lower-real-ℚ x) (upper-real-ℚ x)
 is-dedekind-lower-upper-real-ℚ x =
-  (λ q (H , K) → asymmetric-le-ℚ q x H K) ,
-  located-le-ℚ x
+  ( (λ q (H , K) → asymmetric-le-ℚ q x H K) ,
+    located-le-ℚ x)
 ```
 
 ### The canonical map from `ℚ` to `ℝ lzero`
@@ -161,7 +159,7 @@ all-eq-is-rational-ℝ x p q H H' =
         ( empty-Prop)
         ( pr1 H)
         ( pr2 H')
-        ( is-located-lower-upper-cut-ℝ x p q I))
+        ( is-located-lower-upper-cut-ℝ x I))
 
   right-case : le-ℚ q p → p ＝ q
   right-case I =
@@ -170,7 +168,7 @@ all-eq-is-rational-ℝ x p q H H' =
         ( empty-Prop)
         ( pr1 H')
         ( pr2 H)
-        ( is-located-lower-upper-cut-ℝ x q p I))
+        ( is-located-lower-upper-cut-ℝ x I))
 
 is-prop-rational-real : {l : Level} (x : ℝ l) → is-prop (Σ ℚ (is-rational-ℝ x))
 is-prop-rational-real x =
@@ -240,7 +238,7 @@ opaque
       ( lower-cut-ℝ x p)
       ( id)
       ( ex-falso ∘ q∉ux)
-      ( is-located-lower-upper-cut-ℝ x p q p<q)
+      ( is-located-lower-upper-cut-ℝ x p<q)
 
 eq-real-rational-is-rational-ℝ :
   (x : ℝ lzero) (q : ℚ) (H : is-rational-ℝ x q) → real-ℚ q ＝ x
diff --git a/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
new file mode 100644
index 00000000000..037e54c2f2c
--- /dev/null
+++ b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,71 @@
+# Saturation of inequality of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.saturation-inequality-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-nonnegative-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.saturation-inequality-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
+```
+
+</details>
+
+## Idea
+
+If `x ≤ y + ε` for [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) `x` and `y` and every
+[positive rational](elementary-number-theory.positive-rational-numbers.md) `ε`,
+then `x ≤ y`.
+
+Despite being a property of
+[inequality of nonnegative real numbers](real-numbers.inequality-nonnegative-real-numbers.md),
+this is much easier to prove via
+[strict inequality](real-numbers.strict-inequality-nonnegative-real-numbers.md),
+so it is moved to its own file to prevent circular dependency.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    saturated-leq-ℝ⁰⁺ :
+      ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (y +ℝ⁰⁺ nonnegative-real-ℚ⁺ ε)) →
+      leq-ℝ⁰⁺ x y
+    saturated-leq-ℝ⁰⁺ = saturated-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+```
+
+## Corollaries
+
+### If a nonnegative real number is less than or equal to all positive rational numbers, it is similar to zero
+
+```agda
+sim-zero-le-positive-rational-ℝ⁰⁺ :
+  {l : Level} (x : ℝ⁰⁺ l) →
+  ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (nonnegative-real-ℚ⁺ ε)) →
+  sim-zero-ℝ⁰⁺ x
+sim-zero-le-positive-rational-ℝ⁰⁺ x H =
+  sim-sim-leq-ℝ
+    ( leq-zero-ℝ⁰⁺ x ,
+      saturated-leq-ℝ⁰⁺
+        ( x)
+        ( zero-ℝ⁰⁺)
+        ( λ ε → inv-tr (leq-ℝ⁰⁺ x) (left-unit-law-add-ℝ⁰⁺ _) (H ε)))
+```
diff --git a/src/real-numbers/saturation-inequality-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
index 6657f97f72c..437a1a7a324 100644
--- a/src/real-numbers/saturation-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
@@ -12,12 +12,16 @@ module real-numbers.saturation-inequality-real-numbers where
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
 
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.propositional-truncations
 open import foundation.universe-levels
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -38,6 +42,29 @@ moved to its own file to prevent circular dependency.
 
 ## Proof
 
+### If `x < y + ε` for every positive rational `ε`, then `x ≤ y`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  where
+
+  abstract
+    saturated-le-ℝ : ((ε : ℚ⁺) → le-ℝ x (y +ℝ real-ℚ⁺ ε)) → leq-ℝ x y
+    saturated-le-ℝ H =
+      leq-not-le-ℝ y x
+        ( λ y<x →
+          let open do-syntax-trunc-Prop empty-Prop
+          in do
+            (ε , y+ε<x) ←
+              exists-positive-rational-separation-le-ℝ {x = y} {y = x} y<x
+            irreflexive-le-ℝ
+              ( x)
+              ( transitive-le-ℝ x (y +ℝ real-ℚ⁺ ε) x y+ε<x (H ε)))
+```
+
+### If `x ≤ y + ε` for every positive rational `ε`, then `x ≤ y`
+
 ```agda
 module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
@@ -57,5 +84,5 @@ module _
               ( y)
               ( real-ℚ⁺ (mediant-zero-ℚ⁺ ε))
               ( real-ℚ⁺ ε)
-              ( preserves-le-real-ℚ _ _ (le-mediant-zero-ℚ⁺ ε))))
+              ( preserves-le-real-ℚ (le-mediant-zero-ℚ⁺ ε))))
 ```
diff --git a/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
new file mode 100644
index 00000000000..3686eb0a01d
--- /dev/null
+++ b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,70 @@
+# Similarity of nonnegative real numbers
+
+```agda
+module real-numbers.similarity-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.identity-types
+open import foundation.logical-equivalences
+open import foundation.propositions
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+Two [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are
+{{#concept "similar" Disambiguation="nonnegative real numbers" Agda=sim-ℝ⁰⁺}} if
+they are [similar](real-numbers.similarity-real-numbers.md) as real numbers.
+
+## Definition
+
+### Similarity of nonnegative real numbers
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  sim-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
+  sim-prop-ℝ⁰⁺ = sim-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+
+  sim-ℝ⁰⁺ : UU (l1 ⊔ l2)
+  sim-ℝ⁰⁺ = sim-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+
+infix 6 _~ℝ⁰⁺_
+_~ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → UU (l1 ⊔ l2)
+_~ℝ⁰⁺_ = sim-ℝ⁰⁺
+
+sim-zero-prop-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → Prop l
+sim-zero-prop-ℝ⁰⁺ = sim-prop-ℝ⁰⁺ zero-ℝ⁰⁺
+
+sim-zero-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → UU l
+sim-zero-ℝ⁰⁺ = sim-ℝ⁰⁺ zero-ℝ⁰⁺
+
+eq-sim-ℝ⁰⁺ : {l : Level} (x y : ℝ⁰⁺ l) → sim-ℝ⁰⁺ x y → x ＝ y
+eq-sim-ℝ⁰⁺ x y x~y = eq-ℝ⁰⁺ x y (eq-sim-ℝ {x = real-ℝ⁰⁺ x} {y = real-ℝ⁰⁺ y} x~y)
+```
+
+#### Similarity is symmetric
+
+```agda
+abstract
+  symmetric-sim-ℝ⁰⁺ :
+    {l1 l2 : Level} → (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) → x ~ℝ⁰⁺ y → y ~ℝ⁰⁺ x
+  symmetric-sim-ℝ⁰⁺ _ _ = symmetric-sim-ℝ
+```
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index a9576b242d4..154237168d1 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -54,6 +54,7 @@ open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.squares-real-numbers
 ```
@@ -184,7 +185,7 @@ module _
         (p , p<r² , x<p) ←
           forward-implication (is-rounded-upper-cut-ℝ⁰⁺ x (r *ℚ r)) x<r²
         let
-          is-pos-p = is-positive-is-in-upper-cut-ℝ⁰⁺ x p x<p
+          is-pos-p = is-positive-is-in-upper-cut-ℝ⁰⁺ x x<p
         (q⁺@(q , is-pos-q) , q<r , p<q²) ←
           rounded-above-square-le-ℚ⁺ (r , is-pos-r) (p , is-pos-p) p<r²
         intro-exists
@@ -270,8 +271,6 @@ module _
             ( λ x<q² → (is-positive-le-ℚ⁰⁺ (p , is-nonneg-p) q p<q , x<q²))
             ( is-located-lower-upper-cut-ℝ
               ( real-ℝ⁰⁺ x)
-              ( p *ℚ p)
-              ( q *ℚ q)
               ( preserves-le-square-ℚ⁰⁺
                 ( p , is-nonneg-p)
                 ( q , is-nonnegative-le-ℚ⁰⁺ (p , is-nonneg-p) q p<q)
@@ -533,7 +532,7 @@ opaque
     leq-ℝ' (real-sqrt-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
   leq-unique-sqrt-ℝ⁰⁺' x⁰⁺@(x , _) y⁰⁺@(y , is-nonneg-y) (_ , x≤y²) q y<q =
     let
-      is-pos-q = is-positive-is-in-upper-cut-ℝ⁰⁺ y⁰⁺ q y<q
+      is-pos-q = is-positive-is-in-upper-cut-ℝ⁰⁺ y⁰⁺ y<q
       q⁺ = (q , is-pos-q)
       p⁻@(p , is-neg-p) = neg-ℚ⁺ q⁺
       p≤q = leq-negative-positive-ℚ p⁻ q⁺
@@ -554,7 +553,7 @@ opaque
               ( ([p,q] , p<y , y<q) , ([p,q] , p<y , y<q))
               ( leq-max-leq-both-ℚ _ _ _
                 ( leq-max-leq-both-ℚ _ _ _
-                  ( leq-eq-ℚ _ _ (negative-law-mul-ℚ q q))
+                  ( leq-eq-ℚ (negative-law-mul-ℚ q q))
                   ( leq-negative-positive-ℚ
                     ( mul-negative-positive-ℚ p⁻ q⁺)
                     ( q⁺ *ℚ⁺ q⁺)))
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 7c9c42cc2d9..36ce763cb8c 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -193,7 +193,7 @@ abstract
       ( square-ℝ x)
       ( x *ℝ y)
       ( square-ℝ y)
-      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ x y (leq-le-ℝ x y x<y))
+      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ x y (leq-le-ℝ x<y))
       ( preserves-le-right-mul-ℝ⁺ (y , is-positive-le-ℝ⁰⁺ x⁰⁺ y x<y) x y x<y)
 ```
 
@@ -207,10 +207,9 @@ abstract
   is-in-lower-cut-square-ℝ x q⁰⁺@(q , _) q∈Lx =
     let
       qℝ = nonnegative-real-ℚ⁰⁺ q⁰⁺
-      q<x = le-real-is-in-lower-cut-ℚ q x q∈Lx
+      q<x = le-real-is-in-lower-cut-ℚ x q∈Lx
     in
       is-in-lower-cut-le-real-ℚ
-        ( square-ℚ q)
         ( square-ℝ x)
         ( tr
           ( λ y → le-ℝ y (square-ℝ x))
@@ -230,7 +229,6 @@ abstract
     is-in-upper-cut-ℝ⁰⁺ (square-ℝ⁰⁺ x) (square-ℚ q)
   is-in-upper-cut-square-ℝ x⁰⁺@(x , _) q q∈Ux =
     is-in-upper-cut-le-real-ℚ
-      ( square-ℚ q)
       ( square-ℝ x)
       ( tr
         ( le-ℝ (square-ℝ x))
@@ -238,8 +236,8 @@ abstract
         ( preserves-le-square-ℝ⁰⁺
           ( x⁰⁺)
           ( nonnegative-real-ℚ⁺
-            ( q , is-positive-is-in-upper-cut-ℝ⁰⁺ x⁰⁺ q q∈Ux))
-          ( le-real-is-in-upper-cut-ℚ q x q∈Ux)))
+            ( q , is-positive-is-in-upper-cut-ℝ⁰⁺ x⁰⁺ q∈Ux))
+          ( le-real-is-in-upper-cut-ℚ x q∈Ux)))
 ```
 
 ### If a rational `q` is in the upper cut of both `x` and `-x`, `q²` is in the upper cut of `x²`
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
new file mode 100644
index 00000000000..978356cd75b
--- /dev/null
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -0,0 +1,289 @@
+# Strict inequalities about addition and subtraction on the real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.strict-inequalities-addition-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
+open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.conjunction
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.logical-equivalences
+open import foundation.propositional-truncations
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.arithmetically-located-dedekind-cuts
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+This file describes lemmas about
+[strict inequalities](real-numbers.strict-inequality-real-numbers.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md) related to
+[addition](real-numbers.addition-real-numbers.md) and
+[subtraction](real-numbers.difference-real-numbers.md).
+
+## Lemmas
+
+### Strict inequality on the real numbers is translation invariant
+
+```agda
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  opaque
+    unfolding add-ℝ le-ℝ
+
+    preserves-le-right-add-ℝ : le-ℝ x y → le-ℝ (x +ℝ z) (y +ℝ z)
+    preserves-le-right-add-ℝ x<y =
+      let
+        open
+          do-syntax-trunc-Prop
+            ( ∃ ℚ (λ r → upper-cut-ℝ (x +ℝ z) r ∧ lower-cut-ℝ (y +ℝ z) r))
+      in do
+        ( p , x<p , p<y) ← x<y
+        ( q , p<q , q<y) ← forward-implication (is-rounded-lower-cut-ℝ y p) p<y
+        ( (r , s) , s<r+q-p , r<z , z<s) ←
+          is-arithmetically-located-ℝ
+            ( z)
+            ( positive-diff-le-ℚ p<q)
+        let
+          p-q+s<r : le-ℚ ((p -ℚ q) +ℚ s) r
+          p-q+s<r =
+            tr
+              ( le-ℚ ((p -ℚ q) +ℚ s))
+              ( equational-reasoning
+                  (p -ℚ q) +ℚ (r +ℚ (q -ℚ p))
+                  ＝ (p -ℚ q) +ℚ (r -ℚ (p -ℚ q))
+                    by
+                      ap
+                        ( λ t → (p -ℚ q) +ℚ (r +ℚ t))
+                        ( inv (distributive-neg-diff-ℚ p q))
+                  ＝ r by is-identity-right-conjugation-add-ℚ (p -ℚ q) r)
+              ( preserves-le-right-add-ℚ (p -ℚ q) s (r +ℚ (q -ℚ p)) s<r+q-p)
+        intro-exists
+          ( p +ℚ s)
+          ( intro-exists (p , s) (x<p , z<s , refl) ,
+            intro-exists
+              ( q , (p -ℚ q) +ℚ s)
+              ( q<y ,
+                le-lower-cut-ℝ z ((p -ℚ q) +ℚ s) r p-q+s<r r<z ,
+                ( equational-reasoning
+                    p +ℚ s
+                    ＝ (q +ℚ (p -ℚ q)) +ℚ s
+                      by
+                        ap
+                          ( _+ℚ s)
+                          ( inv (is-identity-right-conjugation-add-ℚ q p))
+                    ＝ q +ℚ ((p -ℚ q) +ℚ s) by associative-add-ℚ _ _ _)))
+
+    preserves-le-left-add-ℝ : le-ℝ x y → le-ℝ (z +ℝ x) (z +ℝ y)
+    preserves-le-left-add-ℝ x<y =
+      binary-tr
+        ( le-ℝ)
+        ( commutative-add-ℝ x z)
+        ( commutative-add-ℝ y z)
+        ( preserves-le-right-add-ℝ x<y)
+
+abstract
+  preserves-le-diff-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    le-ℝ x y → le-ℝ (x -ℝ z) (y -ℝ z)
+  preserves-le-diff-ℝ z = preserves-le-right-add-ℝ (neg-ℝ z)
+
+  reverses-le-diff-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    le-ℝ x y → le-ℝ (z -ℝ y) (z -ℝ x)
+  reverses-le-diff-ℝ z x y x<y =
+    preserves-le-left-add-ℝ z _ _ (neg-le-ℝ x<y)
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  abstract
+    reflects-le-right-add-ℝ : le-ℝ (x +ℝ z) (y +ℝ z) → le-ℝ x y
+    reflects-le-right-add-ℝ x+z<y+z =
+      preserves-le-sim-ℝ
+        ( (x +ℝ z) +ℝ neg-ℝ z)
+        ( x)
+        ( (y +ℝ z) +ℝ neg-ℝ z)
+        ( y)
+        ( cancel-right-add-diff-ℝ x z)
+        ( cancel-right-add-diff-ℝ y z)
+        ( preserves-le-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z<y+z)
+
+    reflects-le-left-add-ℝ : le-ℝ (z +ℝ x) (z +ℝ y) → le-ℝ x y
+    reflects-le-left-add-ℝ z+x<z+y =
+      reflects-le-right-add-ℝ
+        ( binary-tr
+          ( le-ℝ)
+          ( commutative-add-ℝ z x)
+          ( commutative-add-ℝ z y)
+          ( z+x<z+y))
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  iff-translate-right-le-ℝ : le-ℝ x y ↔ le-ℝ (x +ℝ z) (y +ℝ z)
+  pr1 iff-translate-right-le-ℝ = preserves-le-right-add-ℝ z x y
+  pr2 iff-translate-right-le-ℝ = reflects-le-right-add-ℝ z x y
+
+  iff-translate-left-le-ℝ : le-ℝ x y ↔ le-ℝ (z +ℝ x) (z +ℝ y)
+  pr1 iff-translate-left-le-ℝ = preserves-le-left-add-ℝ z x y
+  pr2 iff-translate-left-le-ℝ = reflects-le-left-add-ℝ z x y
+
+abstract
+  preserves-le-add-ℝ :
+    {l1 l2 l3 l4 : Level} {a : ℝ l1} {b : ℝ l2} {c : ℝ l3} {d : ℝ l4} →
+    le-ℝ a b → le-ℝ c d → le-ℝ (a +ℝ c) (b +ℝ d)
+  preserves-le-add-ℝ {a = a} {b = b} {c = c} {d = d} a≤b c≤d =
+    transitive-le-ℝ
+      ( a +ℝ c)
+      ( a +ℝ d)
+      ( b +ℝ d)
+      ( preserves-le-right-add-ℝ d a b a≤b)
+      ( preserves-le-left-add-ℝ a c d c≤d)
+```
+
+### `x + y < z` if and only if `x < z - y`
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-left-add-ℝ : le-ℝ (x +ℝ y) z → le-ℝ x (z -ℝ y)
+    le-transpose-left-add-ℝ x+y<z =
+      preserves-le-left-sim-ℝ
+        ( z -ℝ y)
+        ( (x +ℝ y) -ℝ y)
+        ( x)
+        ( cancel-right-add-diff-ℝ x y)
+        ( preserves-le-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y<z)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-left-add-ℝ' : le-ℝ (x +ℝ y) z → le-ℝ y (z -ℝ x)
+    le-transpose-left-add-ℝ' x+y<z =
+      le-transpose-left-add-ℝ y x z
+        ( tr (λ w → le-ℝ w z) (commutative-add-ℝ _ _) x+y<z)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-right-diff-ℝ : le-ℝ x (y -ℝ z) → le-ℝ (x +ℝ z) y
+    le-transpose-right-diff-ℝ x<y-z =
+      preserves-le-right-sim-ℝ
+        ( x +ℝ z)
+        ( (y -ℝ z) +ℝ z)
+        ( y)
+        ( cancel-right-diff-add-ℝ y z)
+        ( preserves-le-right-add-ℝ z x (y -ℝ z) x<y-z)
+
+    le-transpose-right-diff-ℝ' : le-ℝ x (y -ℝ z) → le-ℝ (z +ℝ x) y
+    le-transpose-right-diff-ℝ' x<y-z =
+      tr
+        ( λ w → le-ℝ w y)
+        ( commutative-add-ℝ _ _)
+        ( le-transpose-right-diff-ℝ x<y-z)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  iff-diff-right-le-ℝ : le-ℝ (x +ℝ y) z ↔ le-ℝ x (z -ℝ y)
+  iff-diff-right-le-ℝ =
+    (le-transpose-left-add-ℝ x y z , le-transpose-right-diff-ℝ x z y)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    iff-add-right-le-ℝ : le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ y)
+    iff-add-right-le-ℝ =
+      tr
+        ( λ w → le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ w))
+        ( neg-neg-ℝ y)
+        ( iff-diff-right-le-ℝ x (neg-ℝ y) z)
+
+    le-transpose-left-diff-ℝ : le-ℝ (x -ℝ y) z → le-ℝ x (z +ℝ y)
+    le-transpose-left-diff-ℝ = forward-implication iff-add-right-le-ℝ
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-right-add-ℝ : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ z) y
+    le-transpose-right-add-ℝ = backward-implication (iff-add-right-le-ℝ x z y)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-right-add-ℝ' : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ y) z
+    le-transpose-right-add-ℝ' x<y+z =
+      le-transpose-right-add-ℝ x z y (tr (le-ℝ x) (commutative-add-ℝ _ _) x<y+z)
+```
+
+### If `x < y`, then there is some `ε : ℚ⁺` with `x + ε < y`
+
+```agda
+module _
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} (x<y : le-ℝ x y)
+  where
+
+  abstract
+    exists-positive-rational-separation-le-ℝ :
+      exists ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y)
+    exists-positive-rational-separation-le-ℝ =
+      let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y))
+      in do
+        (q , 0<q , q<y-x) ←
+          dense-rational-le-ℝ zero-ℝ (y -ℝ x)
+            ( preserves-le-left-sim-ℝ
+              ( y -ℝ x)
+              ( x -ℝ x)
+              ( zero-ℝ)
+              ( right-inverse-law-add-ℝ x)
+              ( preserves-le-right-add-ℝ (neg-ℝ x) x y x<y))
+        intro-exists
+          ( q , is-positive-le-zero-ℚ (reflects-le-real-ℚ 0<q))
+          ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
+```
diff --git a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
index 7bf6d53efa0..07f144ab5d9 100644
--- a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
@@ -15,13 +15,19 @@ open import elementary-number-theory.strict-inequality-nonnegative-rational-numb
 
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
+open import foundation.function-types
+open import foundation.logical-equivalences
 open import foundation.propositions
 open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
+open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -59,7 +65,7 @@ abstract
   preserves-le-nonnegative-real-ℚ⁰⁺ :
     (p q : ℚ⁰⁺) →
     le-ℚ⁰⁺ p q → le-ℝ⁰⁺ (nonnegative-real-ℚ⁰⁺ p) (nonnegative-real-ℚ⁰⁺ q)
-  preserves-le-nonnegative-real-ℚ⁰⁺ p q = preserves-le-real-ℚ _ _
+  preserves-le-nonnegative-real-ℚ⁰⁺ p q = preserves-le-real-ℚ
 ```
 
 ### Similarity preserves strict inequality
@@ -75,20 +81,6 @@ module _
       preserves-le-left-sim-ℝ (real-ℝ⁰⁺ z) _ _ x~y
 ```
 
-### Addition preserves strict inequality
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
-  where
-
-  abstract
-    preserves-le-add-ℝ⁰⁺ :
-      le-ℝ⁰⁺ x y → le-ℝ⁰⁺ z w → le-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
-    preserves-le-add-ℝ⁰⁺ =
-      preserves-le-add-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z) (real-ℝ⁰⁺ w)
-```
-
 ### Concatenation of inequality and strict inequality
 
 ```agda
@@ -114,6 +106,24 @@ module _
       exists ℚ⁺ (λ q → le-prop-ℝ⁰⁺ x (nonnegative-real-ℚ⁺ q))
     le-some-positive-rational-ℝ⁰⁺ =
       map-tot-exists
-        ( λ (q , _) x<q → le-real-is-in-upper-cut-ℚ q (real-ℝ⁰⁺ x) x<q)
+        ( λ (q , _) x<q → le-real-is-in-upper-cut-ℚ (real-ℝ⁰⁺ x) x<q)
         ( exists-ℚ⁺-in-upper-cut-ℝ⁰⁺ x)
 ```
+
+### If `x` is less than the same positive rational numbers `y` is less than, then `x` and `y` are similar
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    sim-le-same-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) →
+        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
+      sim-ℝ⁰⁺ x y
+    sim-le-same-positive-rational-ℝ⁰⁺ H =
+      sim-sim-leq-ℝ
+        ( leq-le-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
+          leq-le-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
+```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index a68d6e8cbd1..dc20d798d6f 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -81,20 +81,16 @@ le-prop-ℝ x y = (le-ℝ x y , is-prop-le-ℝ x y)
 ### Strict inequality on the reals implies inequality
 
 ```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
-  where
-
-  opaque
-    unfolding le-ℝ leq-ℝ
-
-    leq-le-ℝ : le-ℝ x y → leq-ℝ x y
-    leq-le-ℝ x<y p p<x =
-      elim-exists
-        ( lower-cut-ℝ y p)
-        ( λ q (x<q , q<y) →
-          le-lower-cut-ℝ y p q (le-lower-upper-cut-ℝ x p q p<x x<q) q<y)
-        ( x<y)
+opaque
+  unfolding le-ℝ leq-ℝ
+
+  leq-le-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → leq-ℝ x y
+  leq-le-ℝ {x = x} {y = y} x<y p p<x =
+    elim-exists
+      ( lower-cut-ℝ y p)
+      ( λ q (x<q , q<y) →
+        le-lower-cut-ℝ y p q (le-lower-upper-cut-ℝ x p q p<x x<q) q<y)
+      ( x<y)
 ```
 
 ### Strict inequality on the reals is irreflexive
@@ -119,9 +115,7 @@ module _
 
 ```agda
 module _
-  {l1 l2 : Level}
-  (x : ℝ l1)
-  (y : ℝ l2)
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
   opaque
@@ -176,7 +170,7 @@ module _
 
 ```agda
 module _
-  (x y : ℚ)
+  {x y : ℚ}
   where
 
   opaque
@@ -226,7 +220,7 @@ module _
 
 ```agda
 module _
-  {l : Level} (q : ℚ) (x : ℝ l)
+  {l : Level} {q : ℚ} (x : ℝ l)
   where
 
   opaque
@@ -247,7 +241,7 @@ module _
 
 ```agda
 module _
-  {l : Level} (q : ℚ) (x : ℝ l)
+  {l : Level} {q : ℚ} (x : ℝ l)
   where
 
   opaque
@@ -276,9 +270,9 @@ module _
   is-located-le-ℝ : disjunction-type (le-ℝ (real-ℚ p) x) (le-ℝ x (real-ℚ q))
   is-located-le-ℝ =
     map-disjunction
-      ( le-real-is-in-lower-cut-ℚ p x)
-      ( le-real-is-in-upper-cut-ℚ q x)
-      ( is-located-lower-upper-cut-ℝ x p q p<q)
+      ( le-real-is-in-lower-cut-ℚ x)
+      ( le-real-is-in-upper-cut-ℚ x)
+      ( is-located-lower-upper-cut-ℝ x p<q)
 ```
 
 ### Every real is less than a rational number
@@ -292,7 +286,7 @@ module _
     le-some-rational-ℝ : exists ℚ (λ q → le-prop-ℝ x (real-ℚ q))
     le-some-rational-ℝ =
       map-tot-exists
-        ( λ q → le-real-is-in-upper-cut-ℚ q x)
+        ( λ q → le-real-is-in-upper-cut-ℚ x)
         ( is-inhabited-upper-cut-ℝ x)
 ```
 
@@ -311,7 +305,7 @@ module _
         open do-syntax-trunc-Prop (∃ (ℝ lzero) (λ y → le-prop-ℝ y x))
       in do
         ( q , q<x) ← is-inhabited-lower-cut-ℝ x
-        intro-exists (real-ℚ q) (le-real-is-in-lower-cut-ℚ q x q<x)
+        intro-exists (real-ℚ q) (le-real-is-in-lower-cut-ℚ x q<x)
 
     exists-greater-ℝ : exists (ℝ lzero) (λ y → le-prop-ℝ x y)
     exists-greater-ℝ =
@@ -319,7 +313,7 @@ module _
         open do-syntax-trunc-Prop (∃ (ℝ lzero) (le-prop-ℝ x))
       in do
         ( q , x<q) ← is-inhabited-upper-cut-ℝ x
-        intro-exists (real-ℚ q) (le-real-is-in-upper-cut-ℚ q x x<q)
+        intro-exists (real-ℚ q) (le-real-is-in-upper-cut-ℚ x x<q)
 ```
 
 ### Negation reverses the strict ordering of real numbers
@@ -327,8 +321,7 @@ module _
 ```agda
 module _
   {l1 l2 : Level}
-  (x : ℝ l1)
-  (y : ℝ l2)
+  {x : ℝ l1} {y : ℝ l2}
   where
 
   opaque
@@ -385,7 +378,7 @@ module _
           ( lower-cut-ℝ x p)
           ( id)
           ( λ x<q → reductio-ad-absurdum (intro-exists q (x<q , q<y)) x≮y)
-          ( is-located-lower-upper-cut-ℝ x p q p<q)
+          ( is-located-lower-upper-cut-ℝ x p<q)
 ```
 
 ### If `x` is less than or equal to `y`, then `y` is not less than `x`
@@ -476,7 +469,7 @@ opaque
       map-disjunction
         ( λ p<z → intro-exists p (x<p , p<z))
         ( λ z<q → intro-exists q (z<q , q<y))
-        ( is-located-lower-upper-cut-ℝ z p q p<q)
+        ( is-located-lower-upper-cut-ℝ z p<q)
 ```
 
 ### Strict inequality on the real numbers is invariant under similarity
@@ -517,265 +510,6 @@ module _
       ( preserves-le-right-sim-ℝ x1 y1 y2 y1~y2 x1<y1)
 ```
 
-### Strict inequality on the real numbers is translation invariant
-
-```agda
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  opaque
-    unfolding add-ℝ le-ℝ
-
-    preserves-le-right-add-ℝ : le-ℝ x y → le-ℝ (x +ℝ z) (y +ℝ z)
-    preserves-le-right-add-ℝ x<y =
-      let
-        open
-          do-syntax-trunc-Prop
-            ( ∃ ℚ (λ r → upper-cut-ℝ (x +ℝ z) r ∧ lower-cut-ℝ (y +ℝ z) r))
-      in do
-        ( p , x<p , p<y) ← x<y
-        ( q , p<q , q<y) ← forward-implication (is-rounded-lower-cut-ℝ y p) p<y
-        ( (r , s) , s<r+q-p , r<z , z<s) ←
-          is-arithmetically-located-ℝ
-            ( z)
-            ( positive-diff-le-ℚ p q p<q)
-        let
-          p-q+s<r : le-ℚ ((p -ℚ q) +ℚ s) r
-          p-q+s<r =
-            tr
-              ( le-ℚ ((p -ℚ q) +ℚ s))
-              ( equational-reasoning
-                  (p -ℚ q) +ℚ (r +ℚ (q -ℚ p))
-                  ＝ (p -ℚ q) +ℚ (r -ℚ (p -ℚ q))
-                    by
-                      ap
-                        ( λ t → (p -ℚ q) +ℚ (r +ℚ t))
-                        ( inv (distributive-neg-diff-ℚ p q))
-                  ＝ r by is-identity-right-conjugation-add-ℚ (p -ℚ q) r)
-              ( preserves-le-right-add-ℚ (p -ℚ q) s (r +ℚ (q -ℚ p)) s<r+q-p)
-        intro-exists
-          ( p +ℚ s)
-          ( intro-exists (p , s) (x<p , z<s , refl) ,
-            intro-exists
-              ( q , (p -ℚ q) +ℚ s)
-              ( q<y ,
-                le-lower-cut-ℝ z ((p -ℚ q) +ℚ s) r p-q+s<r r<z ,
-                ( equational-reasoning
-                    p +ℚ s
-                    ＝ (q +ℚ (p -ℚ q)) +ℚ s
-                      by
-                        ap
-                          ( _+ℚ s)
-                          ( inv (is-identity-right-conjugation-add-ℚ q p))
-                    ＝ q +ℚ ((p -ℚ q) +ℚ s) by associative-add-ℚ _ _ _)))
-
-    preserves-le-left-add-ℝ : le-ℝ x y → le-ℝ (z +ℝ x) (z +ℝ y)
-    preserves-le-left-add-ℝ x<y =
-      binary-tr
-        ( le-ℝ)
-        ( commutative-add-ℝ x z)
-        ( commutative-add-ℝ y z)
-        ( preserves-le-right-add-ℝ x<y)
-
-abstract
-  preserves-le-diff-ℝ :
-    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
-    le-ℝ x y → le-ℝ (x -ℝ z) (y -ℝ z)
-  preserves-le-diff-ℝ z = preserves-le-right-add-ℝ (neg-ℝ z)
-
-  reverses-le-diff-ℝ :
-    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
-    le-ℝ x y → le-ℝ (z -ℝ y) (z -ℝ x)
-  reverses-le-diff-ℝ z x y x<y =
-    preserves-le-left-add-ℝ z _ _ (neg-le-ℝ _ _ x<y)
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  abstract
-    reflects-le-right-add-ℝ : le-ℝ (x +ℝ z) (y +ℝ z) → le-ℝ x y
-    reflects-le-right-add-ℝ x+z<y+z =
-      preserves-le-sim-ℝ
-        ( (x +ℝ z) +ℝ neg-ℝ z)
-        ( x)
-        ( (y +ℝ z) +ℝ neg-ℝ z)
-        ( y)
-        ( cancel-right-add-diff-ℝ x z)
-        ( cancel-right-add-diff-ℝ y z)
-        ( preserves-le-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z<y+z)
-
-    reflects-le-left-add-ℝ : le-ℝ (z +ℝ x) (z +ℝ y) → le-ℝ x y
-    reflects-le-left-add-ℝ z+x<z+y =
-      reflects-le-right-add-ℝ
-        ( binary-tr
-          ( le-ℝ)
-          ( commutative-add-ℝ z x)
-          ( commutative-add-ℝ z y)
-          ( z+x<z+y))
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  iff-translate-right-le-ℝ : le-ℝ x y ↔ le-ℝ (x +ℝ z) (y +ℝ z)
-  pr1 iff-translate-right-le-ℝ = preserves-le-right-add-ℝ z x y
-  pr2 iff-translate-right-le-ℝ = reflects-le-right-add-ℝ z x y
-
-  iff-translate-left-le-ℝ : le-ℝ x y ↔ le-ℝ (z +ℝ x) (z +ℝ y)
-  pr1 iff-translate-left-le-ℝ = preserves-le-left-add-ℝ z x y
-  pr2 iff-translate-left-le-ℝ = reflects-le-left-add-ℝ z x y
-
-abstract
-  preserves-le-add-ℝ :
-    {l1 l2 l3 l4 : Level} (a : ℝ l1) (b : ℝ l2) (c : ℝ l3) (d : ℝ l4) →
-    le-ℝ a b → le-ℝ c d → le-ℝ (a +ℝ c) (b +ℝ d)
-  preserves-le-add-ℝ a b c d a≤b c≤d =
-    transitive-le-ℝ
-      ( a +ℝ c)
-      ( a +ℝ d)
-      ( b +ℝ d)
-      ( preserves-le-right-add-ℝ d a b a≤b)
-      ( preserves-le-left-add-ℝ a c d c≤d)
-```
-
-### `x + y < z` if and only if `x < z - y`
-
-```agda
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-left-add-ℝ : le-ℝ (x +ℝ y) z → le-ℝ x (z -ℝ y)
-    le-transpose-left-add-ℝ x+y<z =
-      preserves-le-left-sim-ℝ
-        ( z -ℝ y)
-        ( (x +ℝ y) -ℝ y)
-        ( x)
-        ( cancel-right-add-diff-ℝ x y)
-        ( preserves-le-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y<z)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-left-add-ℝ' : le-ℝ (x +ℝ y) z → le-ℝ y (z -ℝ x)
-    le-transpose-left-add-ℝ' x+y<z =
-      le-transpose-left-add-ℝ y x z
-        ( tr (λ w → le-ℝ w z) (commutative-add-ℝ _ _) x+y<z)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-right-diff-ℝ : le-ℝ x (y -ℝ z) → le-ℝ (x +ℝ z) y
-    le-transpose-right-diff-ℝ x<y-z =
-      preserves-le-right-sim-ℝ
-        ( x +ℝ z)
-        ( (y -ℝ z) +ℝ z)
-        ( y)
-        ( cancel-right-diff-add-ℝ y z)
-        ( preserves-le-right-add-ℝ z x (y -ℝ z) x<y-z)
-
-    le-transpose-right-diff-ℝ' : le-ℝ x (y -ℝ z) → le-ℝ (z +ℝ x) y
-    le-transpose-right-diff-ℝ' x<y-z =
-      tr
-        ( λ w → le-ℝ w y)
-        ( commutative-add-ℝ _ _)
-        ( le-transpose-right-diff-ℝ x<y-z)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  iff-diff-right-le-ℝ : le-ℝ (x +ℝ y) z ↔ le-ℝ x (z -ℝ y)
-  iff-diff-right-le-ℝ =
-    (le-transpose-left-add-ℝ x y z , le-transpose-right-diff-ℝ x z y)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    iff-add-right-le-ℝ : le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ y)
-    iff-add-right-le-ℝ =
-      tr
-        ( λ w → le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ w))
-        ( neg-neg-ℝ y)
-        ( iff-diff-right-le-ℝ x (neg-ℝ y) z)
-
-    le-transpose-left-diff-ℝ : le-ℝ (x -ℝ y) z → le-ℝ x (z +ℝ y)
-    le-transpose-left-diff-ℝ = forward-implication iff-add-right-le-ℝ
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-right-add-ℝ : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ z) y
-    le-transpose-right-add-ℝ = backward-implication (iff-add-right-le-ℝ x z y)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-right-add-ℝ' : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ y) z
-    le-transpose-right-add-ℝ' x<y+z =
-      le-transpose-right-add-ℝ x z y (tr (le-ℝ x) (commutative-add-ℝ _ _) x<y+z)
-```
-
-### If `x < y`, then there is some `ε : ℚ⁺` with `x + ε < y`
-
-```agda
-module _
-  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} (x<y : le-ℝ x y)
-  where
-
-  abstract
-    exists-positive-rational-separation-le-ℝ :
-      exists ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y)
-    exists-positive-rational-separation-le-ℝ =
-      let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y))
-      in do
-        (q , 0<q , q<y-x) ←
-          dense-rational-le-ℝ zero-ℝ (y -ℝ x)
-            ( preserves-le-left-sim-ℝ
-              ( y -ℝ x)
-              ( x -ℝ x)
-              ( zero-ℝ)
-              ( right-inverse-law-add-ℝ x)
-              ( preserves-le-right-add-ℝ (neg-ℝ x) x y x<y))
-        intro-exists
-          ( q , is-positive-le-zero-ℚ (reflects-le-real-ℚ _ _ 0<q))
-          ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
-```
-
-### If `x < y + ε` for every positive rational `ε`, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
-  where
-
-  abstract
-    saturated-le-ℝ : ((ε : ℚ⁺) → le-ℝ x (y +ℝ real-ℚ⁺ ε)) → leq-ℝ x y
-    saturated-le-ℝ H =
-      leq-not-le-ℝ y x
-        ( λ y<x →
-          let open do-syntax-trunc-Prop empty-Prop
-          in do
-            (ε , y+ε<x) ←
-              exists-positive-rational-separation-le-ℝ {x = y} {y = x} y<x
-            irreflexive-le-ℝ
-              ( x)
-              ( transitive-le-ℝ x (y +ℝ real-ℚ⁺ ε) x y+ε<x (H ε)))
-```
-
 ### If `x` is less than each rational number `y` is less than, then `x ≤ y`
 
 ```agda
@@ -791,8 +525,8 @@ module _
     leq-le-rational-ℝ H =
       leq-leq'-ℝ _ _
         ( λ q y<q →
-          is-in-upper-cut-le-real-ℚ q x
-            ( H q (le-real-is-in-upper-cut-ℚ q y y<q)))
+          is-in-upper-cut-le-real-ℚ x
+            ( H q (le-real-is-in-upper-cut-ℚ y y<q)))
 ```
 
 ### Two real numbers are similar if they are less than the same rational numbers
@@ -832,13 +566,12 @@ module _
     le-split-add-rational-ℝ x+y<p =
       let open do-syntax-trunc-Prop (∃ _ _)
       in do
-        ((q , r) , x<q , y<r , p=q+r) ←
-          is-in-upper-cut-le-real-ℚ p (x +ℝ y) x+y<p
+        ((q , r) , x<q , y<r , p=q+r) ← is-in-upper-cut-le-real-ℚ (x +ℝ y) x+y<p
         intro-exists
           ( q , r)
           ( p=q+r ,
-            le-real-is-in-upper-cut-ℚ q x x<q ,
-            le-real-is-in-upper-cut-ℚ r y y<r)
+            le-real-is-in-upper-cut-ℚ x x<q ,
+            le-real-is-in-upper-cut-ℚ y y<r)
 ```
 
 ## References
diff --git a/src/real-numbers/suprema-families-real-numbers.lagda.md b/src/real-numbers/suprema-families-real-numbers.lagda.md
index 4ef731d25e2..f730c054982 100644
--- a/src/real-numbers/suprema-families-real-numbers.lagda.md
+++ b/src/real-numbers/suprema-families-real-numbers.lagda.md
@@ -42,6 +42,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 ```
@@ -151,7 +152,7 @@ module _
                 ( x-ε<yᵢ)
                 ( le-transpose-left-add-ℝ' _ _ _
                   ( le-transpose-right-diff-ℝ _ _ _
-                    ( le-real-is-in-lower-cut-ℚ ε (x -ℝ z) ε<x-z))))
+                    ( le-real-is-in-lower-cut-ℚ (x -ℝ z) ε<x-z))))
               ( yᵢ≤z i))
     pr2 (is-least-upper-bound-is-supremum-family-ℝ z) x≤z i =
       transitive-leq-ℝ (y i) x z x≤z
@@ -258,7 +259,7 @@ module _
             ( x-ε<yᵢ)
             ( le-transpose-left-add-ℝ' _ _ _
               ( le-transpose-right-diff-ℝ _ _ _
-                ( le-real-is-in-lower-cut-ℚ ε (x -ℝ z) ε<x-z))))
+                ( le-real-is-in-lower-cut-ℚ (x -ℝ z) ε<x-z))))
 
     le-supremum-iff-le-element-family-ℝ :
       {l4 : Level} → (z : ℝ l4) →
diff --git a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
index ac543613a6c..91e5c9c0bef 100644
--- a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
@@ -24,6 +24,7 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -56,7 +57,6 @@ module _
       is-in-lower-cut-ℝ (x -ℝ real-ℚ q) p
     transpose-add-is-in-lower-cut-ℝ p+q<x =
       is-in-lower-cut-le-real-ℚ
-        ( p)
         ( x -ℝ real-ℚ q)
         ( le-transpose-left-add-ℝ
           ( real-ℚ p)
@@ -65,13 +65,12 @@ module _
           ( inv-tr
             ( λ y → le-ℝ y x)
             ( add-real-ℚ p q)
-            ( le-real-is-in-lower-cut-ℚ (p +ℚ q) x p+q<x)))
+            ( le-real-is-in-lower-cut-ℚ x p+q<x)))
 
     transpose-is-in-lower-cut-diff-ℝ :
       is-in-lower-cut-ℝ (x -ℝ real-ℚ p) q → is-in-lower-cut-ℝ x (q +ℚ p)
     transpose-is-in-lower-cut-diff-ℝ x-p<q =
       is-in-lower-cut-le-real-ℚ
-        ( q +ℚ p)
         ( x)
         ( tr
           ( λ y → le-ℝ y x)
@@ -80,7 +79,7 @@ module _
             ( real-ℚ q)
             ( x)
             ( real-ℚ p)
-            ( le-real-is-in-lower-cut-ℚ q (x -ℝ real-ℚ p) x-p<q)))
+            ( le-real-is-in-lower-cut-ℚ (x -ℝ real-ℚ p) x-p<q)))
 
 module _
   {l : Level} (x : ℝ l) (p q : ℚ)
@@ -127,7 +126,6 @@ module _
       is-in-upper-cut-ℝ (x -ℝ real-ℚ q) p
     transpose-add-is-in-upper-cut-ℝ x<p+q =
       is-in-upper-cut-le-real-ℚ
-        ( p)
         ( x -ℝ real-ℚ q)
         ( le-transpose-right-add-ℝ
           ( x)
@@ -136,13 +134,12 @@ module _
           ( inv-tr
             ( le-ℝ x)
             ( add-real-ℚ p q)
-            ( le-real-is-in-upper-cut-ℚ (p +ℚ q) x x<p+q)))
+            ( le-real-is-in-upper-cut-ℚ x x<p+q)))
 
     transpose-is-in-upper-cut-diff-ℝ :
       is-in-upper-cut-ℝ (x -ℝ real-ℚ p) q → is-in-upper-cut-ℝ x (q +ℚ p)
     transpose-is-in-upper-cut-diff-ℝ x-p<q =
       is-in-upper-cut-le-real-ℚ
-        ( q +ℚ p)
         ( x)
         ( tr
           ( le-ℝ x)
@@ -151,7 +148,7 @@ module _
             ( x)
             ( real-ℚ p)
             ( real-ℚ q)
-            ( le-real-is-in-upper-cut-ℚ q (x -ℝ real-ℚ p) x-p<q)))
+            ( le-real-is-in-upper-cut-ℚ (x -ℝ real-ℚ p) x-p<q)))
 
 module _
   {l : Level} (x : ℝ l) (p q : ℚ)

From b5a27af88b7b41a9eb5f549baa1026529d1a31f4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 15:01:52 -0800
Subject: [PATCH 038/134] Progress

---
 ...lue-closed-intervals-real-numbers.lagda.md |  2 --
 .../absolute-value-real-numbers.lagda.md      | 16 +++------
 .../distance-real-numbers.lagda.md            | 16 ++++-----
 ...nequalities-addition-real-numbers.lagda.md |  4 ---
 .../inequality-real-numbers.lagda.md          |  2 +-
 .../metric-space-of-real-numbers.lagda.md     |  8 -----
 ...tiplication-positive-real-numbers.lagda.md |  4 +--
 .../nonnegative-real-numbers.lagda.md         | 27 ++++++++-------
 .../positive-real-numbers.lagda.md            |  4 +--
 .../similarity-positive-real-numbers.lagda.md | 34 +++++++++++++++++++
 ...nequalities-addition-real-numbers.lagda.md |  4 ---
 .../strict-inequality-real-numbers.lagda.md   |  2 +-
 12 files changed, 67 insertions(+), 56 deletions(-)
 create mode 100644 src/real-numbers/similarity-positive-real-numbers.lagda.md

diff --git a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
index f040689b7c2..634f6c90343 100644
--- a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
@@ -49,8 +49,6 @@ abstract
     leq-ℝ (abs-ℝ z) (max-abs-closed-interval-ℝ [x,y])
   leq-max-abs-is-in-closed-interval-ℝ ((x , y) , x≤y) z (x≤z , z≤y) =
     leq-abs-leq-leq-neg-ℝ
-      ( z)
-      ( max-ℝ (neg-ℝ x) y)
       ( transitive-leq-ℝ _ _ _ (leq-right-max-ℝ _ _) z≤y)
       ( transitive-leq-ℝ _ _ _ (leq-left-max-ℝ _ _) (neg-leq-ℝ x≤z))
 ```
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index d31b66d8b89..8d3c38266f8 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -66,7 +66,7 @@ opaque
 ### The absolute value of zero is zero
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ
 
   abs-zero-ℝ : abs-ℝ zero-ℝ ＝ zero-ℝ
@@ -76,7 +76,7 @@ opaque
 ### The absolute value preserves similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ
 
   preserves-sim-abs-ℝ :
@@ -89,7 +89,7 @@ opaque
 ### The absolute value of a real number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ leq-ℝ max-ℝ neg-ℚ neg-ℝ real-ℚ
 
   is-nonnegative-abs-ℝ : {l : Level} → (x : ℝ l) → is-nonnegative-ℝ (abs-ℝ x)
@@ -210,10 +210,10 @@ module _
 
 ```agda
 module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding abs-ℝ
 
     leq-abs-leq-leq-neg-ℝ : leq-ℝ x y → leq-ℝ (neg-ℝ x) y → leq-ℝ (abs-ℝ x) y
@@ -231,8 +231,6 @@ module _
     triangle-inequality-abs-ℝ : leq-ℝ (abs-ℝ (x +ℝ y)) (abs-ℝ x +ℝ abs-ℝ y)
     triangle-inequality-abs-ℝ =
       leq-abs-leq-leq-neg-ℝ
-        ( x +ℝ y)
-        ( abs-ℝ x +ℝ abs-ℝ y)
         ( preserves-leq-add-ℝ (leq-abs-ℝ x) (leq-abs-ℝ y))
         ( inv-tr
           ( λ z → leq-ℝ z (abs-ℝ x +ℝ abs-ℝ y))
@@ -259,8 +257,6 @@ module _
         ( abs-ℝ x)
         ( abs-ℝ y)
         ( leq-abs-leq-leq-neg-ℝ
-          ( x)
-          ( abs-ℝ y +ℝ real-ℚ⁺ d)
           ( transitive-leq-ℝ
             ( x)
             ( y +ℝ real-ℚ⁺ d)
@@ -286,8 +282,6 @@ module _
               ( x)
               ( right-leq-real-bound-neighborhood-ℝ d x y I))))
         ( leq-abs-leq-leq-neg-ℝ
-          ( y)
-          ( abs-ℝ x +ℝ real-ℚ⁺ d)
           ( transitive-leq-ℝ
             ( y)
             ( x +ℝ real-ℚ⁺ d)
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index df3a005ee36..1f779167f02 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -83,11 +83,13 @@ nonnegative-dist-ℝ x y = (dist-ℝ x y , is-nonnegative-dist-ℝ x y)
 
 ### Relationship to the metric space of real numbers
 
-Two real numbers `x` and `y` are in an `ε`-neighborhood of each other if and
-only if their distance is at most `ε`.
+Two real numbers `x` and `y` are in an `ε`-neighborhood of each other in the
+[metric space of real numbers](real-numbers.metric-space-of-real-numbers.md)
+[if and only if](foundation.logical-equivalences.md) their distance is
+[at most](real-numbers.inequality-real-numbers.md) `ε`.
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ neighborhood-ℝ
 
   diff-bound-neighborhood-ℝ :
@@ -125,8 +127,6 @@ abstract
     dist-ℝ x y ≤-ℝ real-ℚ⁺ d
   leq-dist-neighborhood-ℝ d⁺@(d , _) x y H =
     leq-abs-leq-leq-neg-ℝ
-      ( x -ℝ y)
-      ( real-ℚ d)
       ( diff-bound-neighborhood-ℝ d⁺ x y H)
       ( inv-tr
         ( λ z → leq-ℝ z (real-ℚ d))
@@ -236,8 +236,6 @@ abstract
     dist-ℝ x y ≤-ℝ z
   leq-dist-leq-diff-ℝ x y z x-y≤z y-x≤z =
     leq-abs-leq-leq-neg-ℝ
-      ( _)
-      ( z)
       ( x-y≤z)
       ( inv-tr (λ w → leq-ℝ w z) (distributive-neg-diff-ℝ _ _) y-x≤z)
 
@@ -259,7 +257,7 @@ abstract
   preserves-dist-left-add-ℝ :
     {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
     sim-ℝ
-      ( dist-ℝ (add-ℝ x y) (add-ℝ x z))
+      ( dist-ℝ (x +ℝ y) (x +ℝ z))
       ( dist-ℝ y z)
   preserves-dist-left-add-ℝ x y z =
     similarity-reasoning-ℝ
@@ -280,7 +278,7 @@ abstract
   preserves-dist-right-add-ℝ :
     {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
     sim-ℝ
-      ( dist-ℝ (add-ℝ x z) (add-ℝ y z))
+      ( dist-ℝ (x +ℝ z) (y +ℝ z))
       ( dist-ℝ x y)
   preserves-dist-right-add-ℝ z x y =
     similarity-reasoning-ℝ
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
index 654f922bc10..aa5f3ad62ec 100644
--- a/src/real-numbers/inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
@@ -73,10 +73,6 @@ module _
     reflects-leq-right-add-ℝ : leq-ℝ (x +ℝ z) (y +ℝ z) → leq-ℝ x y
     reflects-leq-right-add-ℝ x+z≤y+z =
       preserves-leq-sim-ℝ
-        ( (x +ℝ z) -ℝ z)
-        ( x)
-        ( (y +ℝ z) -ℝ z)
-        ( y)
         ( cancel-right-add-diff-ℝ x z)
         ( cancel-right-add-diff-ℝ y z)
         ( preserves-leq-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z≤y+z)
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index d9b18435f16..d0a0cd049c4 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -337,7 +337,7 @@ module _
 
 module _
   {l1 l2 l3 l4 : Level}
-  (x1 : ℝ l1) (x2 : ℝ l2) (y1 : ℝ l3) (y2 : ℝ l4)
+  {x1 : ℝ l1} {x2 : ℝ l2} {y1 : ℝ l3} {y2 : ℝ l4}
   (x1~x2 : sim-ℝ x1 x2) (y1~y2 : sim-ℝ y1 y2)
   where
 
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index 5afe6b7b568..7a648355720 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -355,10 +355,6 @@ module _
       ( x')
       ( y')
       ( preserves-leq-sim-ℝ
-        ( x)
-        ( x')
-        ( y +ℝ real-ℚ⁺ d)
-        ( y' +ℝ real-ℚ⁺ d)
         ( x~x')
         ( preserves-sim-right-add-ℝ
           ( real-ℚ⁺ d)
@@ -367,10 +363,6 @@ module _
           ( y~y'))
         ( left-leq-real-bound-neighborhood-ℝ d x y H))
       ( preserves-leq-sim-ℝ
-        ( y)
-        ( y')
-        ( x +ℝ real-ℚ⁺ d)
-        ( x' +ℝ real-ℚ⁺ d)
         ( y~y')
         ( preserves-sim-right-add-ℝ
           ( real-ℚ⁺ d)
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index 0c7aeb53076..ab63080b992 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -123,7 +123,7 @@ abstract
     {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
     le-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z) → le-ℝ y z
   reflects-le-left-mul-ℝ⁺ x y z xy<xz =
-    preserves-le-sim-ℝ _ _ _ _
+    preserves-le-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
       ( preserves-le-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy<xz)
@@ -169,7 +169,7 @@ abstract
     {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
     leq-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z) → leq-ℝ y z
   reflects-leq-left-mul-ℝ⁺ x y z xy≤xz =
-    preserves-leq-sim-ℝ _ _ _ _
+    preserves-leq-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
       ( preserves-leq-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy≤xz)
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 6b5a11da376..999eff4ee7e 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -24,7 +24,9 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
+open import foundation.functoriality-propositional-truncation
 open import foundation.function-types
+open import foundation.inhabited-subtypes
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
@@ -43,10 +45,8 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.saturation-inequality-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
-open import real-numbers.subsets-real-numbers
 ```
 
 </details>
@@ -102,8 +102,12 @@ upper-cut-ℝ⁰⁺ (x , _) = upper-cut-ℝ x
 is-in-upper-cut-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → ℚ → UU l
 is-in-upper-cut-ℝ⁰⁺ (x , _) = is-in-upper-cut-ℝ x
 
+is-inhabited-upper-cut-ℝ⁰⁺ :
+  {l : Level} (x : ℝ⁰⁺ l) → is-inhabited-subtype (upper-cut-ℝ⁰⁺ x)
+is-inhabited-upper-cut-ℝ⁰⁺ (x , _) = is-inhabited-upper-cut-ℝ x
+
 is-rounded-upper-cut-ℝ⁰⁺ :
-  {l : Level} → (x : ℝ⁰⁺ l) (r : ℚ) →
+  {l : Level} (x : ℝ⁰⁺ l) (r : ℚ) →
   (is-in-upper-cut-ℝ⁰⁺ x r ↔ exists ℚ (λ q → le-ℚ-Prop q r ∧ upper-cut-ℝ⁰⁺ x q))
 is-rounded-upper-cut-ℝ⁰⁺ (x , _) = is-rounded-upper-cut-ℝ x
 ```
@@ -163,8 +167,8 @@ abstract
           ( 0≤x)
           ( le-real-is-in-upper-cut-ℚ x x<q)))
 
-opaque
-  unfolding leq-ℝ leq-ℝ' real-ℚ
+abstract opaque
+  unfolding leq-ℝ' real-ℚ
 
   is-nonnegative-is-positive-upper-cut-ℝ :
     {l : Level} → (x : ℝ l) → (upper-cut-ℝ x ⊆ is-positive-prop-ℚ) →
@@ -176,7 +180,7 @@ opaque
 ### A real number is nonnegative if and only if every negative rational number is in its lower cut
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ real-ℚ
 
   is-nonnegative-leq-negative-lower-cut-ℝ :
@@ -200,10 +204,9 @@ abstract
     {l : Level} → (x : ℝ⁰⁺ l) →
     exists ℚ⁺ (λ q → upper-cut-ℝ⁰⁺ x (rational-ℚ⁺ q))
   exists-ℚ⁺-in-upper-cut-ℝ⁰⁺ x =
-    let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → upper-cut-ℝ⁰⁺ x (rational-ℚ⁺ q)))
-    in do
-      (q , x<q) ← is-inhabited-upper-cut-ℝ (real-ℝ⁰⁺ x)
-      intro-exists (q , is-positive-is-in-upper-cut-ℝ⁰⁺ x x<q) x<q
+    map-trunc-Prop
+      ( λ (q , x<q) → ((q , is-positive-is-in-upper-cut-ℝ⁰⁺ x x<q) , x<q))
+      ( is-inhabited-upper-cut-ℝ⁰⁺ x)
 ```
 
 ### Nonpositive rational numbers are not less than or equal to nonnegative real numbers
@@ -219,9 +222,9 @@ module _
     not-le-leq-zero-rational-ℝ⁰⁺ q≤0 x<q =
       irreflexive-le-ℝ
         ( real-ℝ⁰⁺ x)
-        ( concatenate-le-leq-ℝ _ (real-ℚ q) _
+        ( concatenate-le-leq-ℝ _ _ _
           ( x<q)
-          ( transitive-leq-ℝ (real-ℚ q) zero-ℝ (real-ℝ⁰⁺ x)
+          ( transitive-leq-ℝ _ _ _
             ( is-nonnegative-real-ℝ⁰⁺ x)
             ( preserves-leq-real-ℚ q≤0)))
 ```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 3e3881f93cf..95cc41a6621 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -153,7 +153,7 @@ module _
   {l : Level} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     exists-ℚ⁺-in-lower-cut-is-positive-ℝ :
@@ -186,7 +186,7 @@ exists-ℚ⁺-in-lower-cut-ℝ⁺ = ind-Σ exists-ℚ⁺-in-lower-cut-is-positiv
 ### Addition with a positive real number is a strictly inflationary map
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ le-ℝ
 
   le-left-add-real-ℝ⁺ :
diff --git a/src/real-numbers/similarity-positive-real-numbers.lagda.md b/src/real-numbers/similarity-positive-real-numbers.lagda.md
new file mode 100644
index 00000000000..7253e344364
--- /dev/null
+++ b/src/real-numbers/similarity-positive-real-numbers.lagda.md
@@ -0,0 +1,34 @@
+# Similarity of positive real numbers
+
+```agda
+module real-numbers.similarity-positive-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.universe-levels
+open import foundation.propositions
+open import real-numbers.positive-real-numbers
+open import real-numbers.similarity-real-numbers
+open import foundation.dependent-pair-types
+```
+
+</details>
+
+## Idea
+
+Two [positive](real-numbers.positive-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are
+{{#concept "similar" Disambiguation="positive real numbers" Agda=sim-ℝ⁺}} if
+they are [similar](real-numbers.similarity-real-numbers.md) as real numbers.
+
+## Definition
+
+```agda
+sim-prop-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → Prop (l1 ⊔ l2)
+sim-prop-ℝ⁺ (x , _) (y , _) = sim-prop-ℝ x y
+
+sim-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → UU (l1 ⊔ l2)
+sim-ℝ⁺ (x , _) (y , _) = sim-ℝ x y
+```
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 978356cd75b..e28ec7bda1b 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -131,10 +131,6 @@ module _
     reflects-le-right-add-ℝ : le-ℝ (x +ℝ z) (y +ℝ z) → le-ℝ x y
     reflects-le-right-add-ℝ x+z<y+z =
       preserves-le-sim-ℝ
-        ( (x +ℝ z) +ℝ neg-ℝ z)
-        ( x)
-        ( (y +ℝ z) +ℝ neg-ℝ z)
-        ( y)
         ( cancel-right-add-diff-ℝ x z)
         ( cancel-right-add-diff-ℝ y z)
         ( preserves-le-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z<y+z)
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index dc20d798d6f..d2a97facc63 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -496,7 +496,7 @@ module _
 
 module _
   {l1 l2 l3 l4 : Level}
-  (x1 : ℝ l1) (x2 : ℝ l2) (y1 : ℝ l3) (y2 : ℝ l4)
+  {x1 : ℝ l1} {x2 : ℝ l2} {y1 : ℝ l3} {y2 : ℝ l4}
   (x1~x2 : sim-ℝ x1 x2) (y1~y2 : sim-ℝ y1 y2)
   where
 

From 5545c34d011b95f5514e13fc0429bc6f218282f5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 15:07:38 -0800
Subject: [PATCH 039/134] Progress

---
 .../powers-positive-rational-numbers.lagda.md  |  4 ++--
 ...ic-spaces-are-uniformly-continuous.lagda.md | 18 ++++++++++--------
 src/real-numbers.lagda.md                      |  1 +
 .../nonnegative-real-numbers.lagda.md          |  4 ++--
 .../similarity-positive-real-numbers.lagda.md  |  5 +++--
 5 files changed, 18 insertions(+), 14 deletions(-)

diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 1f3932919c6..08a63343eba 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -197,8 +197,8 @@ abstract
     Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n q)))
   bound-unbounded-power-greater-than-one-ℚ⁺ q⁺@(q , _) b 1<q =
     let
-      q-1⁺ = positive-diff-le-ℚ one-ℚ q 1<q
-      (n , b<⟨1+q-1⟩ⁿ) = bound-unbounded-power-one-plus-ℚ⁺ q-1⁺ b
+      (n , b<⟨1+q-1⟩ⁿ) =
+        bound-unbounded-power-one-plus-ℚ⁺ (positive-diff-le-ℚ 1<q) b
     in
       ( n ,
         tr
diff --git a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
index 4a4fb29d7ce..c33453db283 100644
--- a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
@@ -32,7 +32,9 @@ open import order-theory.large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.metric-space-of-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
 ```
@@ -79,15 +81,15 @@ module _
           ≤ ρ' x y' +ℝ ρ' y' y
             by is-triangular-is-metric-of-Metric-Space M ρ H x y' y
           ≤ ρ' y' y +ℝ ρ' x y'
-            by leq-eq-ℝ _ _ (commutative-add-ℝ _ _)
+            by leq-eq-ℝ (commutative-add-ℝ _ _)
           ≤ ρ' y y' +ℝ ρ' x y'
-            by leq-eq-ℝ _ _ (ap-add-ℝ (commutative-ρ' y' y) refl))
+            by leq-eq-ℝ (ap-add-ℝ (commutative-ρ' y' y) refl))
         ( chain-of-inequalities
           ρ' x y'
           ≤ ρ' x y +ℝ ρ' y y'
             by is-triangular-is-metric-of-Metric-Space M ρ H x y y'
           ≤ ρ' y y' +ℝ ρ' x y
-            by leq-eq-ℝ _ _ (commutative-add-ℝ _ _))
+            by leq-eq-ℝ (commutative-add-ℝ _ _))
 
     modulus-of-uniform-continuity-metric-of-Metric-Space : ℚ⁺ → ℚ⁺
     modulus-of-uniform-continuity-metric-of-Metric-Space =
@@ -112,7 +114,7 @@ module _
               by triangle-inequality-dist-ℝ _ _ _
             ≤ dist-ℝ (ρ' x y) (ρ' x y') +ℝ dist-ℝ (ρ' y' x) (ρ' y' x')
               by
-                leq-eq-ℝ _ _
+                leq-eq-ℝ
                   ( ap-add-ℝ
                     ( refl)
                     ( ap-binary
@@ -121,18 +123,18 @@ module _
                       ( commutative-ρ' x' y')))
             ≤ ρ' y y' +ℝ ρ' x x'
               by
-                preserves-leq-add-ℝ _ _ _ _
+                preserves-leq-add-ℝ
                   ( dist-metric-leq-metric-of-Metric-Space x y y')
                   ( dist-metric-leq-metric-of-Metric-Space y' x x')
             ≤ real-ℚ⁺ ε' +ℝ real-ℚ⁺ ε'
               by
-                preserves-leq-add-ℝ _ _ _ _
+                preserves-leq-add-ℝ
                   ( forward-implication (H ε' y y') Nε'yy')
                   ( forward-implication (H ε' x x') Nε'xx')
             ≤ real-ℚ⁺ (ε' +ℚ⁺ ε')
-              by leq-eq-ℝ _ _ (add-real-ℚ _ _)
+              by leq-eq-ℝ (add-real-ℚ _ _)
             ≤ real-ℚ⁺ ε
-              by preserves-leq-real-ℚ _ _ (leq-le-ℚ 2ε'<ε))
+              by preserves-leq-real-ℚ (leq-le-ℚ 2ε'<ε))
 
     is-uniformly-continuous-metric-of-Metric-Space :
       is-uniformly-continuous-function-Metric-Space
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 1a10077ffcd..aa970733de4 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -74,6 +74,7 @@ open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
 open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
+open import real-numbers.similarity-positive-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.square-roots-nonnegative-real-numbers public
 open import real-numbers.squares-real-numbers public
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 999eff4ee7e..c2f3447dd1e 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -24,10 +24,10 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
-open import foundation.functoriality-propositional-truncation
 open import foundation.function-types
-open import foundation.inhabited-subtypes
+open import foundation.functoriality-propositional-truncation
 open import foundation.identity-types
+open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
diff --git a/src/real-numbers/similarity-positive-real-numbers.lagda.md b/src/real-numbers/similarity-positive-real-numbers.lagda.md
index 7253e344364..0dbfb733ac9 100644
--- a/src/real-numbers/similarity-positive-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-positive-real-numbers.lagda.md
@@ -7,11 +7,12 @@ module real-numbers.similarity-positive-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.universe-levels
+open import foundation.dependent-pair-types
 open import foundation.propositions
+open import foundation.universe-levels
+
 open import real-numbers.positive-real-numbers
 open import real-numbers.similarity-real-numbers
-open import foundation.dependent-pair-types
 ```
 
 </details>

From 007d5876181591352e3536ca324d4fc69dcb13e1 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 15:59:13 -0800
Subject: [PATCH 040/134] Progress

---
 ...ddition-positive-rational-numbers.lagda.md |   8 +-
 ...ive-and-negative-rational-numbers.lagda.md |   8 +-
 ...closed-intervals-rational-numbers.lagda.md |  44 ++---
 .../negative-rational-numbers.lagda.md        |  18 +-
 ...ive-and-negative-rational-numbers.lagda.md |  12 +-
 .../positive-rational-numbers.lagda.md        |   9 +-
 ...wers-nonnegative-rational-numbers.lagda.md |   2 +-
 .../squares-rational-numbers.lagda.md         |   2 +-
 ...quality-positive-rational-numbers.lagda.md |   6 +-
 ...trict-inequality-rational-numbers.lagda.md |  11 +-
 .../located-metric-spaces.lagda.md            |   5 +-
 src/real-numbers.lagda.md                     |   1 +
 .../addition-real-numbers.lagda.md            | 104 +++--------
 .../binary-maximum-real-numbers.lagda.md      | 161 ++---------------
 ...lication-nonnegative-real-numbers.lagda.md |   3 +-
 ...tiplication-positive-real-numbers.lagda.md |   6 +-
 .../multiplication-real-numbers.lagda.md      |  32 ++--
 ...ve-inverses-positive-real-numbers.lagda.md |  17 +-
 .../negation-real-numbers.lagda.md            |  13 --
 .../nonnegative-real-numbers.lagda.md         |   7 -
 ...ional-lower-dedekind-real-numbers.lagda.md |   2 +-
 ...ional-upper-dedekind-real-numbers.lagda.md |   2 +-
 ...-from-lower-dedekind-real-numbers.lagda.md |   8 +-
 ...-from-upper-dedekind-real-numbers.lagda.md |  11 +-
 ...short-binary-maximum-real-numbers.lagda.md | 166 ++++++++++++++++++
 ...re-roots-nonnegative-real-numbers.lagda.md |   8 +-
 .../squares-real-numbers.lagda.md             |   2 -
 .../strict-inequality-real-numbers.lagda.md   |   5 +-
 28 files changed, 303 insertions(+), 370 deletions(-)
 create mode 100644 src/real-numbers/short-binary-maximum-real-numbers.lagda.md

diff --git a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
index 7e1cf13e5b6..78b872ffe94 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -334,16 +334,16 @@ module _
     ((d : ℚ⁺) → leq-ℚ x (y +ℚ (rational-ℚ⁺ d))) → leq-ℚ x y
   leq-leq-add-positive-ℚ H =
     rec-coproduct
-      ( λ I →
+      ( λ y<x →
         ex-falso
           ( not-leq-le-ℚ
             ( mediant-ℚ y x)
             ( x)
-            ( le-right-mediant-ℚ y x I)
+            ( le-right-mediant-ℚ y<x)
             ( tr
               ( leq-ℚ x)
-              ( right-law-positive-diff-le-ℚ (le-left-mediant-ℚ y x I))
-              ( H (positive-diff-le-ℚ (le-left-mediant-ℚ y x I))))))
+              ( right-law-positive-diff-le-ℚ (le-left-mediant-ℚ y<x))
+              ( H (positive-diff-le-ℚ (le-left-mediant-ℚ y<x))))))
       ( id)
       ( decide-le-leq-ℚ y x)
 
diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
index 88ac145a32a..8d8b52b30f9 100644
--- a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -101,8 +101,8 @@ abstract
 ```agda
 abstract
   is-positive-le-ℚ⁰⁺ :
-    (p : ℚ⁰⁺) (q : ℚ) → le-ℚ (rational-ℚ⁰⁺ p) q → is-positive-ℚ q
-  is-positive-le-ℚ⁰⁺ (p , nonneg-p) q p<q =
+    (p : ℚ⁰⁺) {q : ℚ} → le-ℚ (rational-ℚ⁰⁺ p) q → is-positive-ℚ q
+  is-positive-le-ℚ⁰⁺ (p , nonneg-p) p<q =
     is-positive-le-zero-ℚ
       ( concatenate-leq-le-ℚ _ _ _ (leq-zero-is-nonnegative-ℚ nonneg-p) p<q)
 ```
@@ -112,8 +112,8 @@ abstract
 ```agda
 abstract
   is-negative-le-ℚ⁰⁻ :
-    (q : ℚ⁰⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁰⁻ q) → is-negative-ℚ p
-  is-negative-le-ℚ⁰⁻ (q , nonpos-q) p p<q =
+    (q : ℚ⁰⁻) {p : ℚ} → le-ℚ p (rational-ℚ⁰⁻ q) → is-negative-ℚ p
+  is-negative-le-ℚ⁰⁻ (q , nonpos-q) {p} p<q =
     is-negative-le-zero-ℚ
       ( concatenate-le-leq-ℚ p q zero-ℚ
         ( p<q)
diff --git a/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
index b05bb91cc05..ed0f43ca734 100644
--- a/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
@@ -1,6 +1,8 @@
 # Multiplication of interior intervals of closed intervals of rational numbers
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module elementary-number-theory.multiplication-interior-closed-intervals-rational-numbers where
 ```
 
@@ -325,12 +327,10 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=a'c')
-            is-pos-b' =
-              is-positive-le-ℚ⁰⁺ (a' , is-nonneg-a') b' a'<b'
-            is-pos-b = is-positive-le-ℚ⁺ (b' , is-pos-b') b b'<b
-            is-pos-d' =
-              is-positive-le-ℚ⁰⁺ (c' , is-nonneg-c') d' c'<d'
-            is-pos-d = is-positive-le-ℚ⁺ (d' , is-pos-d') d d'<d
+            is-pos-b' = is-positive-le-ℚ⁰⁺ (a' , is-nonneg-a') a'<b'
+            is-pos-b = is-positive-le-ℚ⁺ (b' , is-pos-b') b'<b
+            is-pos-d' = is-positive-le-ℚ⁰⁺ (c' , is-nonneg-c') c'<d'
+            is-pos-d = is-positive-le-ℚ⁺ (d' , is-pos-d') d'<d
             is-nonneg-min' =
               inv-tr
                 ( is-nonnegative-ℚ)
@@ -377,11 +377,7 @@ abstract
                         ( le-ℚ (a *ℚ c'))
                         ( min'=a'c')
                         ( preserves-le-right-mul-ℚ⁺
-                          ( c' ,
-                            is-positive-le-ℚ⁰⁺
-                              ( c , is-nonneg-c)
-                              ( c')
-                              ( c<c'))
+                          ( c' , is-positive-le-ℚ⁰⁺ (c , is-nonneg-c) c<c')
                           ( a)
                           ( a')
                           ( a<a'))))
@@ -403,10 +399,8 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=a'd')
-            is-neg-a =
-              is-negative-le-ℚ⁰⁻ (a' , is-nonpos-a') a a<a'
-            is-pos-d =
-              is-positive-le-ℚ⁰⁺ (d' , is-nonneg-d') d d'<d
+            is-neg-a = is-negative-le-ℚ⁰⁻ (a' , is-nonpos-a') a<a'
+            is-pos-d = is-positive-le-ℚ⁰⁺ (d' , is-nonneg-d') d'<d
           in
             concatenate-leq-le-ℚ
               ( min)
@@ -444,8 +438,8 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=b'c')
-            is-pos-b = is-positive-le-ℚ⁰⁺ (b' , is-nonneg-b') b b'<b
-            is-neg-c = is-negative-le-ℚ⁰⁻ (c' , is-nonpos-c') c c<c'
+            is-pos-b = is-positive-le-ℚ⁰⁺ (b' , is-nonneg-b') b'<b
+            is-neg-c = is-negative-le-ℚ⁰⁻ (c' , is-nonpos-c') c<c'
           in
             concatenate-le-leq-ℚ
               ( min)
@@ -481,12 +475,10 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=b'd')
-            is-neg-a' =
-              is-negative-le-ℚ⁰⁻ (b' , is-nonpos-b') a' a'<b'
-            is-neg-a = is-negative-le-ℚ⁻ (a' , is-neg-a') a a<a'
-            is-neg-c' =
-              is-negative-le-ℚ⁰⁻ (d' , is-nonpos-d') c' c'<d'
-            is-neg-c = is-negative-le-ℚ⁻ (c' , is-neg-c') c c<c'
+            is-neg-a' = is-negative-le-ℚ⁰⁻ (b' , is-nonpos-b') a'<b'
+            is-neg-a = is-negative-le-ℚ⁻ (a' , is-neg-a') a<a'
+            is-neg-c' = is-negative-le-ℚ⁰⁻ (d' , is-nonpos-d') c'<d'
+            is-neg-c = is-negative-le-ℚ⁻ (c' , is-neg-c') c<c'
             is-nonneg-min' =
               inv-tr
                 ( is-nonnegative-ℚ)
@@ -533,11 +525,7 @@ abstract
                           ( le-ℚ (b' *ℚ d))
                           ( min'=b'd')
                           ( reverses-le-left-mul-ℚ⁻
-                            ( b' ,
-                              is-negative-le-ℚ⁰⁻
-                                ( b , is-nonpos-b)
-                                ( b')
-                                ( b'<b))
+                            ( b' , is-negative-le-ℚ⁰⁻ (b , is-nonpos-b) b'<b)
                             ( d')
                             ( d)
                             ( d'<d)))))
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index f5754c5d866..7a41b1a752e 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -201,7 +201,7 @@ module _
   {x y : ℚ} (H : le-ℚ x y)
   where
 
-  opaque
+  abstract opaque
     unfolding is-negative-ℚ
 
     is-negative-diff-le-ℚ : is-negative-ℚ (x -ℚ y)
@@ -218,7 +218,7 @@ module _
 ### Negative rational numbers are nonzero
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ
 
   is-nonzero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → is-nonzero-ℚ x
@@ -247,30 +247,30 @@ le-ℚ⁻ p q = le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁻ q)
 ```agda
 abstract
   is-negative-leq-ℚ⁻ :
-    (q : ℚ⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
-  is-negative-leq-ℚ⁻ (q , neg-q) p p≤q =
+    (q : ℚ⁻) {p : ℚ} → leq-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
+  is-negative-leq-ℚ⁻ (q , neg-q) {p} p≤q =
     is-negative-le-zero-ℚ
       ( concatenate-leq-le-ℚ p q zero-ℚ p≤q (le-zero-is-negative-ℚ neg-q))
 
   is-negative-le-ℚ⁻ :
-    (q : ℚ⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
-  is-negative-le-ℚ⁻ q p p<q = is-negative-leq-ℚ⁻ q p (leq-le-ℚ p<q)
+    (q : ℚ⁻) {p : ℚ} → le-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
+  is-negative-le-ℚ⁻ q p<q = is-negative-leq-ℚ⁻ q (leq-le-ℚ p<q)
 ```
 
 ### There is no greatest negative rational number
 
 ```agda
-opaque
+abstract opaque
   mediant-zero-ℚ⁻ : ℚ⁻ → ℚ⁻
   mediant-zero-ℚ⁻ (q , is-neg-q) =
     ( mediant-ℚ q zero-ℚ ,
       is-negative-le-zero-ℚ
-      ( le-right-mediant-ℚ _ _ (le-zero-is-negative-ℚ is-neg-q)))
+      ( le-right-mediant-ℚ (le-zero-is-negative-ℚ is-neg-q)))
 
   le-mediant-zero-ℚ⁻ :
     (q : ℚ⁻) → le-ℚ (rational-ℚ⁻ q) (rational-ℚ⁻ (mediant-zero-ℚ⁻ q))
   le-mediant-zero-ℚ⁻ (q , is-neg-q) =
-    le-left-mediant-ℚ _ _ (le-zero-is-negative-ℚ is-neg-q)
+    le-left-mediant-ℚ (le-zero-is-negative-ℚ is-neg-q)
 
 rational-mediant-zero-ℚ⁻ : ℚ⁻ → ℚ
 rational-mediant-zero-ℚ⁻ q = rational-ℚ⁻ (mediant-zero-ℚ⁻ q)
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index e0447a64bbc..021ef28f500 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -125,7 +125,7 @@ decide-is-negative-is-positive-is-nonzero-ℚ {q} q≠0 =
 #### Any positive rational number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding is-nonnegative-ℚ is-positive-ℚ
 
   is-nonnegative-is-positive-ℚ : {q : ℚ} → is-positive-ℚ q → is-nonnegative-ℚ q
@@ -138,7 +138,7 @@ nonnegative-ℚ⁺ (q , H) = (q , is-nonnegative-is-positive-ℚ H)
 ### Any negative rational number is nonpositive
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ is-nonpositive-ℚ
 
   is-nonpositive-is-negative-ℚ : {q : ℚ} → is-negative-ℚ q → is-nonpositive-ℚ q
@@ -153,7 +153,7 @@ nonpositive-ℚ⁻ (q , H) = (q , is-nonpositive-is-negative-ℚ H)
 #### The negation of a positive rational number is negative
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ
 
   is-negative-neg-is-positive-ℚ :
@@ -168,7 +168,7 @@ neg-ℚ⁺ (q , is-pos-q) =
 #### The negation of a negative rational number is positive
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ
 
   is-positive-neg-is-negative-ℚ :
@@ -182,7 +182,7 @@ neg-ℚ⁻ (q , is-neg-q) = (neg-ℚ q , is-positive-neg-is-negative-ℚ is-neg-
 #### The negation of a nonnegative rational number is nonpositive
 
 ```agda
-opaque
+abstract opaque
   unfolding is-nonpositive-ℚ
 
   is-nonpositive-neg-is-nonnegative-ℚ :
@@ -197,7 +197,7 @@ neg-ℚ⁰⁺ (q , is-nonneg-q) =
 #### The negation of a nonpositive rational number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding is-nonpositive-ℚ
 
   is-nonnegative-neg-is-nonpositive-ℚ :
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index dec1ae2814b..f146b5f15f9 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -284,13 +284,12 @@ nonzero-ℚ⁺ (x , P) = (x , is-nonzero-is-positive-ℚ P)
 ```agda
 abstract
   is-positive-leq-ℚ⁺ :
-    (p : ℚ⁺) (q : ℚ) → leq-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
-  is-positive-leq-ℚ⁺ (p , pos-p) q p≤q =
+    (p : ℚ⁺) {q : ℚ} → leq-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
+  is-positive-leq-ℚ⁺ (p , pos-p) p≤q =
     is-positive-le-zero-ℚ
       ( concatenate-le-leq-ℚ _ _ _ (le-zero-is-positive-ℚ pos-p) p≤q)
 
   is-positive-le-ℚ⁺ :
-    (p : ℚ⁺) (q : ℚ) → le-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
-  is-positive-le-ℚ⁺ p q p<q =
-    is-positive-leq-ℚ⁺ p q (leq-le-ℚ p<q)
+    (p : ℚ⁺) {q : ℚ} → le-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
+  is-positive-le-ℚ⁺ p p<q = is-positive-leq-ℚ⁺ p (leq-le-ℚ p<q)
 ```
diff --git a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
index 36c1c13e24c..535de721529 100644
--- a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
@@ -122,7 +122,7 @@ abstract
       ( power-ℚ⁰⁺ (succ-ℕ n) q)
       ( preserves-leq-left-mul-ℚ⁰⁺ (power-ℚ⁰⁺ n p) _ _ (leq-le-ℚ p<q))
       ( preserves-le-right-mul-ℚ⁺
-        ( rational-ℚ⁰⁺ q , is-positive-le-ℚ⁰⁺ p _ p<q)
+        ( rational-ℚ⁰⁺ q , is-positive-le-ℚ⁰⁺ p p<q)
         ( rational-ℚ⁰⁺ (power-ℚ⁰⁺ n p))
         ( rational-ℚ⁰⁺ (power-ℚ⁰⁺ n q))
         ( preserves-le-power-ℚ⁰⁺ n p q p<q (is-nonzero-succ-ℕ _)))
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index d9c561a0178..ada7062ba55 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -236,7 +236,7 @@ abstract
       ( p *ℚ q)
       ( square-ℚ q)
       ( preserves-leq-left-mul-ℚ⁰⁺ p⁰⁺ p q (leq-le-ℚ p<q))
-      ( preserves-le-right-mul-ℚ⁺ (q , is-positive-le-ℚ⁰⁺ p⁰⁺ q p<q) p q p<q)
+      ( preserves-le-right-mul-ℚ⁺ (q , is-positive-le-ℚ⁰⁺ p⁰⁺ p<q) p q p<q)
 ```
 
 ### Squaring nonnegative rational numbers reflects inequality
diff --git a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
index 83f68e3629e..4e2a1657a36 100644
--- a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
@@ -82,20 +82,16 @@ strict-preorder-ℚ⁺ =
 ### There is no least positive rational number
 
 ```agda
-opaque
+abstract opaque
   mediant-zero-ℚ⁺ : ℚ⁺ → ℚ⁺
   mediant-zero-ℚ⁺ x =
     ( mediant-ℚ zero-ℚ (rational-ℚ⁺ x) ,
       is-positive-le-zero-ℚ
         ( le-left-mediant-ℚ
-          ( zero-ℚ)
-          ( rational-ℚ⁺ x)
           ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x))))
 
   le-mediant-zero-ℚ⁺ : (x : ℚ⁺) → le-ℚ⁺ (mediant-zero-ℚ⁺ x) x
   le-mediant-zero-ℚ⁺ x =
     le-right-mediant-ℚ
-      ( zero-ℚ)
-      ( rational-ℚ⁺ x)
       ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x))
 ```
diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
index db95edd8b78..51e9f6df63f 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -519,12 +519,11 @@ abstract
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (H : le-ℚ x y)
   where
 
-  opaque
-    unfolding le-ℚ-Prop
-    unfolding mediant-ℚ
+  abstract opaque
+    unfolding le-ℚ-Prop mediant-ℚ
 
     le-left-mediant-ℚ : le-ℚ x (mediant-ℚ x y)
     le-left-mediant-ℚ =
@@ -543,7 +542,7 @@ module _
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (x<y : le-ℚ x y)
   where
 
   abstract
@@ -551,7 +550,7 @@ module _
     dense-le-ℚ =
       intro-exists
         ( mediant-ℚ x y)
-        ( le-left-mediant-ℚ x y H , le-right-mediant-ℚ x y H)
+        ( le-left-mediant-ℚ x<y , le-right-mediant-ℚ x<y)
 ```
 
 ### Strict inequality on the rational numbers is located
diff --git a/src/metric-spaces/located-metric-spaces.lagda.md b/src/metric-spaces/located-metric-spaces.lagda.md
index 1fef763229a..a5d4416fc0f 100644
--- a/src/metric-spaces/located-metric-spaces.lagda.md
+++ b/src/metric-spaces/located-metric-spaces.lagda.md
@@ -137,8 +137,7 @@ module _
           let
             r = mediant-ℚ p q
             p⁺ = (p , is-positive-le-zero-ℚ 0<p)
-            p<r = le-left-mediant-ℚ p q p<q
-            r<q = le-right-mediant-ℚ p q p<q
+            p<r = le-left-mediant-ℚ p<q
             r⁺ =
               ( r ,
                 is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ p r p<r 0<p))
@@ -150,7 +149,7 @@ module _
                   ((ε⁺@(ε , _) , Nεxy) , ε<p) ← p∈U
                   ¬Npxy
                     ( monotonic-neighborhood-Metric-Space M x y ε⁺ p⁺ ε<p Nεxy))
-              ( λ Nrxy → intro-exists (r⁺ , Nrxy) r<q)
+              ( λ Nrxy → intro-exists (r⁺ , Nrxy) (le-right-mediant-ℚ p<q))
               ( L x y p⁺ r⁺ p<r))
         ( λ p≤0 →
           inl-disjunction
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index aa970733de4..a80d2a8bf81 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -73,6 +73,7 @@ open import real-numbers.real-numbers-from-lower-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
 open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
+open import real-numbers.short-binary-maximum-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-positive-real-numbers public
 open import real-numbers.similarity-real-numbers public
diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index 70d1e41be98..fa74b6a4c7d 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -11,7 +11,6 @@ module real-numbers.addition-real-numbers where
 ```agda
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
-open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -20,10 +19,8 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.cartesian-product-types
-open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.empty-types
-open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
@@ -33,12 +30,6 @@ open import foundation.propositional-truncations
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
-open import group-theory.commutative-monoids
-open import group-theory.groups
-open import group-theory.monoids
-open import group-theory.semigroups
-
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.addition-lower-dedekind-real-numbers
@@ -46,7 +37,6 @@ open import real-numbers.addition-upper-dedekind-real-numbers
 open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
-open import real-numbers.negation-lower-upper-dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
@@ -156,48 +146,41 @@ ap-add-ℝ x=x' y=y' = ap-binary add-ℝ x=x' y=y'
 ### Addition is commutative
 
 ```agda
-module _
-  {l1 l2 : Level}
-  (x : ℝ l1) (y : ℝ l2)
-  where
+abstract opaque
+  unfolding add-ℝ
 
-  opaque
-    unfolding add-ℝ
-
-    commutative-add-ℝ : x +ℝ y ＝ y +ℝ x
-    commutative-add-ℝ =
-      eq-eq-lower-real-ℝ
-        ( x +ℝ y)
-        ( y +ℝ x)
-        ( commutative-add-lower-ℝ (lower-real-ℝ x) (lower-real-ℝ y))
+  commutative-add-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → x +ℝ y ＝ y +ℝ x
+  commutative-add-ℝ x y =
+    eq-eq-lower-real-ℝ
+      ( x +ℝ y)
+      ( y +ℝ x)
+      ( commutative-add-lower-ℝ (lower-real-ℝ x) (lower-real-ℝ y))
 ```
 
 ### Addition is associative
 
 ```agda
-module _
-  {l1 l2 l3 : Level}
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
+abstract opaque
+  unfolding add-ℝ
 
-  opaque
-    unfolding add-ℝ
-
-    associative-add-ℝ : (x +ℝ y) +ℝ z ＝ x +ℝ (y +ℝ z)
-    associative-add-ℝ =
-      eq-eq-lower-real-ℝ
-        ( (x +ℝ y) +ℝ z)
-        ( x +ℝ (y +ℝ z))
-        ( associative-add-lower-ℝ
-          ( lower-real-ℝ x)
-          ( lower-real-ℝ y)
-          ( lower-real-ℝ z))
+  associative-add-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    (x +ℝ y) +ℝ z ＝ x +ℝ (y +ℝ z)
+  associative-add-ℝ x y z =
+    eq-eq-lower-real-ℝ
+      ( (x +ℝ y) +ℝ z)
+      ( x +ℝ (y +ℝ z))
+      ( associative-add-lower-ℝ
+        ( lower-real-ℝ x)
+        ( lower-real-ℝ y)
+        ( lower-real-ℝ z))
 ```
 
 ### Unit laws for addition
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ real-ℚ
 
   left-unit-law-add-ℝ : {l : Level} → (x : ℝ l) → zero-ℝ +ℝ x ＝ x
@@ -218,7 +201,7 @@ opaque
 ### Inverse laws for addition
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ neg-ℝ
 
   right-inverse-law-add-ℝ :
@@ -283,7 +266,7 @@ module _
   (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ sim-ℝ
 
     preserves-sim-right-add-ℝ : sim-ℝ x y → sim-ℝ (x +ℝ z) (y +ℝ z)
@@ -406,7 +389,7 @@ module _
 ### The inclusion of rational numbers preserves addition
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ real-ℚ
 
   add-real-ℚ : (p q : ℚ) → real-ℚ p +ℝ real-ℚ q ＝ real-ℚ (p +ℚ q)
@@ -502,41 +485,6 @@ module _
             by sim-eq-ℝ (ap (_+ℝ neg-ℝ y) (left-unit-law-add-ℝ _)))
 ```
 
-### The Abelian group of real numbers at `lzero` under addition
-
-```agda
-semigroup-add-ℝ-lzero : Semigroup (lsuc lzero)
-semigroup-add-ℝ-lzero =
-  ( ℝ-Set lzero ,
-    add-ℝ ,
-    associative-add-ℝ)
-
-monoid-add-ℝ-lzero : Monoid (lsuc lzero)
-monoid-add-ℝ-lzero =
-  ( semigroup-add-ℝ-lzero ,
-    zero-ℝ ,
-    left-unit-law-add-ℝ ,
-    right-unit-law-add-ℝ)
-
-commutative-monoid-add-ℝ-lzero : Commutative-Monoid (lsuc lzero)
-commutative-monoid-add-ℝ-lzero =
-  ( monoid-add-ℝ-lzero ,
-    commutative-add-ℝ)
-
-group-add-ℝ-lzero : Group (lsuc lzero)
-group-add-ℝ-lzero =
-  ( ( semigroup-add-ℝ-lzero) ,
-    ( zero-ℝ , left-unit-law-add-ℝ , right-unit-law-add-ℝ) ,
-    ( neg-ℝ ,
-      eq-sim-ℝ ∘ left-inverse-law-add-ℝ ,
-      eq-sim-ℝ ∘ right-inverse-law-add-ℝ))
-
-abelian-group-add-ℝ-lzero : Ab (lsuc lzero)
-abelian-group-add-ℝ-lzero =
-  ( group-add-ℝ-lzero ,
-    commutative-add-ℝ)
-```
-
 ### If `x + y` is similar to `0`, then `y` is similar to `-x` and `x` is similar to `-y`
 
 ```agda
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 2482c2f8552..34cafb59ec7 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -11,7 +11,6 @@ module real-numbers.binary-maximum-real-numbers where
 ```agda
 open import elementary-number-theory.positive-rational-numbers
 
-open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
@@ -22,9 +21,6 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import metric-spaces.metric-space-of-short-functions-metric-spaces
-open import metric-spaces.short-functions-metric-spaces
-
 open import order-theory.join-semilattices
 open import order-theory.large-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
@@ -37,7 +33,6 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
 open import real-numbers.maximum-upper-dedekind-real-numbers
-open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
@@ -125,7 +120,7 @@ module _
   (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ max-ℝ
 
     is-least-binary-upper-bound-max-ℝ :
@@ -177,28 +172,24 @@ module _
 ### The binary maximum is commutative
 
 ```agda
-module _
-  {l1 l2 : Level}
-  (x : ℝ l1) (y : ℝ l2)
-  where
+abstract opaque
+  unfolding leq-ℝ sim-ℝ
 
-  opaque
-    unfolding leq-ℝ sim-ℝ
-
-    commutative-max-ℝ : max-ℝ x y ＝ max-ℝ y x
-    commutative-max-ℝ =
-      eq-sim-ℝ
-        ( sim-is-least-binary-upper-bound-Large-Poset
+  commutative-max-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → max-ℝ x y ＝ max-ℝ y x
+  commutative-max-ℝ x y =
+    eq-sim-ℝ
+      ( sim-is-least-binary-upper-bound-Large-Poset
+        ( ℝ-Large-Poset)
+        ( x)
+        ( y)
+        ( is-least-binary-upper-bound-max-ℝ x y)
+        ( is-binary-least-upper-bound-swap-Large-Poset
           ( ℝ-Large-Poset)
-          ( x)
           ( y)
-          ( is-least-binary-upper-bound-max-ℝ x y)
-          ( is-binary-least-upper-bound-swap-Large-Poset
-            ( ℝ-Large-Poset)
-            ( y)
-            ( x)
-            ( max-ℝ y x)
-            ( is-least-binary-upper-bound-max-ℝ y x)))
+          ( x)
+          ( max-ℝ y x)
+          ( is-least-binary-upper-bound-max-ℝ y x)))
 ```
 
 ### The large poset of real numbers has joins
@@ -306,126 +297,6 @@ abstract
   preserves-sim-right-max-ℝ a = preserves-sim-max-ℝ a a (refl-sim-ℝ a)
 ```
 
-### The binary maximum preserves lower neighborhoods
-
-```agda
-module _
-  {l1 l2 l3 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    preserves-lower-neighborhood-leq-left-max-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ
-        ( max-ℝ x y)
-        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d =
-      leq-is-least-binary-upper-bound-Large-Poset
-        ( ℝ-Large-Poset)
-        ( x)
-        ( y)
-        ( is-least-binary-upper-bound-max-ℝ x y)
-        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
-        ( ( transitive-leq-ℝ
-            ( x)
-            ( max-ℝ x z)
-            ( max-ℝ x z +ℝ real-ℚ⁺ d)
-            ( leq-le-ℝ
-              ( le-left-add-real-ℝ⁺
-                ( max-ℝ x z)
-                ( positive-real-ℚ⁺ d)))
-            ( leq-left-max-ℝ x z)) ,
-          ( transitive-leq-ℝ
-            ( y)
-            ( z +ℝ real-ℚ⁺ d)
-            ( max-ℝ x z +ℝ real-ℚ⁺ d)
-            ( preserves-leq-right-add-ℝ
-              ( real-ℚ⁺ d)
-              ( z)
-              ( max-ℝ x z)
-              ( leq-right-max-ℝ x z))
-            ( z≤y+d)))
-
-    preserves-lower-neighborhood-leq-right-max-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ
-        ( max-ℝ y x)
-        ( (max-ℝ z x) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-right-max-ℝ z≤y+d =
-      binary-tr
-        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
-        ( commutative-max-ℝ x y)
-        ( commutative-max-ℝ x z)
-        ( preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d)
-```
-
-### The maximum with a real number is a short function `ℝ → ℝ`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1)
-  where
-
-  abstract
-    is-short-function-left-max-ℝ :
-      is-short-function-Metric-Space
-        ( metric-space-ℝ l2)
-        ( metric-space-ℝ (l1 ⊔ l2))
-        ( max-ℝ x)
-    is-short-function-left-max-ℝ d y z Nyz =
-      neighborhood-real-bound-each-leq-ℝ
-        ( d)
-        ( max-ℝ x y)
-        ( max-ℝ x z)
-        ( preserves-lower-neighborhood-leq-left-max-ℝ d x y z
-          ( left-leq-real-bound-neighborhood-ℝ d y z Nyz))
-        ( preserves-lower-neighborhood-leq-left-max-ℝ d x z y
-          ( right-leq-real-bound-neighborhood-ℝ d y z Nyz))
-
-  short-left-max-ℝ :
-    short-function-Metric-Space
-      ( metric-space-ℝ l2)
-      ( metric-space-ℝ (l1 ⊔ l2))
-  short-left-max-ℝ =
-    (max-ℝ x , is-short-function-left-max-ℝ)
-```
-
-### The binary maximum is a short function from `ℝ` to the metric space of short functions `ℝ → ℝ`
-
-```agda
-module _
-  {l1 l2 : Level}
-  where
-
-  abstract
-    is-short-function-short-left-max-ℝ :
-      is-short-function-Metric-Space
-        ( metric-space-ℝ l1)
-        ( metric-space-of-short-functions-Metric-Space
-          ( metric-space-ℝ l2)
-          ( metric-space-ℝ (l1 ⊔ l2)))
-        ( short-left-max-ℝ)
-    is-short-function-short-left-max-ℝ d x y Nxy z =
-      neighborhood-real-bound-each-leq-ℝ
-        ( d)
-        ( max-ℝ x z)
-        ( max-ℝ y z)
-        ( preserves-lower-neighborhood-leq-right-max-ℝ d z x y
-          ( left-leq-real-bound-neighborhood-ℝ d x y Nxy))
-        ( preserves-lower-neighborhood-leq-right-max-ℝ d z y x
-          ( right-leq-real-bound-neighborhood-ℝ d x y Nxy))
-
-  short-max-ℝ :
-    short-function-Metric-Space
-      ( metric-space-ℝ l1)
-      ( metric-space-of-short-functions-Metric-Space
-        ( metric-space-ℝ l2)
-        ( metric-space-ℝ (l1 ⊔ l2)))
-  short-max-ℝ =
-    (short-left-max-ℝ , is-short-function-short-left-max-ℝ)
-```
-
 ### For any `ε : ℚ⁺`, `(max-ℝ x y - ε < x) ∨ (max-ℝ x y - ε < y)`
 
 ```agda
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index dd18d3c8e50..a20149edf8d 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -47,7 +47,7 @@ nonnegative.
 ## Definition
 
 ```agda
-opaque
+abstract opaque
   unfolding mul-ℝ
 
   is-nonnegative-mul-ℝ :
@@ -67,7 +67,6 @@ opaque
             d⁺ = (d , is-positive-is-in-upper-cut-ℝ⁰⁺ (y , 0≤y) y<d)
           is-positive-le-ℚ⁺
             ( b⁺ *ℚ⁺ d⁺)
-            ( q)
             ( concatenate-leq-le-ℚ
               ( b *ℚ d)
               ( upper-bound-mul-closed-interval-ℚ [a,b] [c,d])
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index ab63080b992..63176527d9b 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -53,7 +53,7 @@ module _
   {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding mul-ℝ
 
     is-positive-mul-ℝ :
@@ -67,9 +67,9 @@ module _
         (d , y<d) ← is-inhabited-upper-cut-ℝ y
         let
           a<b = le-lower-upper-cut-ℝ x a b a<x x<b
-          b⁺ = (b , is-positive-le-ℚ⁺ a⁺ b a<b)
+          b⁺ = (b , is-positive-le-ℚ⁺ a⁺ a<b)
           c<d = le-lower-upper-cut-ℝ y c d c<y y<d
-          d⁺ = (d , is-positive-le-ℚ⁺ c⁺ d c<d)
+          d⁺ = (d , is-positive-le-ℚ⁺ c⁺ c<d)
           [a,b] = ((a , b) , leq-le-ℚ a<b)
           [c,d] = ((c , d) , leq-le-ℚ c<d)
         is-positive-exists-ℚ⁺-in-lower-cut-ℝ
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index e26e0fb2186..53040c6a1a7 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import elementary-number-theory.maximum-natural-numbers
 open import elementary-number-theory.maximum-nonnegative-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minimum-positive-rational-numbers
-open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
 open import elementary-number-theory.multiplication-interior-closed-intervals-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
@@ -36,7 +35,6 @@ open import elementary-number-theory.poset-closed-intervals-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
-open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 open import elementary-number-theory.unit-fractions-rational-numbers
 
@@ -52,19 +50,16 @@ open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.similarity-subtypes
 open import foundation.subtypes
 open import foundation.transport-along-identifications
-open import foundation.univalence
 open import foundation.universe-levels
 
 open import group-theory.groups
 
 open import logic.functoriality-existential-quantification
 
-open import order-theory.large-posets
 open import order-theory.posets
 
 open import real-numbers.addition-real-numbers
@@ -77,7 +72,6 @@ open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.upper-dedekind-real-numbers
 ```
 
@@ -255,7 +249,7 @@ module _
           do-syntax-trunc-Prop (∃ ℚ (λ r → le-ℚ-Prop q r ∧ lower-cut-mul-ℝ r))
       in do
         ((a<x<b , c<y<d) , q<[a,b][c,d]) ← q∈L
-        (r , q<r , r<[a,b][c,d]) ← dense-le-ℚ q _ q<[a,b][c,d]
+        (r , q<r , r<[a,b][c,d]) ← dense-le-ℚ q<[a,b][c,d]
         intro-exists r (q<r , intro-exists (a<x<b , c<y<d) r<[a,b][c,d])
 
     forward-implication-is-rounded-upper-cut-mul-ℝ :
@@ -267,7 +261,7 @@ module _
           do-syntax-trunc-Prop (∃ ℚ (λ q → le-ℚ-Prop q r ∧ upper-cut-mul-ℝ q))
       in do
         ((a<x<b , c<y<d) , [a,b][c,d]<r) ← r∈U
-        (q , [a,b][c,d]<q , q<r) ← dense-le-ℚ _ r [a,b][c,d]<r
+        (q , [a,b][c,d]<q , q<r) ← dense-le-ℚ [a,b][c,d]<r
         intro-exists q (q<r , intro-exists (a<x<b , c<y<d) [a,b][c,d]<q)
 
     backward-implication-is-rounded-lower-cut-mul-ℝ :
@@ -616,7 +610,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding mul-ℝ
 
     enclosing-rational-range-mul-ℝ :
@@ -717,13 +711,13 @@ module _
 ### Commutativity of multiplication
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ mul-ℝ
 
-  leq-commute-ℝ :
+  leq-commute-mul-ℝ :
     {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) →
     leq-ℝ (x *ℝ y) (y *ℝ x)
-  leq-commute-ℝ x y q q<xy =
+  leq-commute-mul-ℝ x y q q<xy =
     let open do-syntax-trunc-Prop (lower-cut-mul-ℝ y x q)
     in do
       ((a<x<b@([a,b] , _ , _) , c<y<d@([c,d] , _ , _)) , q<[a,b][c,d]) ← q<xy
@@ -737,7 +731,7 @@ opaque
 abstract
   commutative-mul-ℝ : {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → x *ℝ y ＝ y *ℝ x
   commutative-mul-ℝ x y =
-    antisymmetric-leq-ℝ _ _ (leq-commute-ℝ x y) (leq-commute-ℝ y x)
+    antisymmetric-leq-ℝ _ _ (leq-commute-mul-ℝ x y) (leq-commute-mul-ℝ y x)
 ```
 
 ### Associativity of multiplication
@@ -747,7 +741,7 @@ module _
   {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ mul-ℝ
 
     leq-associative-mul-ℝ : leq-ℝ ((x *ℝ y) *ℝ z) (x *ℝ (y *ℝ z))
@@ -811,7 +805,7 @@ module _
   {l : Level} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' mul-ℝ real-ℚ
 
     leq-right-unit-law-mul-ℝ : leq-ℝ (x *ℝ one-ℝ) x
@@ -875,7 +869,7 @@ module _
 ### Distributivity of multiplication over addition
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ leq-ℝ' mul-ℝ add-ℝ
 
   leq-left-distributive-mul-add-ℝ :
@@ -1044,7 +1038,7 @@ module _
   {l : Level} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' mul-ℝ real-ℚ
 
     leq-left-zero-law-mul-ℝ : leq-ℝ (zero-ℝ *ℝ x) zero-ℝ
@@ -1109,7 +1103,7 @@ module _
 ### The inclusion of rational numbers preserves multiplication
 
 ```agda
-opaque
+abstract opaque
   unfolding mul-ℝ real-ℚ
 
   mul-real-ℚ : (p q : ℚ) → real-ℚ p *ℝ real-ℚ q ＝ real-ℚ (p *ℚ q)
@@ -1157,7 +1151,7 @@ opaque
 ### Multiplication on the real numbers preserves similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ leq-ℝ' mul-ℝ sim-ℝ
 
   leq-sim-right-mul-ℝ :
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index d17a3591165..7a3823d1019 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -188,7 +188,7 @@ module _
               ( is-rounded-lower-cut-ℝ⁺ x (rational-inv-ℚ⁺ q⁺))
               ( q⁻¹<x)
           intro-exists
-            ( r , is-positive-le-ℚ⁺ (inv-ℚ⁺ q⁺) r q⁻¹<r)
+            ( r , is-positive-le-ℚ⁺ (inv-ℚ⁺ q⁺) q⁻¹<r)
             ( r<x ,
               tr
                 ( λ s → le-ℚ s (q *ℚ r))
@@ -316,7 +316,7 @@ module _
       in do
         (q' , q<q' , q'∈L) ← ∃q'
         (r'⁺@(r' , _) , x<r' , q'r'<1) ←
-          q'∈L (is-positive-le-ℚ⁺ (q , is-pos-q) q' q<q')
+          q'∈L (is-positive-le-ℚ⁺ (q , is-pos-q) q<q')
         intro-exists
           ( r'⁺)
           ( x<r' ,
@@ -347,7 +347,7 @@ module _
         (r⁺@(r , _) , r<x , 1<qr) ← ∃r
         (r' , r<r' , r'<x) ←
           forward-implication (is-rounded-lower-cut-ℝ⁺ x r) r<x
-        let r'⁺ = (r' , is-positive-le-ℚ⁺ r⁺ r' r<r')
+        let r'⁺ = (r' , is-positive-le-ℚ⁺ r⁺ r<r')
         intro-exists
           ( rational-ℚ⁺ (inv-ℚ⁺ r'⁺))
           ( transitive-le-ℚ
@@ -379,7 +379,7 @@ module _
       in do
         (q' , q'<q , is-pos-q' , ∃r') ← ∃q'
         (r'⁺@(r' , _) , x<r' , 1<q'r') ← ∃r'
-        ( is-positive-le-ℚ⁺ (q' , is-pos-q') q q'<q ,
+        ( is-positive-le-ℚ⁺ (q' , is-pos-q') q'<q ,
           intro-exists
             ( r'⁺)
             ( x<r' ,
@@ -421,7 +421,7 @@ module _
         ( λ is-pos-p →
           let
             p⁺ = (p , is-pos-p)
-            is-pos-q = is-positive-le-ℚ⁺ p⁺ q p<q
+            is-pos-q = is-positive-le-ℚ⁺ p⁺ p<q
             q⁺ = (q , is-pos-q)
           in
             elim-disjunction
@@ -513,8 +513,7 @@ module _
         a' = max-ℚ a (rational-inv-ℚ⁺ d⁺)
         a'<x =
           is-in-lower-cut-max-ℚ (real-ℝ⁺ x) a (rational-inv-ℚ⁺ d⁺) a<x d⁻¹<x
-        is-pos-a' =
-          is-positive-leq-ℚ⁺ (inv-ℚ⁺ d⁺) a' (leq-right-max-ℚ _ _)
+        is-pos-a' = is-positive-leq-ℚ⁺ (inv-ℚ⁺ d⁺) (leq-right-max-ℚ _ _)
         a'⁺ = (a' , is-pos-a')
         c⁺ = (c , is-pos-c)
         x<c⁻¹ : is-in-upper-cut-ℝ⁺ x (rational-inv-ℚ⁺ c⁺)
@@ -635,10 +634,10 @@ module _
             (c'⁺@(c' , is-pos-c') , c'<x⁻¹) ← exists-ℚ⁺-in-lower-cut-inv-ℝ⁺ x
             let
               a'' = max-ℚ a a'
-              is-pos-a'' = is-positive-leq-ℚ⁺ a'⁺ a'' (leq-right-max-ℚ _ _)
+              is-pos-a'' = is-positive-leq-ℚ⁺ a'⁺ (leq-right-max-ℚ _ _)
               a''<x = is-in-lower-cut-max-ℚ (real-ℝ⁺ x) a a' a<x a'<x
               c'' = max-ℚ c c'
-              is-pos-c'' = is-positive-leq-ℚ⁺ c'⁺ c'' (leq-right-max-ℚ _ _)
+              is-pos-c'' = is-positive-leq-ℚ⁺ c'⁺ (leq-right-max-ℚ _ _)
               c''<x⁻¹ = is-in-lower-cut-max-ℚ (real-inv-ℝ⁺ x) c c' c<x⁻¹ c'<x⁻¹
               [a'',b] =
                 ( (a'' , b) , leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'' b a''<x x<b)
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 183ddbcd090..71573c09b72 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -9,37 +9,24 @@ module real-numbers.negation-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
-open import foundation.cartesian-product-types
-open import foundation.conjunction
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjoint-subtypes
 open import foundation.disjunction
-open import foundation.empty-types
-open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.negation
-open import foundation.propositional-truncations
-open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.negation-lower-upper-dedekind-real-numbers
-open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.rational-upper-dedekind-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.upper-dedekind-real-numbers
 ```
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index c2f3447dd1e..fcb7e126e0e 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.nonnegative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-nonnegative-rational-numbers
-open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
@@ -20,17 +18,13 @@ open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.conjunction
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
-open import foundation.empty-types
 open import foundation.existential-quantification
-open import foundation.function-types
 open import foundation.functoriality-propositional-truncation
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
@@ -45,7 +39,6 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
index 5176279b294..3dbae48f3b7 100644
--- a/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
@@ -50,7 +50,7 @@ module _
   pr1 (is-rounded-cut-lower-real-ℚ p) p<q =
     intro-exists
       ( mediant-ℚ p q)
-      ( le-left-mediant-ℚ p q p<q , le-right-mediant-ℚ p q p<q)
+      ( le-left-mediant-ℚ p<q , le-right-mediant-ℚ p<q)
   pr2 (is-rounded-cut-lower-real-ℚ p) =
     elim-exists
       ( le-ℚ-Prop p q)
diff --git a/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
index 5e897208f5d..ca0acca1dc2 100644
--- a/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
@@ -50,7 +50,7 @@ module _
   pr1 (is-rounded-cut-upper-real-ℚ p) q<p =
     intro-exists
       ( mediant-ℚ q p)
-      ( le-right-mediant-ℚ q p q<p , le-left-mediant-ℚ q p q<p)
+      ( le-right-mediant-ℚ q<p , le-left-mediant-ℚ q<p)
   pr2 (is-rounded-cut-upper-real-ℚ p) =
     elim-exists
       ( le-ℚ-Prop q p)
diff --git a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
index 53cc1f1c7d6..227fa5c7afc 100644
--- a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
@@ -94,8 +94,8 @@ module _
         (p , p<q , p∉L) ← q∈U
         intro-exists
           ( mediant-ℚ p q)
-          ( le-right-mediant-ℚ p q p<q ,
-            intro-exists p (le-left-mediant-ℚ p q p<q , p∉L))
+          ( le-right-mediant-ℚ p<q ,
+            intro-exists p (le-left-mediant-ℚ p<q , p∉L))
     pr2 (is-rounded-upper-cut-real-lower-ℝ q) ∃p =
       let open do-syntax-trunc-Prop (upper-cut-real-lower-ℝ q)
       in do
@@ -131,8 +131,8 @@ module _
       in
         map-disjunction
           ( id)
-          ( λ r∉L → intro-exists r ( le-right-mediant-ℚ p q p<q , r∉L))
-          ( L p r (le-left-mediant-ℚ p q p<q))
+          ( λ r∉L → intro-exists r ( le-right-mediant-ℚ p<q , r∉L))
+          ( L p r (le-left-mediant-ℚ p<q))
 
   real-lower-ℝ : ℝ l
   real-lower-ℝ =
diff --git a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
index 622ad1d0a47..80c6cacfa97 100644
--- a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
@@ -41,7 +41,8 @@ open import real-numbers.upper-dedekind-real-numbers
 Given an
 [upper Dedekind real number](real-numbers.upper-dedekind-real-numbers.md) `x`,
 we can convert `x` to a
-[Dedekind real number](real-numbers.dedekind-real-numbers.md) if and only if the
+[Dedekind real number](real-numbers.dedekind-real-numbers.md)
+[if and only if](foundation.logical-equivalences.md) the
 [complement](foundation.complements-subtypes.md) of the cut of `x` is
 [inhabited](foundation.inhabited-subtypes.md) and for any `p q : ℚ` with
 `p < q`, `p` is in the [complement](foundation.complements-subtypes.md) of the
@@ -89,8 +90,8 @@ module _
         (r , q<r , r∉U) ← q∈L
         intro-exists
           ( mediant-ℚ q r)
-          ( le-left-mediant-ℚ q r q<r ,
-            intro-exists r (le-right-mediant-ℚ q r q<r , r∉U))
+          ( le-left-mediant-ℚ q<r ,
+            intro-exists r (le-right-mediant-ℚ q<r , r∉U))
     pr2 (is-rounded-lower-cut-real-upper-ℝ q) ∃r =
       let open do-syntax-trunc-Prop (lower-cut-real-upper-ℝ q)
       in do
@@ -125,9 +126,9 @@ module _
         elim-disjunction
           (lower-cut-real-upper-ℝ p ∨ cut-upper-ℝ x q)
           ( λ r∉U →
-            inl-disjunction (intro-exists r (le-left-mediant-ℚ p q p<q , r∉U)))
+            inl-disjunction (intro-exists r (le-left-mediant-ℚ p<q , r∉U)))
           ( inr-disjunction)
-          ( L r q (le-right-mediant-ℚ p q p<q))
+          ( L r q (le-right-mediant-ℚ p<q))
 
   real-upper-ℝ : ℝ l
   real-upper-ℝ =
diff --git a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
new file mode 100644
index 00000000000..2ac42192c9c
--- /dev/null
+++ b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
@@ -0,0 +1,166 @@
+## The binary maximum of real numbers is a short function
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.short-binary-maximum-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.metric-space-of-short-functions-metric-spaces
+open import metric-spaces.short-functions-metric-spaces
+
+open import order-theory.least-upper-bounds-large-posets
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.binary-maximum-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+For any `a : ℝ`, the
+[binary maximum](real-numbers.binary-maximum-real-numbers.md) with `a` is a
+[short function](metric-spaces.short-functions-metric-spaces.md) `ℝ → ℝ` for the
+[standard real metric structure](real-numbers.metric-space-of-real-numbers.md).
+Moreover, the map `x ↦ max-ℝ x` is a short function from `ℝ` into the
+[metric space of short functions](metric-spaces.metric-space-of-short-functions-metric-spaces.md)
+of `ℝ`.
+
+## Proof
+
+### The binary maximum preserves lower neighborhoods
+
+```agda
+module _
+  {l1 l2 l3 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    preserves-lower-neighborhood-leq-left-max-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ
+        ( max-ℝ x y)
+        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d =
+      leq-is-least-binary-upper-bound-Large-Poset
+        ( ℝ-Large-Poset)
+        ( x)
+        ( y)
+        ( is-least-binary-upper-bound-max-ℝ x y)
+        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
+        ( ( transitive-leq-ℝ
+            ( x)
+            ( max-ℝ x z)
+            ( max-ℝ x z +ℝ real-ℚ⁺ d)
+            ( leq-le-ℝ
+              ( le-left-add-real-ℝ⁺
+                ( max-ℝ x z)
+                ( positive-real-ℚ⁺ d)))
+            ( leq-left-max-ℝ x z)) ,
+          ( transitive-leq-ℝ
+            ( y)
+            ( z +ℝ real-ℚ⁺ d)
+            ( max-ℝ x z +ℝ real-ℚ⁺ d)
+            ( preserves-leq-right-add-ℝ
+              ( real-ℚ⁺ d)
+              ( z)
+              ( max-ℝ x z)
+              ( leq-right-max-ℝ x z))
+            ( z≤y+d)))
+
+    preserves-lower-neighborhood-leq-right-max-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ
+        ( max-ℝ y x)
+        ( (max-ℝ z x) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-right-max-ℝ z≤y+d =
+      binary-tr
+        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
+        ( commutative-max-ℝ x y)
+        ( commutative-max-ℝ x z)
+        ( preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d)
+```
+
+### The maximum with a real number is a short function `ℝ → ℝ`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1)
+  where
+
+  abstract
+    is-short-function-left-max-ℝ :
+      is-short-function-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( max-ℝ x)
+    is-short-function-left-max-ℝ d y z Nyz =
+      neighborhood-real-bound-each-leq-ℝ
+        ( d)
+        ( max-ℝ x y)
+        ( max-ℝ x z)
+        ( preserves-lower-neighborhood-leq-left-max-ℝ d x y z
+          ( left-leq-real-bound-neighborhood-ℝ d y z Nyz))
+        ( preserves-lower-neighborhood-leq-left-max-ℝ d x z y
+          ( right-leq-real-bound-neighborhood-ℝ d y z Nyz))
+
+  short-left-max-ℝ :
+    short-function-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+  short-left-max-ℝ =
+    (max-ℝ x , is-short-function-left-max-ℝ)
+```
+
+### The binary maximum is a short function from `ℝ` to the metric space of short functions `ℝ → ℝ`
+
+```agda
+module _
+  {l1 l2 : Level}
+  where
+
+  abstract
+    is-short-function-short-left-max-ℝ :
+      is-short-function-Metric-Space
+        ( metric-space-ℝ l1)
+        ( metric-space-of-short-functions-Metric-Space
+          ( metric-space-ℝ l2)
+          ( metric-space-ℝ (l1 ⊔ l2)))
+        ( short-left-max-ℝ)
+    is-short-function-short-left-max-ℝ d x y Nxy z =
+      neighborhood-real-bound-each-leq-ℝ
+        ( d)
+        ( max-ℝ x z)
+        ( max-ℝ y z)
+        ( preserves-lower-neighborhood-leq-right-max-ℝ d z x y
+          ( left-leq-real-bound-neighborhood-ℝ d x y Nxy))
+        ( preserves-lower-neighborhood-leq-right-max-ℝ d z y x
+          ( right-leq-real-bound-neighborhood-ℝ d x y Nxy))
+
+  short-max-ℝ :
+    short-function-Metric-Space
+      ( metric-space-ℝ l1)
+      ( metric-space-of-short-functions-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2)))
+  short-max-ℝ =
+    (short-left-max-ℝ , is-short-function-short-left-max-ℝ)
+```
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index 154237168d1..dce82321dee 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -164,7 +164,6 @@ module _
                 is-pos-p =
                   is-positive-le-ℚ⁺
                     ( q *ℚ q , is-positive-mul-ℚ is-pos-q is-pos-q)
-                    ( p)
                     ( q²<p)
               (q'⁺@(q' , _) , q<q' , q'²<p) ←
                 rounded-below-square-le-ℚ⁺ (q , is-pos-q) (p , is-pos-p) q²<p
@@ -203,7 +202,7 @@ module _
         let
           is-nonneg-r =
             is-nonnegative-is-positive-ℚ
-              ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) r q<r)
+              ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) q<r)
         le-lower-cut-ℝ
           ( real-ℝ⁰⁺ x)
           ( q *ℚ q)
@@ -218,7 +217,7 @@ module _
       let open do-syntax-trunc-Prop (upper-cut-sqrt-ℝ⁰⁺ r)
       in do
         (q , q<r , is-pos-q , x<q²) ← ∃q
-        let is-pos-r = is-positive-le-ℚ⁺ (q , is-pos-q) r q<r
+        let is-pos-r = is-positive-le-ℚ⁺ (q , is-pos-q) q<r
         ( is-pos-r ,
           le-upper-cut-ℝ
             ( real-ℝ⁰⁺ x)
@@ -268,7 +267,7 @@ module _
         ( λ is-nonneg-p →
           map-disjunction
             ( λ p²<x _ → p²<x)
-            ( λ x<q² → (is-positive-le-ℚ⁰⁺ (p , is-nonneg-p) q p<q , x<q²))
+            ( λ x<q² → (is-positive-le-ℚ⁰⁺ (p , is-nonneg-p) p<q , x<q²))
             ( is-located-lower-upper-cut-ℝ
               ( real-ℝ⁰⁺ x)
               ( preserves-le-square-ℚ⁰⁺
@@ -338,7 +337,6 @@ module _
               is-pos-xy x y {r} lb1 lb2 =
                 is-positive-le-ℚ⁰⁺
                   ( q , is-nonneg-q)
-                  ( x *ℚ y)
                   ( concatenate-le-leq-ℚ
                     ( q)
                     ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 36ce763cb8c..1b6faca2fd5 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -119,7 +119,6 @@ opaque
               in
                 is-positive-le-ℚ⁺
                   ( a'⁻ *ℚ⁻ a'⁻)
-                  ( q)
                   ( concatenate-leq-le-ℚ
                     ( a' *ℚ a')
                     ( upper-bound-mul-closed-interval-ℚ [a',b'] [a',b'])
@@ -134,7 +133,6 @@ opaque
               in
                 is-positive-le-ℚ⁺
                   ( b'⁺ *ℚ⁺ b'⁺)
-                  ( q)
                   ( concatenate-leq-le-ℚ
                     ( b' *ℚ b')
                     ( upper-bound-mul-closed-interval-ℚ [a',b'] [a',b'])
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index d2a97facc63..b8256aa198a 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -177,10 +177,7 @@ module _
     unfolding le-ℝ real-ℚ
 
     preserves-le-real-ℚ : le-ℚ x y → le-ℝ (real-ℚ x) (real-ℚ y)
-    preserves-le-real-ℚ x<y =
-      intro-exists
-        ( mediant-ℚ x y)
-        ( le-left-mediant-ℚ x y x<y , le-right-mediant-ℚ x y x<y)
+    preserves-le-real-ℚ = dense-le-ℚ
 
     reflects-le-real-ℚ : le-ℝ (real-ℚ x) (real-ℚ y) → le-ℚ x y
     reflects-le-real-ℚ =

From 6241f8cb6c5ff0c574e58e803dbf8e8ff444af4c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:01:54 -0800
Subject: [PATCH 041/134] Progress

---
 .../metrics-of-metric-spaces.lagda.md         |   1 +
 .../absolute-value-real-numbers.lagda.md      |   2 +-
 .../addition-real-numbers.lagda.md            |   4 +-
 ...thmetically-located-dedekind-cuts.lagda.md |   6 +-
 ...ompleteness-dedekind-real-numbers.lagda.md |   8 --
 .../dedekind-real-numbers.lagda.md            |   9 +-
 .../distance-real-numbers.lagda.md            |   2 +-
 ...d-rational-intervals-real-numbers.lagda.md |  13 +-
 ...nequalities-addition-real-numbers.lagda.md |   2 +-
 ...ality-lower-dedekind-real-numbers.lagda.md |   2 -
 .../inequality-real-numbers.lagda.md          |  14 +-
 ...nuity-multiplication-real-numbers.lagda.md |   2 -
 .../metric-space-of-real-numbers.lagda.md     |   5 +-
 ...tiplication-negative-real-numbers.lagda.md |  12 +-
 ...lication-nonnegative-real-numbers.lagda.md |  14 +-
 ...tiplication-positive-real-numbers.lagda.md |  32 ++---
 .../multiplication-real-numbers.lagda.md      |  24 ++--
 ...ve-inverses-positive-real-numbers.lagda.md |  31 ++--
 ...lower-upper-dedekind-real-numbers.lagda.md |  21 ++-
 .../negation-real-numbers.lagda.md            |   4 +-
 .../positive-real-numbers.lagda.md            |   8 +-
 .../rational-real-numbers.lagda.md            |   2 +-
 .../similarity-real-numbers.lagda.md          |  14 +-
 ...re-roots-nonnegative-real-numbers.lagda.md |  34 ++---
 .../squares-real-numbers.lagda.md             |  10 +-
 ...nequalities-addition-real-numbers.lagda.md |  33 ++++-
 .../strict-inequality-real-numbers.lagda.md   | 135 +++++-------------
 27 files changed, 180 insertions(+), 264 deletions(-)

diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index 7a8cea0f775..d113ff5bac4 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -28,6 +28,7 @@ open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.saturation-inequality-nonnegative-real-numbers
 open import real-numbers.similarity-nonnegative-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index 8d3c38266f8..0ebbeb29d43 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -107,7 +107,7 @@ nonnegative-abs-ℝ x = (abs-ℝ x , is-nonnegative-abs-ℝ x)
 ### The absolute value of the negation of a real number is its absolute value
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ
 
   abs-neg-ℝ : {l : Level} → (x : ℝ l) → abs-ℝ (neg-ℝ x) ＝ abs-ℝ x
diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index fa74b6a4c7d..fef1ddfc687 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -88,8 +88,8 @@ module _
               { qx}
               { py}
               { qy}
-              ( le-lower-upper-cut-ℝ x px qx px<x x<qx)
-              ( le-lower-upper-cut-ℝ y py qy py<y y<qy)))
+              ( le-lower-upper-cut-ℝ x px<x x<qx)
+              ( le-lower-upper-cut-ℝ y py<y y<qy)))
 
     is-arithmetically-located-lower-upper-add-ℝ :
       is-arithmetically-located-lower-upper-ℝ lower-real-add-ℝ upper-real-add-ℝ
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index ffca8483f91..f70d4ca8b46 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -358,7 +358,7 @@ abstract
                 ( chain-of-inequalities
                     a
                     ≤ q
-                      by leq-lower-upper-cut-ℝ x a q a<x x<q
+                      by leq-lower-upper-cut-ℝ x a<x x<q
                     ≤ rational-abs-ℚ q
                       by leq-abs-ℚ q
                     ≤ ε +ℚ rational-abs-ℚ q
@@ -387,7 +387,7 @@ abstract
                           ( neg-ℚ b)
                           ( neg-ℚ p)
                           ( neg-leq-ℚ
-                            ( leq-lower-upper-cut-ℝ x p b p<x x<b))
+                            ( leq-lower-upper-cut-ℝ x p<x x<b))
                     ≤ rational-abs-ℚ (ε -ℚ p)
                       by leq-abs-ℚ _
                     ≤ rational-abs-ℚ ε +ℚ rational-abs-ℚ p
@@ -419,7 +419,7 @@ abstract
                 ( chain-of-inequalities
                     neg-ℚ b
                     ≤ neg-ℚ a
-                      by neg-leq-ℚ (leq-lower-upper-cut-ℝ x a b a<x x<b)
+                      by neg-leq-ℚ (leq-lower-upper-cut-ℝ x a<x x<b)
                     ≤ rational-abs-ℚ a
                       by neg-leq-abs-ℚ a
                     ≤ rational-abs-ℚ a +ℚ ε
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index 2a3e5294511..57f47ac4dda 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -207,8 +207,6 @@ module _
             ( ε⁺ , θ⁺)
             ( le-lower-cut-ℝ
               ( xε)
-              ( q +ℚ (ε +ℚ θ))
-              ( r +ℚ (ε +ℚ θ))
               ( preserves-le-left-add-ℚ (ε +ℚ θ) q r q<r)
               ( r+ε+θ<xε))
 
@@ -274,8 +272,6 @@ module _
             ( ε⁺ , θ⁺)
             ( le-upper-cut-ℝ
               ( xε)
-              ( p -ℚ (ε +ℚ θ))
-              ( q -ℚ (ε +ℚ θ))
               ( preserves-le-left-add-ℚ (neg-ℚ (ε +ℚ θ)) p q p<q)
               ( xε<p-ε-θ))
 
@@ -344,8 +340,6 @@ module _
           ( q -ℚ εu)
           ( le-lower-cut-ℝ
               ( xεu)
-              ( q -ℚ εu)
-              ( (q -ℚ εu) +ℚ θl)
               ( le-right-add-rational-ℚ⁺ (q -ℚ εu) θl⁺)
               ( q-εu+θl<xεu) ,
             tr
@@ -359,8 +353,6 @@ module _
                 ＝ q -ℚ εu by is-section-diff-ℚ θu _)
               ( le-upper-cut-ℝ
                 ( xεu)
-                ( q -ℚ (εu +ℚ θu))
-                ( (q -ℚ (εu +ℚ θu)) +ℚ θu)
                 ( le-right-add-rational-ℚ⁺ (q -ℚ (εu +ℚ θu)) θu⁺)
                 ( xεu<q-εu-θu)))
 
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index a6183b2bc47..0c1531e91fc 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -222,7 +222,7 @@ abstract
 
 ```agda
 module _
-  {l : Level} (x : ℝ l) (p q : ℚ)
+  {l : Level} (x : ℝ l) {p q : ℚ}
   where
 
   le-lower-cut-ℝ :
@@ -254,7 +254,7 @@ module _
 
 ```agda
 module _
-  {l : Level} (x : ℝ l) (p q : ℚ)
+  {l : Level} (x : ℝ l) {p q : ℚ}
   where
 
   abstract
@@ -265,10 +265,7 @@ module _
     le-lower-upper-cut-ℝ H H' =
       rec-coproduct
         ( id)
-        ( λ I →
-          ex-falso
-            ( is-disjoint-cut-ℝ x p
-                ( H , leq-upper-cut-ℝ x q p I H')))
+        ( λ I → ex-falso (is-disjoint-cut-ℝ x p (H , leq-upper-cut-ℝ x I H')))
         ( decide-le-leq-ℚ p q)
 
     leq-lower-upper-cut-ℝ :
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 1f779167f02..eff38740f50 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -149,7 +149,7 @@ abstract
           ( transpose-add-is-in-lower-cut-ℝ x q d q+d<x))
         ( swap-right-diff-leq-ℝ x y (real-ℚ d) x-y≤d))
 
-opaque
+abstract opaque
   unfolding neighborhood-ℝ
 
   neighborhood-dist-ℝ :
diff --git a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
index 1f9272b8db4..77d3ab8c6b5 100644
--- a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
@@ -8,15 +8,11 @@ module real-numbers.enclosing-closed-rational-intervals-real-numbers where
 
 ```agda
 open import elementary-number-theory.closed-intervals-rational-numbers
-open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.interior-closed-intervals-rational-numbers
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers
-open import elementary-number-theory.maximum-rational-numbers
-open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.rational-numbers
 
 open import foundation.conjunction
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.inhabited-types
@@ -24,7 +20,6 @@ open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
@@ -81,7 +76,7 @@ module _
         (p , p<x) ← is-inhabited-lower-cut-ℝ x
         (q , x<q) ← is-inhabited-upper-cut-ℝ x
         intro-exists
-          ( (p , q) , leq-lower-upper-cut-ℝ x p q p<x x<q)
+          ( (p , q) , leq-lower-upper-cut-ℝ x p<x x<q)
           ( p<x , x<q)
 ```
 
@@ -117,7 +112,7 @@ module _
         (b' , b'<b , x<b') ←
           forward-implication (is-rounded-upper-cut-ℝ x b) x<b
         intro-exists
-          ( (a' , b') , leq-lower-upper-cut-ℝ x a' b' a'<x x<b')
+          ( (a' , b') , leq-lower-upper-cut-ℝ x a'<x x<b')
           ( ( a'<x , x<b') , (a<a' , b'<b))
 ```
 
@@ -136,8 +131,8 @@ module _
       intersect-closed-interval-ℚ [a,b] [c,d]
     intersect-enclosing-closed-rational-interval-ℝ
       [a,b]@((a , b) , _) [c,d]@((c , d) , _) (a<x , x<b) (c<x , x<d) =
-      ( leq-lower-upper-cut-ℝ x a d a<x x<d ,
-        leq-lower-upper-cut-ℝ x c b c<x x<b)
+      ( leq-lower-upper-cut-ℝ x a<x x<d ,
+        leq-lower-upper-cut-ℝ x c<x x<b)
 
     is-enclosing-intersection-enclosing-closed-rational-interval-ℝ :
       ([a,b] [c,d] : closed-interval-ℚ) →
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
index aa5f3ad62ec..b5af4c0787f 100644
--- a/src/real-numbers/inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
@@ -48,7 +48,7 @@ module _
   {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ leq-ℝ
 
     preserves-leq-right-add-ℝ : leq-ℝ x y → leq-ℝ (x +ℝ z) (y +ℝ z)
diff --git a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
index 5f31c608057..7e9c14d77b2 100644
--- a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -18,8 +18,6 @@ open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.logical-equivalences
-open import foundation.negation
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.unit-type
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index d0a0cd049c4..89c0fbbb574 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -10,20 +10,15 @@ module real-numbers.inequality-real-numbers where
 
 ```agda
 open import elementary-number-theory.inequality-rational-numbers
-open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.action-on-identifications-functions
-open import foundation.binary-transport
 open import foundation.cartesian-product-types
 open import foundation.complements-subtypes
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.function-types
-open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
@@ -33,25 +28,18 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import order-theory.large-posets
 open import order-theory.large-preorders
 open import order-theory.posets
 open import order-theory.preorders
 open import order-theory.similarity-of-elements-large-posets
 
-open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-lower-dedekind-real-numbers
 open import real-numbers.inequality-upper-dedekind-real-numbers
-open import real-numbers.lower-dedekind-real-numbers
-open import real-numbers.negation-lower-upper-dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
@@ -62,7 +50,7 @@ The {{#concept "standard ordering" Disambiguation="real numbers" Agda=leq-ℝ}}
 the [real numbers](real-numbers.dedekind-real-numbers.md) is defined as the
 [lower cut](real-numbers.lower-dedekind-real-numbers.md) of one being a
 [subset](foundation-core.subtypes.md) of the lower cut of the other. I.e.,
-`x ≤ y` if `lower-cut x ⊆ lower-cut y `. This is the definition used in
+`x ≤ y` if `lower-cut-ℝ x ⊆ lower-cut-ℝ y `. This is the definition used in
 {{#cite UF13}}, section 11.2.1.
 
 Inequality of the real numbers is equivalently described by the _upper_ cut of
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index 7b3d8f8c669..ed960956fe8 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -258,8 +258,6 @@ module _
                   by
                     preserves-leq-right-mul-ℝ⁰⁺
                       ( nonnegative-real-ℚ⁺ ε)
-                      ( my +ℝ mx)
-                      ( real-ℚ q)
                       ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ (my +ℝ mx) my+mx<q))
                 ≤ real-ℚ⁺ (q⁺ *ℚ⁺ ε)
                   by leq-eq-ℝ (mul-real-ℚ q (rational-ℚ⁺ ε))))
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index 7a648355720..d199e3ef54d 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -132,10 +132,7 @@ module _
       diagonal-product
         ( (r : ℚ) →
           is-in-lower-cut-ℝ x (r +ℚ (rational-ℚ⁺ d)) → is-in-lower-cut-ℝ x r)
-        ( λ r →
-          le-lower-cut-ℝ x r
-            ( r +ℚ rational-ℚ⁺ d)
-            ( le-right-add-rational-ℚ⁺ r d))
+        ( λ r → le-lower-cut-ℝ x (le-right-add-rational-ℚ⁺ r d))
 
     is-symmetric-neighborhood-ℝ :
       is-symmetric-Rational-Neighborhood-Relation (neighborhood-prop-ℝ l)
diff --git a/src/real-numbers/multiplication-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
index e13cdd5cc28..7d2e4c8289f 100644
--- a/src/real-numbers/multiplication-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
@@ -69,14 +69,14 @@ _*ℝ⁻_ = mul-ℝ⁻
 ```agda
 abstract
   reverses-le-left-mul-ℝ⁻ :
-    {l1 l2 l3 : Level} (x : ℝ⁻ l1) (y : ℝ l2) (z : ℝ l3) → le-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁻ l1) {y : ℝ l2} {z : ℝ l3} → le-ℝ y z →
     le-ℝ (real-ℝ⁻ x *ℝ z) (real-ℝ⁻ x *ℝ y)
-  reverses-le-left-mul-ℝ⁻ x y z y<z =
+  reverses-le-left-mul-ℝ⁻ x y<z =
     binary-tr
       ( le-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-le-ℝ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-le-ℝ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y<z))
 ```
 
 ### Multiplication by a negative real number reverses inequality
@@ -84,12 +84,12 @@ abstract
 ```agda
 abstract
   reverses-leq-left-mul-ℝ⁻ :
-    {l1 l2 l3 : Level} (x : ℝ⁻ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁻ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (real-ℝ⁻ x *ℝ z) (real-ℝ⁻ x *ℝ y)
-  reverses-leq-left-mul-ℝ⁻ x y z y<z =
+  reverses-leq-left-mul-ℝ⁻ x y<z =
     binary-tr
       ( leq-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-leq-ℝ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-leq-ℝ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y<z))
 ```
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index a20149edf8d..a1bb60ef8e5 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -96,9 +96,9 @@ ap-mul-ℝ⁰⁺ = ap-binary mul-ℝ⁰⁺
 ```agda
 abstract
   preserves-leq-left-mul-ℝ⁰⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (real-ℝ⁰⁺ x *ℝ y) (real-ℝ⁰⁺ x *ℝ z)
-  preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺@(x , 0≤x) y z y≤z =
+  preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺@(x , 0≤x) {y} {z} y≤z =
     leq-is-nonnegative-diff-ℝ
       ( x *ℝ y)
       ( x *ℝ z)
@@ -110,14 +110,14 @@ abstract
           ( is-nonnegative-diff-leq-ℝ y≤z)))
 
   preserves-leq-right-mul-ℝ⁰⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (y *ℝ real-ℝ⁰⁺ x) (z *ℝ real-ℝ⁰⁺ x)
-  preserves-leq-right-mul-ℝ⁰⁺ x y z y≤z =
+  preserves-leq-right-mul-ℝ⁰⁺ x y≤z =
     binary-tr
       ( leq-ℝ)
       ( commutative-mul-ℝ _ _)
       ( commutative-mul-ℝ _ _)
-      ( preserves-leq-left-mul-ℝ⁰⁺ x y z y≤z)
+      ( preserves-leq-left-mul-ℝ⁰⁺ x y≤z)
 
   preserves-leq-mul-ℝ⁰⁺ :
     {l1 l2 l3 l4 : Level} →
@@ -125,6 +125,6 @@ abstract
     leq-ℝ⁰⁺ x x' → leq-ℝ⁰⁺ y y' → leq-ℝ⁰⁺ (x *ℝ⁰⁺ y) (x' *ℝ⁰⁺ y')
   preserves-leq-mul-ℝ⁰⁺ x x' y y' x≤x' y≤y' =
     transitive-leq-ℝ _ _ _
-      ( preserves-leq-right-mul-ℝ⁰⁺ y' (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ x') x≤x')
-      ( preserves-leq-left-mul-ℝ⁰⁺ x (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ y') y≤y')
+      ( preserves-leq-right-mul-ℝ⁰⁺ y' x≤x')
+      ( preserves-leq-left-mul-ℝ⁰⁺ x y≤y')
 ```
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index 63176527d9b..7972741f76c 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -66,9 +66,9 @@ module _
         (c⁺@(c , _) , c<y) ← exists-ℚ⁺-in-lower-cut-is-positive-ℝ y 0<y
         (d , y<d) ← is-inhabited-upper-cut-ℝ y
         let
-          a<b = le-lower-upper-cut-ℝ x a b a<x x<b
+          a<b = le-lower-upper-cut-ℝ x a<x x<b
           b⁺ = (b , is-positive-le-ℚ⁺ a⁺ a<b)
-          c<d = le-lower-upper-cut-ℝ y c d c<y y<d
+          c<d = le-lower-upper-cut-ℝ y c<y y<d
           d⁺ = (d , is-positive-le-ℚ⁺ c⁺ c<d)
           [a,b] = ((a , b) , leq-le-ℚ a<b)
           [c,d] = ((c , d) , leq-le-ℚ c<d)
@@ -110,9 +110,9 @@ abstract
 ```agda
 abstract
   preserves-le-left-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → le-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → le-ℝ y z →
     le-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z)
-  preserves-le-left-mul-ℝ⁺ x⁺@(x , 0<x) y z y<z =
+  preserves-le-left-mul-ℝ⁺ x⁺@(x , 0<x) {y} {z} y<z =
     le-is-positive-diff-ℝ
       ( tr
         ( is-positive-ℝ)
@@ -120,29 +120,29 @@ abstract
         ( is-positive-mul-ℝ 0<x (is-positive-diff-le-ℝ y<z)))
 
   reflects-le-left-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} →
     le-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z) → le-ℝ y z
-  reflects-le-left-mul-ℝ⁺ x y z xy<xz =
+  reflects-le-left-mul-ℝ⁺ x {y} {z} xy<xz =
     preserves-le-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
-      ( preserves-le-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy<xz)
+      ( preserves-le-left-mul-ℝ⁺ (inv-ℝ⁺ x) xy<xz)
 
   preserves-le-right-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → le-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → le-ℝ y z →
     le-ℝ (y *ℝ real-ℝ⁺ x) (z *ℝ real-ℝ⁺ x)
-  preserves-le-right-mul-ℝ⁺ x y z y<z =
+  preserves-le-right-mul-ℝ⁺ x y<z =
     binary-tr
       ( le-ℝ)
       ( commutative-mul-ℝ _ _)
       ( commutative-mul-ℝ _ _)
-      ( preserves-le-left-mul-ℝ⁺ x y z y<z)
+      ( preserves-le-left-mul-ℝ⁺ x y<z)
 
   reflects-le-right-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} →
     le-ℝ (y *ℝ real-ℝ⁺ x) (z *ℝ real-ℝ⁺ x) → le-ℝ y z
-  reflects-le-right-mul-ℝ⁺ x y z yx<zx =
-    reflects-le-left-mul-ℝ⁺ x y z
+  reflects-le-right-mul-ℝ⁺ x yx<zx =
+    reflects-le-left-mul-ℝ⁺ x
       ( binary-tr
         ( le-ℝ)
         ( commutative-mul-ℝ _ _)
@@ -155,12 +155,12 @@ abstract
 ```agda
 abstract
   preserves-leq-left-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z)
   preserves-leq-left-mul-ℝ⁺ x⁺ = preserves-leq-left-mul-ℝ⁰⁺ (nonnegative-ℝ⁺ x⁺)
 
   preserves-leq-right-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (y *ℝ real-ℝ⁺ x) (z *ℝ real-ℝ⁺ x)
   preserves-leq-right-mul-ℝ⁺ x⁺ =
     preserves-leq-right-mul-ℝ⁰⁺ (nonnegative-ℝ⁺ x⁺)
@@ -172,7 +172,7 @@ abstract
     preserves-leq-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
-      ( preserves-leq-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy≤xz)
+      ( preserves-leq-left-mul-ℝ⁺ (inv-ℝ⁺ x) xy≤xz)
 
   reflects-leq-right-mul-ℝ⁺ :
     {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index 53040c6a1a7..b389211c2d2 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -160,8 +160,8 @@ module _
               ( [c',d'])
               ( a<a' , b'<b)
               ( c<c' , d'<d)
-              ( le-lower-upper-cut-ℝ x a' b' a'<x x<b')
-              ( le-lower-upper-cut-ℝ y c' d' c'<y y<d')))
+              ( le-lower-upper-cut-ℝ x a'<x x<b')
+              ( le-lower-upper-cut-ℝ y c'<y y<d')))
 
     leq-upper-cut-mul-ℝ'-upper-cut-mul-ℝ : upper-cut-mul-ℝ' ⊆ upper-cut-mul-ℝ
     leq-upper-cut-mul-ℝ'-upper-cut-mul-ℝ q q∈U' =
@@ -190,8 +190,8 @@ module _
               ( [c',d'])
               ( a<a' , b'<b)
               ( c<c' , d'<d)
-              ( le-lower-upper-cut-ℝ x a' b' a'<x x<b')
-              ( le-lower-upper-cut-ℝ y c' d' c'<y y<d'))
+              ( le-lower-upper-cut-ℝ x a'<x x<b')
+              ( le-lower-upper-cut-ℝ y c'<y y<d'))
             ( [a,b][c,d]≤q))
 
     eq-lower-cut-mul-ℝ' : lower-cut-mul-ℝ ＝ lower-cut-mul-ℝ'
@@ -336,11 +336,11 @@ module _
                     ( c'')
                     ( leq-right-max-ℚ a a' ,
                       leq-max-leq-both-ℚ b' a a'
-                        ( leq-lower-upper-cut-ℝ x a b' a<x x<b')
+                        ( leq-lower-upper-cut-ℝ x a<x x<b')
                         ( a'≤b'))
                     ( leq-right-max-ℚ c c' ,
                       leq-max-leq-both-ℚ d' c c'
-                        ( leq-lower-upper-cut-ℝ y c d' c<y y<d')
+                        ( leq-lower-upper-cut-ℝ y c<y y<d')
                         ( c'≤d'))))
                 ( pr1
                   ( is-in-mul-interval-mul-is-in-closed-interval-ℚ
@@ -351,11 +351,11 @@ module _
                     ( leq-left-max-ℚ a a' ,
                       leq-max-leq-both-ℚ b a a'
                         ( a≤b)
-                        ( leq-lower-upper-cut-ℝ x a' b a'<x x<b))
+                        ( leq-lower-upper-cut-ℝ x a'<x x<b))
                     ( leq-left-max-ℚ c c' ,
                       leq-max-leq-both-ℚ d c c'
                         ( c≤d)
-                        ( leq-lower-upper-cut-ℝ y c' d c'<y y<d)))))
+                        ( leq-lower-upper-cut-ℝ y c'<y y<d)))))
               ( [a',b'][c',d']<q))
             ( q<[a,b][c,d]))
 
@@ -427,8 +427,8 @@ module _
         ((p , q) , q<p+εx , p<x , x<q) ← is-arithmetically-located-ℝ x εx
         ((r , s) , s<r+εy , r<y , y<s) ← is-arithmetically-located-ℝ y εy
         let
-          p≤q = leq-lower-upper-cut-ℝ x p q p<x x<q
-          r≤s = leq-lower-upper-cut-ℝ y r s r<y y<s
+          p≤q = leq-lower-upper-cut-ℝ x p<x x<q
+          r≤s = leq-lower-upper-cut-ℝ y r<y y<s
           q-p<εx : le-ℚ (q -ℚ p) (rational-ℚ⁺ εx)
           q-p<εx =
             le-transpose-right-add-ℚ _ _ _
@@ -815,7 +815,7 @@ module _
         ( ( ax<x<bx@([ax,bx]@((ax , bx) , _) , ax<x , x<bx) ,
             a₁<1<b₁@([a₁,b₁]@((a₁ , b₁) , _) , a₁<1 , 1<b₁)) ,
           q<[ax,bx][a₁,b₁]) ← q<x1
-        le-lower-cut-ℝ x q ax
+        le-lower-cut-ℝ x
           ( concatenate-le-leq-ℚ _ _ _
             ( q<[ax,bx][a₁,b₁])
             ( tr
@@ -840,7 +840,7 @@ module _
             ( ( ax<x<bx@([ax,bx]@((ax , bx) , _) , ax<x , x<bx) ,
                 a₁<1<b₁@([a₁,b₁]@((a₁ , b₁) , _) , a₁<1 , 1<b₁)) ,
               [ax,bx][a₁,b₁]<q) ← x1<q
-            le-upper-cut-ℝ x bx q
+            le-upper-cut-ℝ x
               ( concatenate-leq-le-ℚ _ _ _
                 ( tr
                   ( λ p →
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 7a3823d1019..490dd190442 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -117,8 +117,6 @@ module _
         (r⁺@(r , _) , x<r , qr<1) ← q∈L is-pos-q
         le-upper-cut-ℝ
           ( real-ℝ⁺ x)
-          ( r)
-          ( rational-inv-ℚ⁺ q⁺)
           ( reflects-le-left-mul-ℚ⁺
             ( q⁺)
             ( r)
@@ -140,8 +138,6 @@ module _
           (r⁺@(r , _) , r<x , 1<qr) ← ∃r
           le-lower-cut-ℝ
             ( real-ℝ⁺ x)
-            ( rational-inv-ℚ⁺ q⁺)
-            ( r)
             ( reflects-le-left-mul-ℚ⁺
               ( q⁺)
               ( rational-inv-ℚ⁺ q⁺)
@@ -166,7 +162,7 @@ module _
             ( is-rounded-upper-cut-ℝ⁺ x (rational-inv-ℚ⁺ q⁺))
             ( q∈L' is-pos-q)
         intro-exists
-          ( r , is-positive-is-in-upper-cut-ℝ⁺ x r x<r)
+          ( r , is-positive-is-in-upper-cut-ℝ⁺ x x<r)
           ( x<r ,
             tr
               ( le-ℚ (q *ℚ r))
@@ -216,7 +212,7 @@ module _
       let open do-syntax-trunc-Prop (∃ ℚ⁺ (lower-cut-inv-ℝ⁺ ∘ rational-ℚ⁺))
       in do
         (q , x<q) ← is-inhabited-upper-cut-ℝ⁺ x
-        let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x q x<q)
+        let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x x<q)
         intro-exists
           ( inv-ℚ⁺ q⁺)
           ( leq-lower-cut-inv-ℝ⁺'-lower-cut-inv-ℝ⁺
@@ -274,7 +270,7 @@ module _
                 forward-implication (is-rounded-upper-cut-ℝ⁺ x r) x<r
               let
                 r⁻¹ = inv-ℚ⁺ r⁺
-                r'⁺ = (r' , is-positive-is-in-upper-cut-ℝ⁺ x r' x<r')
+                r'⁺ = (r' , is-positive-is-in-upper-cut-ℝ⁺ x x<r')
                 r'⁻¹ = inv-ℚ⁺ r'⁺
               intro-exists
                 ( rational-ℚ⁺ r'⁻¹)
@@ -411,7 +407,7 @@ module _
             ( _)
             ( _)
             ( transitive-le-ℚ (q *ℚ rₗ) one-ℚ (q *ℚ rᵤ) 1<qrᵤ qrₗ<1))
-          ( le-lower-upper-cut-ℝ⁺ x rᵤ rₗ rᵤ<x x<rₗ)
+          ( le-lower-upper-cut-ℝ⁺ x rᵤ<x x<rₗ)
 
     is-located-lower-upper-cut-inv-ℝ⁺ :
       (p q : ℚ) → le-ℚ p q →
@@ -506,7 +502,7 @@ module _
       [a,b]@((a , b) , a≤b) [c,d]@((c , d) , c≤d) (a<x , x<b) (c<x⁻¹ , x⁻¹<d)
       is-pos-a is-pos-c =
       let
-        is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x b x<b
+        is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x x<b
         (is-pos-d , d⁻¹<x) =
           leq-upper-cut-inv-ℝ⁺-upper-cut-inv-ℝ⁺' x d x⁻¹<d
         d⁺ = (d , is-pos-d)
@@ -531,12 +527,7 @@ module _
             ( inv-leq-ℚ⁺
               ( a'⁺)
               ( inv-ℚ⁺ c⁺)
-              ( leq-lower-upper-cut-ℝ
-                ( real-ℝ⁺ x)
-                ( a')
-                ( rational-inv-ℚ⁺ c⁺)
-                ( a'<x)
-                ( x<c⁻¹)))
+              ( leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'<x x<c⁻¹))
       in
         tr
           ( is-in-closed-interval-ℚ
@@ -548,7 +539,7 @@ module _
             ( a')
             ( rational-inv-ℚ⁺ a'⁺)
             ( leq-left-max-ℚ _ _ ,
-              leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a' b a'<x x<b)
+              leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'<x x<b)
             ( c≤a'⁻¹ , a'⁻¹≤d))
 
     leq-real-right-inverse-law-mul-ℝ⁺ :
@@ -560,7 +551,7 @@ module _
             ([c,d]@((c , d) , c≤d) , c<x⁻¹ , x⁻¹<d)) ,
           q<[a,b][c,d]) ← q<xx⁻¹
         let
-          is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x b x<b
+          is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x x<b
           (is-pos-d , d⁻¹<x) =
             leq-upper-cut-inv-ℝ⁺-upper-cut-inv-ℝ⁺' x d x⁻¹<d
           case-is-nonpos-a is-nonpos-a =
@@ -640,10 +631,10 @@ module _
               is-pos-c'' = is-positive-leq-ℚ⁺ c'⁺ (leq-right-max-ℚ _ _)
               c''<x⁻¹ = is-in-lower-cut-max-ℚ (real-inv-ℝ⁺ x) c c' c<x⁻¹ c'<x⁻¹
               [a'',b] =
-                ( (a'' , b) , leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'' b a''<x x<b)
+                ( (a'' , b) , leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a''<x x<b)
               [c'',d] =
                 ( (c'' , d) ,
-                  leq-lower-upper-cut-ℝ (real-inv-ℝ⁺ x) c'' d c''<x⁻¹ x⁻¹<d)
+                leq-lower-upper-cut-ℝ (real-inv-ℝ⁺ x) c''<x⁻¹ x⁻¹<d)
             concatenate-leq-le-ℚ
               ( one-ℚ)
               ( upper-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -718,7 +709,7 @@ opaque
     let open do-syntax-trunc-Prop (le-prop-ℝ⁺ (inv-ℝ⁺ y) (inv-ℝ⁺ x))
     in do
       (q , x<q , q<y) ← x<y
-      let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x q x<q)
+      let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x x<q)
       intro-exists
         ( rational-inv-ℚ⁺ q⁺)
         ( leq-upper-cut-inv-ℝ⁺'-upper-cut-inv-ℝ⁺
diff --git a/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
index 0bf3ac519c9..ab010a21b38 100644
--- a/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
@@ -19,7 +19,6 @@ open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.retractions
 open import foundation.sections
 open import foundation.similarity-subtypes
@@ -29,8 +28,6 @@ open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
-open import real-numbers.inequality-lower-dedekind-real-numbers
-open import real-numbers.inequality-upper-dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
@@ -177,16 +174,18 @@ abstract
           (λ p → tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)))))
 
   is-equiv-neg-lower-ℝ : (l : Level) → is-equiv (neg-lower-ℝ {l})
-  pr1 (pr1 (is-equiv-neg-lower-ℝ l)) = neg-upper-ℝ
-  pr1 (pr2 (is-equiv-neg-lower-ℝ l)) = neg-upper-ℝ
-  pr2 (pr1 (is-equiv-neg-lower-ℝ l)) = is-section-neg-upper-ℝ l
-  pr2 (pr2 (is-equiv-neg-lower-ℝ l)) = is-retraction-neg-upper-ℝ l
+  is-equiv-neg-lower-ℝ l =
+    is-equiv-is-invertible
+      ( neg-upper-ℝ)
+      ( is-section-neg-upper-ℝ l)
+      ( is-retraction-neg-upper-ℝ l)
 
   is-equiv-neg-upper-ℝ : (l : Level) → is-equiv (neg-upper-ℝ {l})
-  pr1 (pr1 (is-equiv-neg-upper-ℝ l)) = neg-lower-ℝ
-  pr1 (pr2 (is-equiv-neg-upper-ℝ l)) = neg-lower-ℝ
-  pr2 (pr1 (is-equiv-neg-upper-ℝ l)) = is-retraction-neg-upper-ℝ l
-  pr2 (pr2 (is-equiv-neg-upper-ℝ l)) = is-section-neg-upper-ℝ l
+  is-equiv-neg-upper-ℝ l =
+    is-equiv-is-invertible
+      ( neg-lower-ℝ)
+      ( is-retraction-neg-upper-ℝ l)
+      ( is-section-neg-upper-ℝ l)
 ```
 
 ### The negation of a rational projected to a lower real is the projection of its negation as an upper real
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 71573c09b72..514703a9e37 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -87,7 +87,7 @@ module _
 ### The negation function on real numbers is an involution
 
 ```agda
-opaque
+abstract opaque
   unfolding neg-ℝ
 
   neg-neg-ℝ : {l : Level} → (x : ℝ l) → neg-ℝ (neg-ℝ x) ＝ x
@@ -130,7 +130,7 @@ abstract
 ### Negation preserves similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding neg-ℝ
 
   preserves-sim-neg-ℝ :
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 95cc41a6621..dcad10a6766 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -111,7 +111,7 @@ is-rounded-upper-cut-ℝ⁺ :
 is-rounded-upper-cut-ℝ⁺ (x , _) = is-rounded-upper-cut-ℝ x
 
 le-lower-upper-cut-ℝ⁺ :
-  {l : Level} (x : ℝ⁺ l) (p q : ℚ) →
+  {l : Level} (x : ℝ⁺ l) {p q : ℚ} →
   is-in-lower-cut-ℝ⁺ x p → is-in-upper-cut-ℝ⁺ x q → le-ℚ p q
 le-lower-upper-cut-ℝ⁺ (x , _) = le-lower-upper-cut-ℝ x
 ```
@@ -277,8 +277,8 @@ one-ℝ⁺ = positive-real-ℚ⁺ one-ℚ⁺
 ```agda
 abstract
   is-positive-is-in-upper-cut-ℝ⁺ :
-    {l : Level} (x : ℝ⁺ l) (q : ℚ) → is-in-upper-cut-ℝ⁺ x q → is-positive-ℚ q
-  is-positive-is-in-upper-cut-ℝ⁺ x⁺@(x , _) q x<q =
+    {l : Level} (x : ℝ⁺ l) {q : ℚ} → is-in-upper-cut-ℝ⁺ x q → is-positive-ℚ q
+  is-positive-is-in-upper-cut-ℝ⁺ x⁺@(x , _) x<q =
     is-positive-le-zero-ℚ
-      ( le-lower-upper-cut-ℝ x zero-ℚ q (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
+      ( le-lower-upper-cut-ℝ x (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
 ```
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index c6398376914..df01919a624 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -232,7 +232,7 @@ opaque
       ( q)
       ( id)
       ( λ p=q → ex-falso (q∉lx (tr (is-in-lower-cut-ℝ x) p=q p∈lx)))
-      ( λ q<p → ex-falso (q∉lx (le-lower-cut-ℝ x q p q<p p∈lx)))
+      ( λ q<p → ex-falso (q∉lx (le-lower-cut-ℝ x q<p p∈lx)))
   pr2 (sim-rational-ℝ (x , q , q∉lx , q∉ux)) p p<q =
     elim-disjunction
       ( lower-cut-ℝ x p)
diff --git a/src/real-numbers/similarity-real-numbers.lagda.md b/src/real-numbers/similarity-real-numbers.lagda.md
index 3f8a1ef83c4..5664b3a49bc 100644
--- a/src/real-numbers/similarity-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-real-numbers.lagda.md
@@ -70,7 +70,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding sim-ℝ
 
     sim-lower-cut-iff-sim-ℝ :
@@ -85,7 +85,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding sim-ℝ
 
     sim-sim-upper-cut-ℝ : sim-subtype (upper-cut-ℝ x) (upper-cut-ℝ y) → (x ~ℝ y)
@@ -102,7 +102,7 @@ module _
 ### Reflexivity
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   refl-sim-ℝ : {l : Level} → (x : ℝ l) → x ~ℝ x
@@ -115,7 +115,7 @@ opaque
 ### Symmetry
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   symmetric-sim-ℝ :
@@ -127,7 +127,7 @@ opaque
 ### Transitivity
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   transitive-sim-ℝ :
@@ -141,7 +141,7 @@ opaque
 ### Similar real numbers in the same universe are equal
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   eq-sim-ℝ : {l : Level} → {x y : ℝ l} → x ~ℝ y → x ＝ y
@@ -185,7 +185,7 @@ similarity-reasoning-ℝ
 infixl 1 similarity-reasoning-ℝ_
 infixl 0 step-similarity-reasoning-ℝ
 
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   similarity-reasoning-ℝ_ :
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index dce82321dee..d73dc98a0f4 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -105,7 +105,7 @@ module _
         (p⁺@(p , is-pos-p) , q<p²) ← square-le-ℚ⁺' q⁺
         intro-exists
           ( p)
-          ( is-pos-p , le-upper-cut-ℝ (real-ℝ⁰⁺ x) q (p *ℚ p) q<p² x<q)
+          ( is-pos-p , le-upper-cut-ℝ (real-ℝ⁰⁺ x) q<p² x<q)
 
     forward-implication-is-rounded-lower-cut-sqrt-ℝ⁰⁺ :
       (q : ℚ) → is-in-subtype lower-cut-sqrt-ℝ⁰⁺ q →
@@ -152,8 +152,7 @@ module _
                     ( λ r → le-ℚ r p)
                     ( q=0)
                     ( le-zero-is-positive-ℚ is-pos-p) ,
-                  λ _ →
-                    le-lower-cut-ℝ (real-ℝ⁰⁺ x) (p *ℚ p) q' p²<q' q'<x))
+                  λ _ → le-lower-cut-ℝ (real-ℝ⁰⁺ x) p²<q' q'<x))
           ( λ is-pos-q →
             do
               (p , q²<p , p<x) ←
@@ -169,8 +168,7 @@ module _
                 rounded-below-square-le-ℚ⁺ (q , is-pos-q) (p , is-pos-p) q²<p
               intro-exists
                 ( q')
-                ( q<q' ,
-                  λ _ → le-lower-cut-ℝ (real-ℝ⁰⁺ x) (q' *ℚ q') p q'²<p p<x))
+                ( q<q' , λ _ → le-lower-cut-ℝ (real-ℝ⁰⁺ x) q'²<p p<x))
 
     forward-implication-is-rounded-upper-cut-sqrt-ℝ⁰⁺ :
       (r : ℚ) → is-in-subtype upper-cut-sqrt-ℝ⁰⁺ r →
@@ -189,7 +187,7 @@ module _
           rounded-above-square-le-ℚ⁺ (r , is-pos-r) (p , is-pos-p) p<r²
         intro-exists
           ( q)
-          ( q<r , is-pos-q , le-upper-cut-ℝ (real-ℝ⁰⁺ x) p (q *ℚ q) p<q² x<p)
+          ( q<r , is-pos-q , le-upper-cut-ℝ (real-ℝ⁰⁺ x) p<q² x<p)
 
     backward-implication-is-rounded-lower-cut-sqrt-ℝ⁰⁺ :
       (q : ℚ) → exists ℚ (λ r → le-ℚ-Prop q r ∧ lower-cut-sqrt-ℝ⁰⁺ r) →
@@ -205,8 +203,6 @@ module _
               ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) q<r)
         le-lower-cut-ℝ
           ( real-ℝ⁰⁺ x)
-          ( q *ℚ q)
-          ( r *ℚ r)
           ( preserves-le-square-ℚ⁰⁺ (q , is-nonneg-q) (r , is-nonneg-r) q<r)
           ( r<√x is-nonneg-r)
 
@@ -221,8 +217,6 @@ module _
         ( is-pos-r ,
           le-upper-cut-ℝ
             ( real-ℝ⁰⁺ x)
-            ( q *ℚ q)
-            ( r *ℚ r)
             ( preserves-le-square-ℚ⁰⁺
               ( q , is-nonnegative-is-positive-ℚ is-pos-q)
               ( r , is-nonnegative-is-positive-ℚ is-pos-r)
@@ -298,7 +292,7 @@ module _
   {l : Level} (x : ℝ⁰⁺ l)
   where
 
-  opaque
+  abstract opaque
     unfolding real-ℚ real-sqrt-ℝ⁰⁺
 
     is-nonnegative-real-sqrt-ℝ⁰⁺ : is-nonnegative-ℝ (real-sqrt-ℝ⁰⁺ x)
@@ -316,7 +310,7 @@ module _
   {l : Level} (x : ℝ⁰⁺ l)
   where
 
-  opaque
+  abstract opaque
     unfolding mul-ℝ real-sqrt-ℝ⁰⁺ leq-ℝ leq-ℝ'
 
     leq-square-sqrt-ℝ⁰⁺ :
@@ -390,8 +384,6 @@ module _
                   ( linear-leq-ℚ a c)
             le-lower-cut-ℝ
               ( real-ℝ⁰⁺ x)
-              ( q)
-              ( a' *ℚ a')
               ( concatenate-le-leq-ℚ
                 ( q)
                 ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -406,15 +398,11 @@ module _
                     ( leq-left-max-ℚ a c ,
                       leq-lower-upper-cut-ℝ
                         ( real-sqrt-ℝ⁰⁺ x)
-                        ( a')
-                        ( b)
                         ( λ _ → a'²<x)
                         ( √x<b))
                     ( leq-right-max-ℚ a c ,
                       leq-lower-upper-cut-ℝ
                         ( real-sqrt-ℝ⁰⁺ x)
-                        ( a')
-                        ( d)
                         ( λ _ → a'²<x)
                         ( √x<d)))))
               ( a'²<x))
@@ -459,8 +447,6 @@ module _
           √x<b' = (is-pos-b' , x<b'²)
         le-upper-cut-ℝ
           ( real-ℝ⁰⁺ x)
-          ( b' *ℚ b')
-          ( q)
           ( concatenate-leq-le-ℚ
             ( b' *ℚ b')
             ( upper-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -471,9 +457,9 @@ module _
                 ( [c,d])
                 ( b')
                 ( b')
-                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) a b' a<√x √x<b' ,
+                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) a<√x √x<b' ,
                   leq-left-min-ℚ b d)
-                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) c b' c<√x √x<b' ,
+                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) c<√x √x<b' ,
                   leq-right-min-ℚ b d)))
             ( [a,b][c,d]<q))
           ( x<b'²)
@@ -491,7 +477,7 @@ module _
 ### Square roots of nonnegative real numbers are unique
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ leq-ℝ' mul-ℝ sim-ℝ real-sqrt-ℝ⁰⁺
 
   leq-unique-sqrt-ℝ⁰⁺ :
@@ -504,7 +490,7 @@ opaque
     in do
       (r , y<r) ← is-inhabited-upper-cut-ℝ y
       let
-        q≤r = leq-lower-upper-cut-ℝ y q r q<y y<r
+        q≤r = leq-lower-upper-cut-ℝ y q<y y<r
         [q,r] = ((q , r) , q≤r)
         q⁰⁺ = (q , is-nonneg-q)
         r⁰⁺ = (r , is-nonnegative-leq-ℚ⁰⁺ q⁰⁺ r q≤r)
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 1b6faca2fd5..773a8b90668 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -70,7 +70,7 @@ square-ℝ x = x *ℝ x
 ### The square of a real number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding mul-ℝ
 
   is-nonnegative-square-ℝ :
@@ -142,7 +142,7 @@ opaque
                       ( leq-right-max-ℚ _ _))
                     ( [a',b'][a',b']<q)))
             ( located-le-ℚ zero-ℚ a' b'
-              ( le-lower-upper-cut-ℝ x a' b' a'<x x<b')))
+              ( le-lower-upper-cut-ℝ x a'<x x<b')))
 
 nonnegative-square-ℝ : {l : Level} → ℝ l → ℝ⁰⁺ l
 nonnegative-square-ℝ x = (square-ℝ x , is-nonnegative-square-ℝ x)
@@ -191,8 +191,8 @@ abstract
       ( square-ℝ x)
       ( x *ℝ y)
       ( square-ℝ y)
-      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ x y (leq-le-ℝ x<y))
-      ( preserves-le-right-mul-ℝ⁺ (y , is-positive-le-ℝ⁰⁺ x⁰⁺ y x<y) x y x<y)
+      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ (leq-le-ℝ x<y))
+      ( preserves-le-right-mul-ℝ⁺ (y , is-positive-le-ℝ⁰⁺ x⁰⁺ y x<y) x<y)
 ```
 
 ### If a nonnegative rational `q` is in the lower cut of `x`, `q²` is in the lower cut of `x²`
@@ -250,7 +250,7 @@ abstract opaque
     is-in-upper-cut-ℝ (square-ℝ x) (square-ℚ q)
   is-in-upper-cut-square-pos-neg-ℝ x q x<q -q<x =
     let
-      [-q,q] = ((neg-ℚ q , q) , leq-lower-upper-cut-ℝ x (neg-ℚ q) q -q<x x<q)
+      [-q,q] = ((neg-ℚ q , q) , leq-lower-upper-cut-ℝ x -q<x x<q)
     in
       leq-upper-cut-mul-ℝ'-upper-cut-mul-ℝ x x (square-ℚ q)
         ( intro-exists
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index e28ec7bda1b..6db92caf63a 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -18,6 +18,7 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
+open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.empty-types
@@ -25,6 +26,7 @@ open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
+open import foundation.sets
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
@@ -93,7 +95,7 @@ module _
             intro-exists
               ( q , (p -ℚ q) +ℚ s)
               ( q<y ,
-                le-lower-cut-ℝ z ((p -ℚ q) +ℚ s) r p-q+s<r r<z ,
+                le-lower-cut-ℝ z p-q+s<r r<z ,
                 ( equational-reasoning
                     p +ℚ s
                     ＝ (q +ℚ (p -ℚ q)) +ℚ s
@@ -283,3 +285,32 @@ module _
           ( q , is-positive-le-zero-ℚ (reflects-le-real-ℚ 0<q))
           ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
 ```
+
+### If `x + y < p` for some rational `p`, then there exist `q r : ℚ` such that `p = q + r`, `x < q`, `y < r`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) (p : ℚ)
+  where
+
+  opaque
+    unfolding add-ℝ
+
+    le-split-add-rational-ℝ :
+      le-ℝ (x +ℝ y) (real-ℚ p) →
+      exists
+        ( ℚ × ℚ)
+        ( λ (q , r) →
+          Id-Prop ℚ-Set p (q +ℚ r) ∧
+          le-prop-ℝ x (real-ℚ q) ∧
+          le-prop-ℝ y (real-ℚ r))
+    le-split-add-rational-ℝ x+y<p =
+      let open do-syntax-trunc-Prop (∃ _ _)
+      in do
+        ((q , r) , x<q , y<r , p=q+r) ← is-in-upper-cut-le-real-ℚ (x +ℝ y) x+y<p
+        intro-exists
+          ( q , r)
+          ( p=q+r ,
+            le-real-is-in-upper-cut-ℚ x x<q ,
+            le-real-is-in-upper-cut-ℚ y y<r)
+```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index b8256aa198a..96296b0ca74 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -9,16 +9,9 @@ module real-numbers.strict-inequality-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-rational-numbers
-open import elementary-number-theory.additive-group-of-rational-numbers
-open import elementary-number-theory.difference-rational-numbers
-open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.action-on-identifications-functions
-open import foundation.binary-transport
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
@@ -28,26 +21,19 @@ open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-disjunction
-open import foundation.identity-types
 open import foundation.large-binary-relations
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.sets
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
-
 open import logic.functoriality-existential-quantification
 
-open import real-numbers.addition-real-numbers
-open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
@@ -81,34 +67,28 @@ le-prop-ℝ x y = (le-ℝ x y , is-prop-le-ℝ x y)
 ### Strict inequality on the reals implies inequality
 
 ```agda
-opaque
+abstract opaque
   unfolding le-ℝ leq-ℝ
 
   leq-le-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → leq-ℝ x y
   leq-le-ℝ {x = x} {y = y} x<y p p<x =
     elim-exists
       ( lower-cut-ℝ y p)
-      ( λ q (x<q , q<y) →
-        le-lower-cut-ℝ y p q (le-lower-upper-cut-ℝ x p q p<x x<q) q<y)
+      ( λ q (x<q , q<y) → le-lower-cut-ℝ y (le-lower-upper-cut-ℝ x p<x x<q) q<y)
       ( x<y)
 ```
 
 ### Strict inequality on the reals is irreflexive
 
 ```agda
-module _
-  {l : Level}
-  (x : ℝ l)
-  where
-
-  opaque
-    unfolding le-ℝ
+abstract opaque
+  unfolding le-ℝ
 
-    irreflexive-le-ℝ : ¬ (le-ℝ x x)
-    irreflexive-le-ℝ =
-      elim-exists
-        ( empty-Prop)
-        ( λ q (x<q , q<x) → is-disjoint-cut-ℝ x q (q<x , x<q))
+  irreflexive-le-ℝ : {l : Level} (x : ℝ l) → ¬ (le-ℝ x x)
+  irreflexive-le-ℝ x =
+    elim-exists
+      ( empty-Prop)
+      ( λ q (x<q , q<x) → is-disjoint-cut-ℝ x q (q<x , x<q))
 ```
 
 ### Strict inequality on the reals is asymmetric
@@ -118,7 +98,7 @@ module _
   {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ
 
     asymmetric-le-ℝ : le-ℝ x y → ¬ (le-ℝ y x)
@@ -132,11 +112,11 @@ module _
           ( asymmetric-le-ℚ
             ( q)
             ( p)
-            ( le-lower-upper-cut-ℝ x q p q<x x<p))
+            ( le-lower-upper-cut-ℝ x q<x x<p))
           ( not-leq-le-ℚ
             ( p)
             ( q)
-            ( le-lower-upper-cut-ℝ y p q p<y y<q))
+            ( le-lower-upper-cut-ℝ y p<y y<q))
           ( decide-le-leq-ℚ p q)
 ```
 
@@ -150,7 +130,7 @@ module _
   (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ
 
     transitive-le-ℝ : le-ℝ y z → le-ℝ x y → le-ℝ x z
@@ -162,8 +142,7 @@ module _
         ( q , y<q , q<z) ← y<z
         intro-exists
           ( p)
-          ( x<p ,
-            le-lower-cut-ℝ z p q (le-lower-upper-cut-ℝ y p q p<y y<q) q<z)
+          ( x<p , le-lower-cut-ℝ z (le-lower-upper-cut-ℝ y p<y y<q) q<z)
 ```
 
 ### The canonical map from rationals to reals preserves and reflects strict inequality
@@ -173,7 +152,7 @@ module _
   {x y : ℚ}
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     preserves-le-real-ℚ : le-ℚ x y → le-ℝ (real-ℚ x) (real-ℚ y)
@@ -200,7 +179,7 @@ module _
   (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ leq-ℝ leq-ℝ'
 
     concatenate-le-leq-ℝ : le-ℝ x y → leq-ℝ y z → le-ℝ x z
@@ -220,13 +199,12 @@ module _
   {l : Level} {q : ℚ} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     le-real-iff-lower-cut-ℚ : is-in-lower-cut-ℝ x q ↔ le-ℝ (real-ℚ q) x
     le-real-iff-lower-cut-ℚ = is-rounded-lower-cut-ℝ x q
 
-  abstract
     le-real-is-in-lower-cut-ℚ : is-in-lower-cut-ℝ x q → le-ℝ (real-ℚ q) x
     le-real-is-in-lower-cut-ℚ = forward-implication le-real-iff-lower-cut-ℚ
 
@@ -241,7 +219,7 @@ module _
   {l : Level} {q : ℚ} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     le-iff-upper-cut-real-ℚ : is-in-upper-cut-ℝ x q ↔ le-ℝ x (real-ℚ q)
@@ -249,7 +227,6 @@ module _
       iff-tot-exists (λ _ → iff-equiv commutative-product) ∘iff
       is-rounded-upper-cut-ℝ x q
 
-  abstract
     le-real-is-in-upper-cut-ℚ : is-in-upper-cut-ℝ x q → le-ℝ x (real-ℚ q)
     le-real-is-in-upper-cut-ℚ = forward-implication le-iff-upper-cut-real-ℚ
 
@@ -264,12 +241,13 @@ module _
   {l : Level} (x : ℝ l) (p q : ℚ) (p<q : le-ℚ p q)
   where
 
-  is-located-le-ℝ : disjunction-type (le-ℝ (real-ℚ p) x) (le-ℝ x (real-ℚ q))
-  is-located-le-ℝ =
-    map-disjunction
-      ( le-real-is-in-lower-cut-ℚ x)
-      ( le-real-is-in-upper-cut-ℚ x)
-      ( is-located-lower-upper-cut-ℝ x p<q)
+  abstract
+    is-located-le-ℝ : disjunction-type (le-ℝ (real-ℚ p) x) (le-ℝ x (real-ℚ q))
+    is-located-le-ℝ =
+      map-disjunction
+        ( le-real-is-in-lower-cut-ℚ x)
+        ( le-real-is-in-upper-cut-ℚ x)
+        ( is-located-lower-upper-cut-ℝ x p<q)
 ```
 
 ### Every real is less than a rational number
@@ -321,7 +299,7 @@ module _
   {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ neg-ℝ
 
     neg-le-ℝ : le-ℝ x y → le-ℝ (neg-ℝ y) (neg-ℝ x)
@@ -343,7 +321,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ leq-ℝ
 
     not-leq-le-ℝ : le-ℝ x y → ¬ (leq-ℝ y x)
@@ -361,7 +339,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ leq-ℝ
 
     leq-not-le-ℝ : ¬ (le-ℝ x y) → leq-ℝ y x
@@ -410,24 +388,18 @@ module _
   (y : ℝ l2)
   where
 
-  opaque
-    unfolding le-ℝ real-ℚ
+  abstract opaque
+    unfolding le-ℝ
 
     dense-rational-le-ℝ :
       le-ℝ x y →
       exists ℚ (λ z → le-prop-ℝ x (real-ℚ z) ∧ le-prop-ℝ (real-ℚ z) y)
-    dense-rational-le-ℝ x<y =
-      let
-        open
-          do-syntax-trunc-Prop
-            ( ∃ ℚ (λ z → le-prop-ℝ x (real-ℚ z) ∧ le-prop-ℝ (real-ℚ z) y))
-      in do
-        ( q , x<q , q<y) ← x<y
-        ( p , p<q , x<p) ← forward-implication (is-rounded-upper-cut-ℝ x q) x<q
-        ( r , q<r , r<y) ← forward-implication (is-rounded-lower-cut-ℝ y q) q<y
-        intro-exists
-          ( q)
-          ( intro-exists p (x<p , p<q) , intro-exists r (q<r , r<y))
+    dense-rational-le-ℝ =
+      map-tot-exists
+        ( λ q →
+          map-product
+            ( le-real-is-in-upper-cut-ℚ x)
+            ( le-real-is-in-lower-cut-ℚ y))
 ```
 
 ### Strict inequality on the real numbers is dense
@@ -453,7 +425,7 @@ module _
 ### Strict inequality on the real numbers is cotransitive
 
 ```agda
-opaque
+abstract opaque
   unfolding le-ℝ
 
   cotransitive-le-ℝ : is-cotransitive-Large-Relation-Prop ℝ le-prop-ℝ
@@ -476,7 +448,7 @@ module _
   {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) (x~y : sim-ℝ x y)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ sim-ℝ
 
     preserves-le-left-sim-ℝ : le-ℝ x z → le-ℝ y z
@@ -514,7 +486,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ'
 
     leq-le-rational-ℝ :
@@ -542,35 +514,6 @@ module _
           leq-le-rational-ℝ y x (forward-implication ∘ H))
 ```
 
-### If `x + y < p` for some rational `p`, then there exist `q r : ℚ` such that `p = q + r`, `x < q`, `y < r`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) (p : ℚ)
-  where
-
-  opaque
-    unfolding add-ℝ
-
-    le-split-add-rational-ℝ :
-      le-ℝ (x +ℝ y) (real-ℚ p) →
-      exists
-        ( ℚ × ℚ)
-        ( λ (q , r) →
-          Id-Prop ℚ-Set p (q +ℚ r) ∧
-          le-prop-ℝ x (real-ℚ q) ∧
-          le-prop-ℝ y (real-ℚ r))
-    le-split-add-rational-ℝ x+y<p =
-      let open do-syntax-trunc-Prop (∃ _ _)
-      in do
-        ((q , r) , x<q , y<r , p=q+r) ← is-in-upper-cut-le-real-ℚ (x +ℝ y) x+y<p
-        intro-exists
-          ( q , r)
-          ( p=q+r ,
-            le-real-is-in-upper-cut-ℚ x x<q ,
-            le-real-is-in-upper-cut-ℚ y y<r)
-```
-
 ## References
 
 {{#bibliography}}

From 13ddd47231bc1218b7fb161e662e9f26482f0520 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:07:50 -0800
Subject: [PATCH 042/134] Mark more things abstract opaque

---
 .../binary-minimum-real-numbers.lagda.md      |  4 ++--
 ...ompleteness-dedekind-real-numbers.lagda.md |  4 ++--
 .../closed-intervals-real-numbers.lagda.md    |  2 +-
 .../inequality-real-numbers.lagda.md          | 22 +++++++++----------
 4 files changed, 16 insertions(+), 16 deletions(-)

diff --git a/src/real-numbers/binary-minimum-real-numbers.lagda.md b/src/real-numbers/binary-minimum-real-numbers.lagda.md
index 121bf5d42b9..b0ae4c06407 100644
--- a/src/real-numbers/binary-minimum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-minimum-real-numbers.lagda.md
@@ -66,7 +66,7 @@ module _
   (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ min-ℝ
 
     is-greatest-binary-lower-bound-min-ℝ :
@@ -155,7 +155,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding min-ℝ
 
     approximate-above-min-ℝ :
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index 57f47ac4dda..ab073ba3c24 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -299,7 +299,7 @@ module _
     is-inhabited-upper-cut-lim-cauchy-approximation-ℝ ,
     is-rounded-upper-cut-lim-cauchy-approximation-ℝ
 
-  opaque
+  abstract opaque
     unfolding neighborhood-ℝ
 
     is-disjoint-cut-lim-cauchy-approximation-ℝ :
@@ -411,7 +411,7 @@ module _
   {l : Level} (x : cauchy-approximation-ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ lim-cauchy-approximation-ℝ
 
     is-limit-lim-cauchy-approximation-ℝ :
diff --git a/src/real-numbers/closed-intervals-real-numbers.lagda.md b/src/real-numbers/closed-intervals-real-numbers.lagda.md
index 846087aa483..d8460f5bdc8 100644
--- a/src/real-numbers/closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/closed-intervals-real-numbers.lagda.md
@@ -90,7 +90,7 @@ unit-closed-interval-ℝ =
 ### Closed intervals in the real numbers are closed in the metric space of real numbers
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ neighborhood-ℝ
 
   is-closed-subset-closed-interval-ℝ :
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index 89c0fbbb574..4ece5ae59ad 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -127,7 +127,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ'
 
     leq'-leq-ℝ : leq-ℝ x y → leq-ℝ' x y
@@ -175,7 +175,7 @@ module _
 ### Inequality on the real numbers is reflexive
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ
 
   refl-leq-ℝ : {l : Level} (x : ℝ l) → leq-ℝ x x
@@ -184,7 +184,7 @@ opaque
   leq-eq-ℝ : {l : Level} {x y : ℝ l} → x ＝ y → leq-ℝ x y
   leq-eq-ℝ {x = x} refl = refl-leq-ℝ x
 
-opaque
+abstract opaque
   unfolding leq-ℝ sim-ℝ
 
   leq-sim-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → sim-ℝ x y → leq-ℝ x y
@@ -194,7 +194,7 @@ opaque
 ### Inequality on the real numbers is antisymmetric
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ sim-ℝ
 
   sim-antisymmetric-leq-ℝ :
@@ -215,7 +215,7 @@ module _
   (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ
 
     transitive-leq-ℝ : leq-ℝ y z → leq-ℝ x y → leq-ℝ x z
@@ -244,7 +244,7 @@ antisymmetric-leq-Large-Poset ℝ-Large-Poset = antisymmetric-leq-ℝ
 ### Similarity in the large poset of real numbers is equivalent to similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ sim-ℝ
 
   sim-sim-leq-ℝ :
@@ -280,7 +280,7 @@ module _
   {x y : ℚ}
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ real-ℚ
 
     preserves-leq-real-ℚ : leq-ℚ x y → leq-ℝ (real-ℚ x) (real-ℚ y)
@@ -300,7 +300,7 @@ module _
   {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' neg-ℝ
 
     neg-leq-ℝ : leq-ℝ x y → leq-ℝ (neg-ℝ y) (neg-ℝ x)
@@ -314,7 +314,7 @@ module _
   {l1 l2 l3 : Level} {z : ℝ l1} {x : ℝ l2} {y : ℝ l3} (x~y : sim-ℝ x y)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ sim-ℝ
 
     preserves-leq-left-sim-ℝ : leq-ℝ x z → leq-ℝ y z
@@ -343,7 +343,7 @@ module _
   {l : Level} (x : ℝ l) (q : ℚ)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' real-ℚ
 
     not-in-lower-cut-leq-ℝ : leq-ℝ x (real-ℚ q) → ¬ (is-in-lower-cut-ℝ x q)
@@ -376,7 +376,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ'
 
     leq-leq-rational-ℝ :

From 04ccb85e348ae65c7afe3dd091e99585f0a20d4a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:09:00 -0800
Subject: [PATCH 043/134] Fix indent

---
 src/real-numbers/addition-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index fef1ddfc687..356d51654fd 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -97,7 +97,7 @@ module _
       let
         open
           do-syntax-trunc-Prop
-            (∃
+            ( ∃
               ( ℚ × ℚ)
               ( close-bounds-lower-upper-ℝ
                 ( lower-real-add-ℝ)

From 1d65b0a47540f785cc9142871450b6018706fbcb Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:13:56 -0800
Subject: [PATCH 044/134] Fix title

---
 src/real-numbers/short-binary-maximum-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
index 2ac42192c9c..5dd32621e15 100644
--- a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
@@ -1,4 +1,4 @@
-## The binary maximum of real numbers is a short function
+# The binary maximum of real numbers is a short function
 
 ```agda
 {-# OPTIONS --lossy-unification #-}

From 84dec2a5fc910c14f47edcb4457754f5cbe8d797 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:14:54 -0800
Subject: [PATCH 045/134] Fix

---
 src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md | 2 --
 1 file changed, 2 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index f70d4ca8b46..1592876b213 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -298,8 +298,6 @@ module _
             ( p<x)
             ( le-upper-cut-ℝ
               ( x)
-              ( q)
-              ( p +ℚ (nℚ *ℚ ε'))
               ( tr
                 ( le-ℚ q)
                 ( commutative-add-ℚ (nℚ *ℚ ε') p)

From 315b668f71c86598c98134713e013758004af0e4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:20:11 -0800
Subject: [PATCH 046/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 ...equality-positive-rational-numbers.lagda.md |  2 +-
 ...licative-group-of-rational-numbers.lagda.md |  2 +-
 .../powers-positive-rational-numbers.lagda.md  | 18 +++++++++---------
 3 files changed, 11 insertions(+), 11 deletions(-)

diff --git a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
index d967080b3a1..5b51936f6a6 100644
--- a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
@@ -93,7 +93,7 @@ antisymmetric-leq-ℚ⁺ : is-antisymmetric leq-ℚ⁺
 antisymmetric-leq-ℚ⁺ = antisymmetric-leq-Poset poset-ℚ⁺
 ```
 
-### If `x ＝ y`, `x ≤ y`
+### If `x ＝ y` then `x ≤ y`
 
 ```agda
 leq-eq-ℚ⁺ : {x y : ℚ⁺} → x ＝ y → leq-ℚ⁺ x y
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index b6fb3598c70..30f54cea5a3 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -200,7 +200,7 @@ invertible-diff-neq-ℚ : (a b : ℚ) → a ≠ b → ℚˣ
 invertible-diff-neq-ℚ a b a≠b = (b -ℚ a , is-invertible-diff-neq-ℚ a b a≠b)
 ```
 
-### If `|a| < b` for positive `b`, `b - a` is invertible
+### If `|a| < b` and `b` is positive then `b - a` is invertible
 
 ```agda
 is-invertible-diff-le-abs-ℚ :
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index fc414c812b1..9e8992215fb 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -288,11 +288,11 @@ abstract
           by leq-eq-ℚ⁺ (distributive-power-add-ℚ⁺ k m ε)
         ≤ rational-power-ℚ⁺ k one-ℚ⁺ *ℚ rational-power-ℚ⁺ m ε
           by
-            preserves-leq-right-mul-ℚ⁺
-              ( power-ℚ⁺ m ε)
-              ( _)
-              ( _)
-              ( preserves-leq-power-ℚ⁺ k ε one-ℚ⁺ ε≤1)
+          preserves-leq-right-mul-ℚ⁺
+            ( power-ℚ⁺ m ε)
+            ( _)
+            ( _)
+            ( preserves-leq-power-ℚ⁺ k ε one-ℚ⁺ ε≤1)
         ≤ one-ℚ *ℚ rational-power-ℚ⁺ m ε
           by leq-eq-ℚ _ _ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
         ≤ rational-power-ℚ⁺ m ε
@@ -327,12 +327,12 @@ abstract
                   rational-dist-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε)) zero-ℚ
                   ≤ rational-abs-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε))
                     by
-                      leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
+                    leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ n ε)
                     by
-                      leq-eq-ℚ _ _
-                        ( ap rational-ℚ⁰⁺
-                          ( abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ (power-ℚ⁺ n ε))))
+                    leq-eq-ℚ _ _
+                      ( ap rational-ℚ⁰⁺
+                        ( abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ (power-ℚ⁺ n ε))))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ m ε)
                     by leq-power-leq-one-ℚ⁺ ε (leq-le-ℚ ε<1) m n m≤n
                   ≤ rational-ℚ⁺ δ

From ed77ccd5cfae98e22e3ade4d3eee10bfe6fe00e4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:27:11 -0800
Subject: [PATCH 047/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/real-numbers/powers-real-numbers.lagda.md | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index aee3f51d45b..d7812a9177e 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -263,7 +263,7 @@ abstract
           by ap neg-ℝ (ap (λ m → power-ℝ m x) k2+1=n)
 ```
 
-### `|x|ⁿ=|xⁿ|`
+### `|x|ⁿ = |xⁿ|`
 
 ```agda
 abstract
@@ -302,8 +302,10 @@ abstract
         ( sim-raise-ℝ _ one-ℝ))
       ( refl-leq-ℝ one-ℝ)
   preserves-leq-abs-power-ℝ (succ-ℕ n) x y |x|≤|y| =
-    let open inequality-reasoning-Large-Poset ℝ-Large-Poset
-    in chain-of-inequalities
+    let
+      open inequality-reasoning-Large-Poset ℝ-Large-Poset
+    in
+    chain-of-inequalities
       abs-ℝ (power-ℝ (succ-ℕ n) x)
       ≤ abs-ℝ (power-ℝ n x *ℝ x)
         by leq-eq-ℝ _ _ (ap abs-ℝ (power-succ-ℝ n x))

From cbda75f92ba888c60df9751b2215442dacff2542 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 20:55:17 -0800
Subject: [PATCH 048/134] Geometric series in the reals

---
 .../commutative-rings.lagda.md                |  24 ++
 ...etric-sequences-commutative-rings.lagda.md | 145 ++++++++++
 ...metric-sequences-rational-numbers.lagda.md | 130 ++-------
 ...quality-positive-rational-numbers.lagda.md |   2 +-
 .../powers-positive-rational-numbers.lagda.md |   8 +-
 src/literature/100-theorems.lagda.md          |   9 +
 src/real-numbers.lagda.md                     |   5 +
 .../absolute-value-real-numbers.lagda.md      |   2 +-
 .../apartness-real-numbers.lagda.md           |  90 ++++++-
 .../convergent-series-real-numbers.lagda.md   |  40 +++
 .../geometric-sequences-real-numbers.lagda.md | 252 ++++++++++++++++++
 .../isometry-difference-real-numbers.lagda.md |  73 +++++
 .../large-ring-of-real-numbers.lagda.md       |   2 +-
 .../limits-sequences-real-numbers.lagda.md    |   5 +
 ...ric-abelian-group-of-real-numbers.lagda.md |  44 +++
 ...ositive-and-negative-real-numbers.lagda.md |   2 +-
 .../multiplication-real-numbers.lagda.md      |   6 +-
 ...ive-inverses-nonzero-real-numbers.lagda.md |  43 ++-
 .../nonnegative-real-numbers.lagda.md         |  11 +-
 .../nonzero-real-numbers.lagda.md             |  52 ++++
 src/real-numbers/powers-real-numbers.lagda.md |  28 +-
 ...real-sequences-approximating-zero.lagda.md |  16 +-
 src/real-numbers/series-real-numbers.lagda.md |  42 +++
 23 files changed, 875 insertions(+), 156 deletions(-)
 create mode 100644 src/real-numbers/convergent-series-real-numbers.lagda.md
 create mode 100644 src/real-numbers/geometric-sequences-real-numbers.lagda.md
 create mode 100644 src/real-numbers/isometry-difference-real-numbers.lagda.md
 create mode 100644 src/real-numbers/metric-abelian-group-of-real-numbers.lagda.md
 create mode 100644 src/real-numbers/series-real-numbers.lagda.md

diff --git a/src/commutative-algebra/commutative-rings.lagda.md b/src/commutative-algebra/commutative-rings.lagda.md
index a12e07d1feb..2fd1976b3d9 100644
--- a/src/commutative-algebra/commutative-rings.lagda.md
+++ b/src/commutative-algebra/commutative-rings.lagda.md
@@ -200,6 +200,13 @@ module _
   right-subtraction-Commutative-Ring =
     right-subtraction-Ring ring-Commutative-Ring
 
+  ap-right-subtraction-Commutative-Ring :
+    {x x' y y' : type-Commutative-Ring} → x ＝ x' → y ＝ y' →
+    right-subtraction-Commutative-Ring x y ＝
+    right-subtraction-Commutative-Ring x' y'
+  ap-right-subtraction-Commutative-Ring =
+    ap-right-subtraction-Ring ring-Commutative-Ring
+
   is-section-right-subtraction-Commutative-Ring :
     (x : type-Commutative-Ring) →
     ( add-Commutative-Ring' x ∘
@@ -659,3 +666,20 @@ module _
   preserves-concat-add-list-Commutative-Ring =
     preserves-concat-add-list-Ring ring-Commutative-Ring
 ```
+
+### The sum of `x - y` and `y - z` is `x - z`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l)
+  where
+
+  add-right-subtraction-Commutative-Ring :
+    (x y z : type-Commutative-Ring R) →
+    add-Commutative-Ring R
+      ( right-subtraction-Commutative-Ring R x y)
+      ( right-subtraction-Commutative-Ring R y z) ＝
+    right-subtraction-Commutative-Ring R x z
+  add-right-subtraction-Commutative-Ring =
+    add-right-subtraction-Ab (ab-Commutative-Ring R)
+```
diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
index 9343e05ed82..70d3a210a8f 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -10,7 +10,10 @@ module commutative-algebra.geometric-sequences-commutative-rings where
 open import commutative-algebra.commutative-rings
 open import commutative-algebra.commutative-semirings
 open import commutative-algebra.geometric-sequences-commutative-semirings
+open import commutative-algebra.groups-of-units-commutative-rings
+open import commutative-algebra.invertible-elements-commutative-rings
 open import commutative-algebra.powers-of-elements-commutative-rings
+open import commutative-algebra.sums-of-finite-sequences-of-elements-commutative-rings
 
 open import elementary-number-theory.natural-numbers
 
@@ -288,6 +291,11 @@ module _
 ```agda
 module _
   {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  (let _*_ = mul-Commutative-Ring R)
+  (let _+_ = add-Commutative-Ring R)
+  (let _-_ = right-subtraction-Commutative-Ring R)
+  (let zero-R = zero-Commutative-Ring R)
+  (let one-R = one-Commutative-Ring R)
   where
 
   abstract
@@ -307,4 +315,141 @@ module _
         ( commutative-semiring-Commutative-Ring R)
         ( a)
         ( r)
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-Commutative-Ring :
+      (H :
+        is-invertible-element-Commutative-Ring R
+          ( right-subtraction-Commutative-Ring R (one-Commutative-Ring R) r)) →
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-Commutative-Ring R a r n ＝
+      mul-Commutative-Ring R
+        ( mul-Commutative-Ring R
+          ( a)
+          ( inv-is-invertible-element-Commutative-Ring R H))
+        ( right-subtraction-Commutative-Ring R
+          ( one-Commutative-Ring R)
+          ( power-Commutative-Ring R n r))
+    compute-sum-standard-geometric-fin-sequence-Commutative-Ring
+      (1/⟨1-r⟩ , H) 0 =
+      inv
+        ( equational-reasoning
+          (a * 1/⟨1-r⟩) * (one-R - one-R)
+          ＝ (a * 1/⟨1-r⟩) * zero-R
+            by
+              ap-mul-Commutative-Ring R
+                ( refl)
+                ( right-inverse-law-add-Commutative-Ring R one-R)
+          ＝ zero-R
+            by right-zero-law-mul-Commutative-Ring R _)
+    compute-sum-standard-geometric-fin-sequence-Commutative-Ring
+      (1/⟨1-r⟩ , H) (succ-ℕ n) =
+      equational-reasoning
+        sum-standard-geometric-fin-sequence-Commutative-Ring R a r (succ-ℕ n)
+        ＝
+          sum-standard-geometric-fin-sequence-Commutative-Ring R a r n +
+          seq-standard-geometric-sequence-Commutative-Ring R a r n
+          by
+            cons-sum-fin-sequence-type-Commutative-Ring R
+              ( n)
+              ( standard-geometric-fin-sequence-Commutative-Ring R
+                ( a)
+                ( r)
+                ( succ-ℕ n))
+              ( refl)
+        ＝
+          ( (a * 1/⟨1-r⟩) * (one-R - power-Commutative-Ring R n r)) +
+          ( a * power-Commutative-Ring R n r)
+          by
+            ap-add-Commutative-Ring R
+              ( compute-sum-standard-geometric-fin-sequence-Commutative-Ring
+                ( 1/⟨1-r⟩ , H)
+                ( n))
+              ( inv
+                ( htpy-mul-pow-standard-geometric-sequence-Commutative-Ring R
+                  ( a)
+                  ( r)
+                  ( n)))
+        ＝
+          ( a * (1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r))) +
+          ( a * (one-R * power-Commutative-Ring R n r))
+          by
+            ap-add-Commutative-Ring R
+              ( associative-mul-Commutative-Ring R _ _ _)
+              ( ap-mul-Commutative-Ring R
+                ( refl)
+                ( inv (left-unit-law-mul-Commutative-Ring R _)))
+        ＝
+          a *
+          ( (1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r)) +
+            (one-R * power-Commutative-Ring R n r))
+          by inv (left-distributive-mul-add-Commutative-Ring R a _ _)
+        ＝
+          a *
+          ( ( 1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r)) +
+            ( (1/⟨1-r⟩ * (one-R - r)) * power-Commutative-Ring R n r))
+            by
+              ap-mul-Commutative-Ring R
+                ( refl)
+                ( ap-add-Commutative-Ring R
+                  ( refl)
+                  ( ap-mul-Commutative-Ring R (inv (pr2 H)) refl))
+        ＝
+          a *
+          ( ( 1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r)) +
+            ( 1/⟨1-r⟩ * ((one-R - r) * power-Commutative-Ring R n r)))
+          by
+            ap-mul-Commutative-Ring R
+              ( refl)
+              ( ap-add-Commutative-Ring R
+                ( refl)
+                ( associative-mul-Commutative-Ring R _ _ _))
+        ＝
+          a *
+            ( 1/⟨1-r⟩ *
+              ( ( one-R - power-Commutative-Ring R n r) +
+                ( (one-R - r) * power-Commutative-Ring R n r)))
+          by
+            ap-mul-Commutative-Ring R
+              ( refl)
+              ( inv (left-distributive-mul-add-Commutative-Ring R _ _ _))
+        ＝
+          ( a * 1/⟨1-r⟩) *
+          ( ( one-R - power-Commutative-Ring R n r) +
+            ( (one-R - r) * power-Commutative-Ring R n r))
+          by inv (associative-mul-Commutative-Ring R _ _ _)
+        ＝
+          ( a * 1/⟨1-r⟩) *
+          ( ( one-R - power-Commutative-Ring R n r) +
+            ( (one-R * power-Commutative-Ring R n r) -
+              (r * power-Commutative-Ring R n r)))
+          by
+            ap-mul-Commutative-Ring R
+              ( refl)
+              ( ap-add-Commutative-Ring R
+                ( refl)
+                ( right-distributive-mul-right-subtraction-Commutative-Ring R
+                  ( _)
+                  ( _)
+                  ( _)))
+        ＝
+          ( a * 1/⟨1-r⟩) *
+          ( ( one-R - power-Commutative-Ring R n r) +
+            ( power-Commutative-Ring R n r -
+              power-Commutative-Ring R (succ-ℕ n) r))
+          by
+            ap-mul-Commutative-Ring R
+              ( refl)
+              ( ap-add-Commutative-Ring R
+                ( refl)
+                ( ap-right-subtraction-Commutative-Ring R
+                  ( left-unit-law-mul-Commutative-Ring R _)
+                  ( inv (power-succ-Commutative-Ring' R n r))))
+        ＝
+          ( a * 1/⟨1-r⟩) *
+          ( one-R - power-Commutative-Ring R (succ-ℕ n) r)
+          by
+            ap-mul-Commutative-Ring R
+              ( refl)
+              ( add-right-subtraction-Commutative-Ring R _ _ _)
 ```
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 21695e9d15a..87e5c0a29c7 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -115,102 +115,14 @@ module _
     compute-sum-standard-geometric-fin-sequence-ℚ :
       (n : ℕ) →
       sum-standard-geometric-fin-sequence-ℚ a r n ＝
-      ( a *ℚ
-        ( (one-ℚ -ℚ power-ℚ n r) *ℚ
-          rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)))
-    compute-sum-standard-geometric-fin-sequence-ℚ 0 =
-      inv
-        ( equational-reasoning
-          a *ℚ ((one-ℚ -ℚ one-ℚ) *ℚ _)
-          ＝ a *ℚ (zero-ℚ *ℚ _)
-            by
-              ap-mul-ℚ
-                ( refl)
-                ( ap-mul-ℚ (right-inverse-law-add-ℚ one-ℚ) refl)
-          ＝ a *ℚ zero-ℚ
-            by ap-mul-ℚ refl (left-zero-law-mul-ℚ _)
-          ＝ zero-ℚ
-            by right-zero-law-mul-ℚ a)
-    compute-sum-standard-geometric-fin-sequence-ℚ (succ-ℕ n) =
-      let
-        1/⟨1-r⟩ = rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
-      in
-        equational-reasoning
-          sum-standard-geometric-fin-sequence-ℚ a r (succ-ℕ n)
-          ＝
-            sum-standard-geometric-fin-sequence-ℚ a r n +ℚ
-            seq-standard-geometric-sequence-ℚ a r n
-            by
-              cons-sum-fin-sequence-type-Commutative-Ring
-                ( commutative-ring-ℚ)
-                ( n)
-                ( _)
-                ( refl)
-          ＝
-            ( a *ℚ
-              ( (one-ℚ -ℚ power-ℚ n r) *ℚ 1/⟨1-r⟩)) +ℚ
-            ( a *ℚ power-ℚ n r)
-            by
-              ap-add-ℚ
-                ( compute-sum-standard-geometric-fin-sequence-ℚ n)
-                ( compute-standard-geometric-sequence-ℚ a r n)
-          ＝
-            a *ℚ
-            (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/⟨1-r⟩) +ℚ power-ℚ n r)
-            by inv (left-distributive-mul-add-ℚ a _ _)
-          ＝
-            a *ℚ
-            ( (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/⟨1-r⟩) +ℚ
-              (power-ℚ n r *ℚ (one-ℚ -ℚ r)) *ℚ 1/⟨1-r⟩))
-            by
-              ap-mul-ℚ
-                ( refl)
-                ( ap-add-ℚ
-                  ( refl)
-                  ( inv
-                    ( cancel-right-mul-div-ℚˣ _
-                      ( invertible-diff-neq-ℚ r one-ℚ r≠1))))
-          ＝
-            a *ℚ
-            ( ( (one-ℚ -ℚ power-ℚ n r) +ℚ (power-ℚ n r *ℚ (one-ℚ -ℚ r))) *ℚ
-              1/⟨1-r⟩)
-            by
-              ap-mul-ℚ
-                ( refl)
-                ( inv (right-distributive-mul-add-ℚ _ _ 1/⟨1-r⟩))
-          ＝
-            a *ℚ
-            ( ( one-ℚ -ℚ power-ℚ n r +ℚ
-                ((power-ℚ n r *ℚ one-ℚ) -ℚ (power-ℚ n r *ℚ r))) *ℚ
-              1/⟨1-r⟩)
-            by
-              ap-mul-ℚ
-                ( refl)
-                ( ap-mul-ℚ
-                  ( ap-add-ℚ refl (left-distributive-mul-diff-ℚ _ _ _))
-                  ( refl))
-          ＝
-            a *ℚ
-            ( ( one-ℚ -ℚ power-ℚ n r +ℚ
-                ((power-ℚ n r -ℚ power-ℚ (succ-ℕ n) r))) *ℚ
-              1/⟨1-r⟩)
-            by
-              ap-mul-ℚ
-                ( refl)
-                ( ap-mul-ℚ
-                  ( ap-add-ℚ
-                    ( refl)
-                    ( ap-diff-ℚ
-                      ( right-unit-law-mul-ℚ _)
-                      ( inv (power-succ-ℚ n r))))
-                  ( refl))
-          ＝ a *ℚ ((one-ℚ -ℚ power-ℚ (succ-ℕ n) r) *ℚ 1/⟨1-r⟩)
-            by
-              ap-mul-ℚ
-                ( refl)
-                ( ap-mul-ℚ
-                  ( mul-right-div-Group group-add-ℚ _ _ _)
-                  ( refl))
+      ( (a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)) *ℚ
+        (one-ℚ -ℚ power-ℚ n r))
+    compute-sum-standard-geometric-fin-sequence-ℚ =
+      compute-sum-standard-geometric-fin-sequence-Commutative-Ring
+        ( commutative-ring-ℚ)
+        ( a)
+        ( r)
+        ( pr2 (invertible-diff-neq-ℚ r one-ℚ r≠1))
 ```
 
 ### If `|r| < 1`, the sum of the standard geometric sequence `n ↦ arⁿ` is `a/(1-r)`
@@ -245,15 +157,14 @@ module _
           ( inv
             ( eq-htpy (compute-sum-standard-geometric-fin-sequence-ℚ a r r≠1)))
           ( equational-reasoning
-            a *ℚ
-            ( (one-ℚ -ℚ zero-ℚ) *ℚ
-              rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1))
+            a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1) *ℚ
+            (one-ℚ -ℚ zero-ℚ)
+            ＝
+              a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1) *ℚ one-ℚ
+              by ap-mul-ℚ refl (right-zero-law-diff-ℚ _)
             ＝
-              a *ℚ
-              ( one-ℚ *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1))
-              by ap-mul-ℚ refl (ap-mul-ℚ (right-zero-law-diff-ℚ one-ℚ) refl)
-            ＝ a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
-              by ap-mul-ℚ refl (left-unit-law-mul-ℚ _))
+              a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
+              by right-unit-law-mul-ℚ _)
           ( uniformly-continuous-map-limit-sequence-Metric-Space
             ( metric-space-ℚ)
             ( metric-space-ℚ)
@@ -261,14 +172,9 @@ module _
               ( metric-space-ℚ)
               ( metric-space-ℚ)
               ( metric-space-ℚ)
-              ( uniformly-continuous-left-mul-ℚ a)
-              ( comp-uniformly-continuous-function-Metric-Space
-                ( metric-space-ℚ)
-                ( metric-space-ℚ)
-                ( metric-space-ℚ)
-                ( uniformly-continuous-right-mul-ℚ
-                  ( rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)))
-                ( uniformly-continuous-diff-ℚ one-ℚ)))
+              ( uniformly-continuous-left-mul-ℚ
+                ( a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)))
+              ( uniformly-continuous-diff-ℚ one-ℚ))
             ( λ n → power-ℚ n r)
             ( zero-ℚ)
             ( is-zero-limit-power-le-one-abs-ℚ r |r|<1))
diff --git a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
index d967080b3a1..97787b4d3bb 100644
--- a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
@@ -97,5 +97,5 @@ antisymmetric-leq-ℚ⁺ = antisymmetric-leq-Poset poset-ℚ⁺
 
 ```agda
 leq-eq-ℚ⁺ : {x y : ℚ⁺} → x ＝ y → leq-ℚ⁺ x y
-leq-eq-ℚ⁺ x=y = leq-eq-ℚ _ _ (ap rational-ℚ⁺ x=y)
+leq-eq-ℚ⁺ x=y = leq-eq-ℚ (ap rational-ℚ⁺ x=y)
 ```
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index c43c4d3a7ce..aca25ee3b65 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -294,9 +294,9 @@ abstract
               ( _)
               ( preserves-leq-power-ℚ⁺ k ε one-ℚ⁺ ε≤1)
         ≤ one-ℚ *ℚ rational-power-ℚ⁺ m ε
-          by leq-eq-ℚ _ _ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
+          by leq-eq-ℚ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
         ≤ rational-power-ℚ⁺ m ε
-          by leq-eq-ℚ _ _ (left-unit-law-mul-ℚ _)
+          by leq-eq-ℚ (left-unit-law-mul-ℚ _)
 ```
 
 ### If `ε` is a positive rational number less than 1, `εⁿ` approaches 0
@@ -327,10 +327,10 @@ abstract
                   rational-dist-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε)) zero-ℚ
                   ≤ rational-abs-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε))
                     by
-                      leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
+                      leq-eq-ℚ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ n ε)
                     by
-                      leq-eq-ℚ _ _
+                      leq-eq-ℚ
                         ( ap rational-ℚ⁰⁺
                           ( abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ (power-ℚ⁺ n ε))))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ m ε)
diff --git a/src/literature/100-theorems.lagda.md b/src/literature/100-theorems.lagda.md
index a17c9a0db8f..d9ea4bf83ad 100644
--- a/src/literature/100-theorems.lagda.md
+++ b/src/literature/100-theorems.lagda.md
@@ -116,6 +116,15 @@ open import foundation.cantors-theorem using
   ( theorem-Cantor)
 ```
 
+### 66. Sum of a Geometric Series {#66}
+
+**Author:** [Louis Wasserman](https://github.com/lowasser)
+
+```agda
+open import real-numbers.geometric-sequences-real-numbers using
+  ( compute-sum-standard-geometric-series-ℝ)
+```
+
 ### 68. Sum of an arithmetic series {#68}
 
 **Author:** [malarbol](http://www.github.com/malarbol)
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 419351f5c8b..a33ee5156c3 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -18,11 +18,13 @@ open import real-numbers.binary-minimum-real-numbers public
 open import real-numbers.cauchy-completeness-dedekind-real-numbers public
 open import real-numbers.cauchy-sequences-real-numbers public
 open import real-numbers.closed-intervals-real-numbers public
+open import real-numbers.convergent-series-real-numbers public
 open import real-numbers.dedekind-real-numbers public
 open import real-numbers.difference-real-numbers public
 open import real-numbers.distance-real-numbers public
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers public
 open import real-numbers.finitely-enumerable-subsets-real-numbers public
+open import real-numbers.geometric-sequences-real-numbers public
 open import real-numbers.inequalities-addition-real-numbers public
 open import real-numbers.inequality-lower-dedekind-real-numbers public
 open import real-numbers.inequality-nonnegative-real-numbers public
@@ -34,6 +36,7 @@ open import real-numbers.infima-families-real-numbers public
 open import real-numbers.inhabited-finitely-enumerable-subsets-real-numbers public
 open import real-numbers.inhabited-totally-bounded-subsets-real-numbers public
 open import real-numbers.isometry-addition-real-numbers public
+open import real-numbers.isometry-difference-real-numbers public
 open import real-numbers.isometry-negation-real-numbers public
 open import real-numbers.large-additive-group-of-real-numbers public
 open import real-numbers.large-multiplicative-monoid-of-real-numbers public
@@ -45,6 +48,7 @@ open import real-numbers.maximum-finite-families-real-numbers public
 open import real-numbers.maximum-inhabited-finitely-enumerable-subsets-real-numbers public
 open import real-numbers.maximum-lower-dedekind-real-numbers public
 open import real-numbers.maximum-upper-dedekind-real-numbers public
+open import real-numbers.metric-abelian-group-of-real-numbers public
 open import real-numbers.metric-space-of-nonnegative-real-numbers public
 open import real-numbers.metric-space-of-real-numbers public
 open import real-numbers.minimum-finite-families-real-numbers public
@@ -76,6 +80,7 @@ open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
 open import real-numbers.real-sequences-approximating-zero public
 open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
+open import real-numbers.series-real-numbers public
 open import real-numbers.short-binary-maximum-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-positive-real-numbers public
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index ba2031c0029..b8b64174495 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -132,7 +132,7 @@ abstract opaque
           ( zero-ℝ)
           ( x)
           ( 0≤x)
-          ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ _ _ 0≤x))))
+          ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ 0≤x))))
 
   abs-real-ℝ⁺ : {l : Level} (x : ℝ⁺ l) → abs-ℝ (real-ℝ⁺ x) ＝ real-ℝ⁺ x
   abs-real-ℝ⁺ x = abs-real-ℝ⁰⁺ (nonnegative-ℝ⁺ x)
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 023123e63ef..31770cd4ac0 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -8,6 +8,7 @@ module real-numbers.apartness-real-numbers where
 
 ```agda
 open import foundation.apartness-relations
+open import foundation.binary-transport
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
@@ -21,7 +22,12 @@ open import foundation.propositions
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
+open import real-numbers.absolute-value-real-numbers
+open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -45,6 +51,10 @@ module _
 
   apart-ℝ : UU (l1 ⊔ l2)
   apart-ℝ = type-Prop apart-prop-ℝ
+
+apart-le-ℝ :
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → apart-ℝ x y
+apart-le-ℝ = inl-disjunction
 ```
 
 ## Properties
@@ -61,8 +71,8 @@ antireflexive-apart-ℝ x =
 
 ```agda
 symmetric-apart-ℝ :
-  {l1 l2 : Level} → (x : ℝ l1) (y : ℝ l2) → apart-ℝ x y → apart-ℝ y x
-symmetric-apart-ℝ x y =
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → apart-ℝ x y → apart-ℝ y x
+symmetric-apart-ℝ {x = x} {y = y} =
   elim-disjunction (apart-prop-ℝ y x) inr-disjunction inl-disjunction
 ```
 
@@ -94,7 +104,7 @@ apart-prop-Large-Apartness-Relation large-apartness-relation-ℝ =
   apart-prop-ℝ
 antirefl-Large-Apartness-Relation large-apartness-relation-ℝ =
   antireflexive-apart-ℝ
-symmetric-Large-Apartness-Relation large-apartness-relation-ℝ =
+symmetric-Large-Apartness-Relation large-apartness-relation-ℝ _ _ =
   symmetric-apart-ℝ
 cotransitive-Large-Apartness-Relation large-apartness-relation-ℝ =
   cotransitive-apart-ℝ
@@ -107,3 +117,77 @@ nonequal-apart-ℝ : {l : Level} (x y : ℝ l) → apart-ℝ x y → x ≠ y
 nonequal-apart-ℝ x y =
   nonequal-apart-Large-Apartness-Relation large-apartness-relation-ℝ
 ```
+
+### Apartness is preserved by translation
+
+```agda
+abstract
+  preserves-apart-left-add-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    apart-ℝ y z → apart-ℝ (x +ℝ y) (x +ℝ z)
+  preserves-apart-left-add-ℝ x y z =
+    map-disjunction
+      ( preserves-le-left-add-ℝ x y z)
+      ( preserves-le-left-add-ℝ x z y)
+
+  preserves-apart-right-add-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    apart-ℝ y z → apart-ℝ (y +ℝ x) (z +ℝ x)
+  preserves-apart-right-add-ℝ x y z y#z =
+    binary-tr
+      ( apart-ℝ)
+      ( commutative-add-ℝ x y)
+      ( commutative-add-ℝ x z)
+      ( preserves-apart-left-add-ℝ x y z y#z)
+```
+
+### Apartness is preserved by negation
+
+```agda
+abstract
+  preserves-apart-neg-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → apart-ℝ x y →
+    apart-ℝ (neg-ℝ x) (neg-ℝ y)
+  preserves-apart-neg-ℝ x y =
+    elim-disjunction
+      ( apart-prop-ℝ _ _)
+      ( inr-disjunction ∘ neg-le-ℝ)
+      ( inl-disjunction ∘ neg-le-ℝ)
+```
+
+### Apartness is preserved by similarity
+
+```agda
+abstract
+  apart-right-sim-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    sim-ℝ x y → apart-ℝ z x → apart-ℝ z y
+  apart-right-sim-ℝ z x y x~y =
+    map-disjunction
+      ( preserves-le-right-sim-ℝ z x y x~y)
+      ( preserves-le-left-sim-ℝ z x y x~y)
+
+  apart-left-sim-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    sim-ℝ x y → apart-ℝ x z → apart-ℝ y z
+  apart-left-sim-ℝ z x y x~y =
+    map-disjunction
+      ( preserves-le-left-sim-ℝ z x y x~y)
+      ( preserves-le-right-sim-ℝ z x y x~y)
+
+  apart-sim-ℝ :
+    {l1 l2 l3 l4 : Level} {x : ℝ l1} {x' : ℝ l2} {y : ℝ l3} {y' : ℝ l4} →
+    sim-ℝ x x' → sim-ℝ y y' → apart-ℝ x y → apart-ℝ x' y'
+  apart-sim-ℝ {x = x} {x' = x'} {y = y} {y' = y'} x~x' y~y' x#y =
+    apart-left-sim-ℝ
+      ( y')
+      ( x)
+      ( x')
+      ( x~x')
+      ( apart-right-sim-ℝ
+        ( x)
+        ( y)
+        ( y')
+        ( y~y')
+        ( x#y))
+```
diff --git a/src/real-numbers/convergent-series-real-numbers.lagda.md b/src/real-numbers/convergent-series-real-numbers.lagda.md
new file mode 100644
index 00000000000..5b3446be744
--- /dev/null
+++ b/src/real-numbers/convergent-series-real-numbers.lagda.md
@@ -0,0 +1,40 @@
+# Convergent series of real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.convergent-series-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.convergent-series-metric-abelian-groups
+
+open import foundation.propositions
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.series-real-numbers
+```
+
+</details>
+
+## Idea
+
+A [series](real-numbers.series-real-numbers.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md)
+{{#concepts "converges" Disambiguation="series of real numbers" Agda=is-convergent-series-ℝ}}
+to `x` if the sequence of its partial sums
+[converges](metric-spaces.limits-of-sequences-metric-spaces.md) to `x` in the
+[standard metric space of real numbers](real-numbers.metric-space-of-real-numbers.md).
+
+## Definition
+
+```agda
+is-sum-prop-series-ℝ : {l : Level} → series-ℝ l → ℝ l → Prop l
+is-sum-prop-series-ℝ = is-sum-prop-series-Metric-Ab
+
+is-sum-series-ℝ : {l : Level} → series-ℝ l → ℝ l → UU l
+is-sum-series-ℝ = is-sum-series-Metric-Ab
+```
diff --git a/src/real-numbers/geometric-sequences-real-numbers.lagda.md b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
new file mode 100644
index 00000000000..c70f545f881
--- /dev/null
+++ b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
@@ -0,0 +1,252 @@
+# Geometric sequences of real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.geometric-sequences-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.geometric-sequences-commutative-rings
+
+open import elementary-number-theory.natural-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.function-extensionality
+open import foundation.identity-types
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import lists.sequences
+
+open import metric-spaces.limits-of-sequences-metric-spaces
+open import metric-spaces.uniformly-continuous-functions-metric-spaces
+
+open import real-numbers.absolute-value-real-numbers
+open import real-numbers.apartness-real-numbers
+open import real-numbers.convergent-series-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.isometry-difference-real-numbers
+open import real-numbers.large-ring-of-real-numbers
+open import real-numbers.limits-sequences-real-numbers
+open import real-numbers.lipschitz-continuity-multiplication-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.multiplication-real-numbers
+open import real-numbers.multiplicative-inverses-nonzero-real-numbers
+open import real-numbers.nonzero-real-numbers
+open import real-numbers.powers-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.series-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+A
+{{#concept "geometric sequence" Disambiguation="of real numbers" Agda=geometric-sequence-ℝ}}
+of [real numbers](real-numbers.dedekind-real-numbers.md) is a
+[geometric sequence](commutative-algebra.geometric-sequences-commutative-rings.md)
+in the
+[commutative ring of real numbers](real-numbers.large-ring-of-real-numbers.md).
+
+## Definitions
+
+```agda
+geometric-sequence-ℝ : (l : Level) → UU (lsuc l)
+geometric-sequence-ℝ l =
+  geometric-sequence-Commutative-Ring (commutative-ring-ℝ l)
+
+seq-geometric-sequence-ℝ : {l : Level} → geometric-sequence-ℝ l → sequence (ℝ l)
+seq-geometric-sequence-ℝ {l} =
+  seq-geometric-sequence-Commutative-Ring (commutative-ring-ℝ l)
+
+standard-geometric-sequence-ℝ :
+  {l : Level} → ℝ l → ℝ l → geometric-sequence-ℝ l
+standard-geometric-sequence-ℝ {l} =
+  standard-geometric-sequence-Commutative-Ring (commutative-ring-ℝ l)
+
+seq-standard-geometric-sequence-ℝ :
+  {l : Level} → ℝ l → ℝ l → sequence (ℝ l)
+seq-standard-geometric-sequence-ℝ {l} =
+  seq-standard-geometric-sequence-Commutative-Ring (commutative-ring-ℝ l)
+```
+
+## Properties
+
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  {l : Level} (a r : ℝ l)
+  where
+
+  compute-standard-geometric-sequence-ℝ :
+    (n : ℕ) → seq-standard-geometric-sequence-ℝ a r n ＝ a *ℝ power-ℝ n r
+  compute-standard-geometric-sequence-ℝ n =
+    inv
+      ( htpy-mul-pow-standard-geometric-sequence-Commutative-Ring
+        ( commutative-ring-ℝ l)
+        ( a)
+        ( r)
+        ( n))
+```
+
+### If `r` is apart from `1`, the sum of the first `n` elements of the standard geometric sequence `n ↦ arⁿ` is `a(1-rⁿ)/(1-r)`
+
+```agda
+sum-standard-geometric-fin-sequence-ℝ : {l : Level} → ℝ l → ℝ l → ℕ → ℝ l
+sum-standard-geometric-fin-sequence-ℝ {l} =
+  sum-standard-geometric-fin-sequence-Commutative-Ring (commutative-ring-ℝ l)
+
+module _
+  {l : Level} (a r : ℝ l) (r≠1 : apart-ℝ r one-ℝ)
+  where
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-ℝ :
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-ℝ a r n ＝
+      ( ( a *ℝ
+          real-inv-nonzero-ℝ
+            ( nonzero-diff-apart-ℝ one-ℝ r (symmetric-apart-ℝ r≠1))) *ℝ
+        (one-ℝ -ℝ power-ℝ n r))
+    compute-sum-standard-geometric-fin-sequence-ℝ n =
+      let
+        1#r = symmetric-apart-ℝ r≠1
+        1l#r =
+          apart-left-sim-ℝ
+            ( r)
+            ( one-ℝ)
+            ( raise-ℝ l one-ℝ)
+            ( sim-raise-ℝ l one-ℝ)
+            ( 1#r)
+      in
+        equational-reasoning
+          sum-standard-geometric-fin-sequence-ℝ a r n
+          ＝
+            ( a *ℝ
+              real-inv-nonzero-ℝ
+                ( nonzero-diff-apart-ℝ (raise-ℝ l one-ℝ) r 1l#r)) *ℝ
+            ( raise-ℝ l one-ℝ -ℝ power-ℝ n r)
+            by
+              compute-sum-standard-geometric-fin-sequence-Commutative-Ring
+                ( commutative-ring-ℝ l)
+                ( a)
+                ( r)
+                ( is-invertible-is-nonzero-ℝ _
+                  ( is-nonzero-diff-is-apart-ℝ _ _ 1l#r))
+                ( n)
+          ＝
+            ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-apart-ℝ one-ℝ r 1#r)) *ℝ
+            ( one-ℝ -ℝ power-ℝ n r)
+            by
+              ap-mul-ℝ
+                ( ap-mul-ℝ
+                  ( refl)
+                  ( ap
+                    ( real-inv-nonzero-ℝ)
+                    ( eq-nonzero-ℝ _ _
+                      ( eq-sim-ℝ
+                        ( preserves-sim-diff-ℝ
+                          ( symmetric-sim-ℝ (sim-raise-ℝ l one-ℝ))
+                          ( refl-sim-ℝ r))))))
+                ( eq-sim-ℝ
+                  ( preserves-sim-diff-ℝ
+                    ( symmetric-sim-ℝ (sim-raise-ℝ l one-ℝ))
+                    ( refl-sim-ℝ (power-ℝ n r))))
+```
+
+### If `|r| < 1`, then the geometric series `Σₙ arⁿ` converges to `a/(1-r)`
+
+This is the [66th](literature.100-theorems.md#66) theorem on
+[Freek Wiedijk](http://www.cs.ru.nl/F.Wiedijk/)'s list of
+[100 theorems](literature.100-theorems.md) {{#cite 100theorems}}.
+
+```agda
+module _
+  {l : Level} (a r : ℝ l)
+  where
+
+  standard-geometric-series-ℝ : series-ℝ l
+  standard-geometric-series-ℝ =
+    series-terms-ℝ (seq-standard-geometric-sequence-ℝ a r)
+
+  abstract
+    compute-sum-standard-geometric-series-ℝ :
+      (|r|<1 : le-ℝ (abs-ℝ r) one-ℝ) →
+      is-sum-series-ℝ
+        ( standard-geometric-series-ℝ)
+        ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1))
+    compute-sum-standard-geometric-series-ℝ |r|<1 =
+      let
+        r#1 = apart-le-ℝ (concatenate-leq-le-ℝ _ _ _ (leq-abs-ℝ r) |r|<1)
+      in
+        binary-tr
+          ( is-limit-sequence-ℝ)
+          ( inv
+            ( eq-htpy
+              ( λ n →
+                equational-reasoning
+                  sum-standard-geometric-fin-sequence-ℝ a r n
+                  ＝
+                    ( a *ℝ real-inv-nonzero-ℝ _) *ℝ
+                    ( one-ℝ -ℝ power-ℝ n r)
+                    by compute-sum-standard-geometric-fin-sequence-ℝ a r r#1 n
+                  ＝
+                    ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)) *ℝ
+                    ( one-ℝ -ℝ power-ℝ n r)
+                    by
+                      ap
+                        ( λ z →
+                          (a *ℝ real-inv-nonzero-ℝ z) *ℝ (one-ℝ -ℝ power-ℝ n r))
+                        ( eq-nonzero-ℝ
+                          ( nonzero-diff-apart-ℝ
+                            ( one-ℝ)
+                            ( r)
+                            ( symmetric-apart-ℝ r#1))
+                          ( nonzero-diff-le-abs-ℝ |r|<1)
+                          ( refl)))))
+          ( equational-reasoning
+            ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)) *ℝ
+            ( one-ℝ -ℝ raise-ℝ l zero-ℝ)
+            ＝
+              ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)) *ℝ
+              ( one-ℝ -ℝ zero-ℝ)
+              by
+                eq-sim-ℝ
+                  ( preserves-sim-left-mul-ℝ _ _ _
+                    ( preserves-sim-diff-ℝ
+                      ( refl-sim-ℝ one-ℝ)
+                      ( symmetric-sim-ℝ (sim-raise-ℝ l zero-ℝ))))
+            ＝
+              ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)) *ℝ one-ℝ
+              by ap-mul-ℝ refl (right-unit-law-diff-ℝ one-ℝ)
+            ＝ a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)
+              by right-unit-law-mul-ℝ _)
+          ( uniformly-continuous-map-limit-sequence-Metric-Space
+            ( metric-space-ℝ l)
+            ( metric-space-ℝ l)
+            ( comp-uniformly-continuous-function-Metric-Space
+              ( metric-space-ℝ l)
+              ( metric-space-ℝ l)
+              ( metric-space-ℝ l)
+              ( uniformly-continuous-right-mul-ℝ
+                ( l)
+                ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)))
+              ( uniformly-continuous-diff-ℝ one-ℝ))
+            ( λ n → power-ℝ n r)
+            ( raise-ℝ l zero-ℝ)
+            ( is-zero-lim-power-le-one-abs-ℝ r |r|<1))
+```
+
+## References
+
+{{#bibliography}}
diff --git a/src/real-numbers/isometry-difference-real-numbers.lagda.md b/src/real-numbers/isometry-difference-real-numbers.lagda.md
new file mode 100644
index 00000000000..73b411df6a4
--- /dev/null
+++ b/src/real-numbers/isometry-difference-real-numbers.lagda.md
@@ -0,0 +1,73 @@
+# The difference isometry on real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.isometry-difference-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.uniformly-continuous-functions-metric-spaces
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.isometry-addition-real-numbers
+open import real-numbers.isometry-negation-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.negation-real-numbers
+```
+
+</details>
+
+## Idea
+
+[Subtracting](real-numbers.difference-real-numbers.md) from a fixed
+[real number](real-numbers.dedekind-real-numbers.md) is an
+[isometry](metric-spaces.isometries-metric-spaces.md) from the
+[metric space of real numbers](real-numbers.metric-space-of-real-numbers.md) to
+itself.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1)
+  where
+
+  abstract
+    is-isometry-diff-ℝ :
+      is-isometry-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( diff-ℝ x)
+    is-isometry-diff-ℝ =
+      is-isometry-comp-is-isometry-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( add-ℝ x)
+        ( neg-ℝ)
+        ( is-isometry-left-add-ℝ x)
+        ( is-isometry-neg-ℝ)
+
+  isometry-diff-ℝ :
+    isometry-Metric-Space (metric-space-ℝ l2) (metric-space-ℝ (l1 ⊔ l2))
+  isometry-diff-ℝ = (diff-ℝ x , is-isometry-diff-ℝ)
+
+  uniformly-continuous-diff-ℝ :
+    uniformly-continuous-function-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+  uniformly-continuous-diff-ℝ =
+    uniformly-continuous-isometry-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+      ( isometry-diff-ℝ)
+```
diff --git a/src/real-numbers/large-ring-of-real-numbers.lagda.md b/src/real-numbers/large-ring-of-real-numbers.lagda.md
index 5786d7c56cb..6743c00b131 100644
--- a/src/real-numbers/large-ring-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-ring-of-real-numbers.lagda.md
@@ -48,7 +48,7 @@ large-ring-ℝ =
   make-Large-Ring
     ( large-ab-add-ℝ)
     ( mul-ℝ)
-    ( preserves-sim-mul-ℝ)
+    ( λ _ _ a~a' _ _ → preserves-sim-mul-ℝ a~a')
     ( one-ℝ)
     ( associative-mul-ℝ)
     ( left-unit-law-mul-ℝ)
diff --git a/src/real-numbers/limits-sequences-real-numbers.lagda.md b/src/real-numbers/limits-sequences-real-numbers.lagda.md
index f40aab460f5..664e9e49cca 100644
--- a/src/real-numbers/limits-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/limits-sequences-real-numbers.lagda.md
@@ -11,6 +11,7 @@ module real-numbers.limits-sequences-real-numbers where
 ```agda
 open import foundation.dependent-pair-types
 open import foundation.propositional-truncations
+open import foundation.propositions
 open import foundation.universe-levels
 
 open import lists.sequences
@@ -36,6 +37,10 @@ On this page, we describe properties of
 ## Definition
 
 ```agda
+is-limit-prop-sequence-ℝ : {l : Level} → sequence (ℝ l) → ℝ l → Prop l
+is-limit-prop-sequence-ℝ {l} =
+  is-limit-prop-sequence-Metric-Space (metric-space-ℝ l)
+
 is-limit-sequence-ℝ : {l : Level} → sequence (ℝ l) → ℝ l → UU l
 is-limit-sequence-ℝ {l} = is-limit-sequence-Metric-Space (metric-space-ℝ l)
 ```
diff --git a/src/real-numbers/metric-abelian-group-of-real-numbers.lagda.md b/src/real-numbers/metric-abelian-group-of-real-numbers.lagda.md
new file mode 100644
index 00000000000..34024894955
--- /dev/null
+++ b/src/real-numbers/metric-abelian-group-of-real-numbers.lagda.md
@@ -0,0 +1,44 @@
+# The metric abelian group of real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.metric-abelian-group-of-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.metric-abelian-groups
+
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.pseudometric-spaces
+
+open import real-numbers.isometry-addition-real-numbers
+open import real-numbers.isometry-negation-real-numbers
+open import real-numbers.large-additive-group-of-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+```
+
+</details>
+
+## Idea
+
+The [Dedekind real numbers](real-numbers.dedekind-real-numbers.md) form a
+[metric abelian group](analysis.metric-abelian-groups.md) under
+[addition](real-numbers.addition-real-numbers.md) with regards to their
+[standard metric space](real-numbers.metric-space-of-real-numbers.md).
+
+## Definition
+
+```agda
+metric-ab-ℝ : (l : Level) → Metric-Ab (lsuc l) l
+metric-ab-ℝ l =
+  ( ab-add-ℝ l ,
+    structure-Pseudometric-Space (pseudometric-space-ℝ l) ,
+    is-extensional-pseudometric-space-ℝ ,
+    is-isometry-neg-ℝ ,
+    is-isometry-left-add-ℝ)
+```
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 7ecc0768e38..c0fea896952 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -44,7 +44,7 @@ abstract
       ( x *ℝ zero-ℝ)
       ( zero-ℝ)
       ( right-zero-law-mul-ℝ x)
-      ( preserves-le-left-mul-ℝ⁺ (x , is-pos-x) y zero-ℝ is-neg-y)
+      ( preserves-le-left-mul-ℝ⁺ (x , is-pos-x) is-neg-y)
 
 mul-positive-negative-ℝ :
   {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁻ l2 → ℝ⁻ (l1 ⊔ l2)
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index b389211c2d2..3b5d39d0f64 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -1189,10 +1189,10 @@ module _
 abstract
   preserves-sim-mul-ℝ :
     {l1 l2 l3 l4 : Level} →
-    (a : ℝ l1) (a' : ℝ l2) → sim-ℝ a a' →
-    (b : ℝ l3) (b' : ℝ l4) → sim-ℝ b b' →
+    {a : ℝ l1} {a' : ℝ l2} → sim-ℝ a a' →
+    {b : ℝ l3} {b' : ℝ l4} → sim-ℝ b b' →
     sim-ℝ (a *ℝ b) (a' *ℝ b')
-  preserves-sim-mul-ℝ a a' a~a' b b' b~b' =
+  preserves-sim-mul-ℝ {a = a} {a' = a'} a~a' {b = b} {b' = b'} b~b' =
     transitive-sim-ℝ
       ( a *ℝ b)
       ( a *ℝ b')
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index f91c1d2bfcf..6b8f350f16e 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -9,6 +9,8 @@ module real-numbers.multiplicative-inverses-nonzero-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import commutative-algebra.invertible-elements-commutative-rings
+
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
@@ -36,12 +38,14 @@ open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.large-ring-of-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-negative-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
 ```
@@ -124,13 +128,27 @@ module _
   right-inverse-law-mul-nonzero-ℝ = pr2 has-nonzero-inv-nonzero-ℝ
 
   abstract
-    left-inverse-law-nonzero-ℝ :
+    left-inverse-law-mul-nonzero-ℝ :
       sim-ℝ (real-inv-nonzero-ℝ *ℝ real-nonzero-ℝ x) one-ℝ
-    left-inverse-law-nonzero-ℝ =
+    left-inverse-law-mul-nonzero-ℝ =
       tr
         ( λ y → sim-ℝ y one-ℝ)
         ( commutative-mul-ℝ _ _)
         ( right-inverse-law-mul-nonzero-ℝ)
+
+is-invertible-is-nonzero-ℝ :
+  {l : Level} (x : ℝ l) → is-nonzero-ℝ x →
+  is-invertible-element-Commutative-Ring (commutative-ring-ℝ l) x
+is-invertible-is-nonzero-ℝ x x≠0 =
+  ( real-inv-nonzero-ℝ (x , x≠0) ,
+    eq-sim-ℝ
+      ( transitive-sim-ℝ _ _ _
+        ( sim-raise-ℝ _ _)
+        ( right-inverse-law-mul-nonzero-ℝ (x , x≠0))) ,
+    eq-sim-ℝ
+      ( transitive-sim-ℝ _ _ _
+        ( sim-raise-ℝ _ _)
+        ( left-inverse-law-mul-nonzero-ℝ (x , x≠0))))
 ```
 
 ### If a real number has a multiplicative inverse, it is nonzero
@@ -229,3 +247,24 @@ abstract
     unique-right-inv-nonzero-ℝ y x
       ( tr (λ z → sim-ℝ z one-ℝ) (commutative-mul-ℝ _ _) xy=1)
 ```
+
+### If two nonzero real numbers are similar, so are their inverses
+
+```agda
+abstract
+  preserves-sim-inv-nonzero-ℝ :
+    {l1 l2 : Level} (x : nonzero-ℝ l1) (y : nonzero-ℝ l2) →
+    sim-ℝ (real-nonzero-ℝ x) (real-nonzero-ℝ y) →
+    sim-ℝ (real-inv-nonzero-ℝ x) (real-inv-nonzero-ℝ y)
+  preserves-sim-inv-nonzero-ℝ (x , x≠0) (y , y≠0) x~y =
+    symmetric-sim-ℝ
+      ( unique-right-inv-nonzero-ℝ
+        ( x , x≠0)
+        ( inv-nonzero-ℝ (y , y≠0))
+        ( similarity-reasoning-ℝ
+          x *ℝ real-inv-nonzero-ℝ (y , y≠0)
+          ~ℝ y *ℝ real-inv-nonzero-ℝ (y , y≠0)
+            by preserves-sim-right-mul-ℝ _ _ _ x~y
+          ~ℝ one-ℝ
+            by right-inverse-law-mul-nonzero-ℝ (y , y≠0)))
+```
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 1f2bd04d059..8d0ea479a28 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -40,6 +40,7 @@ open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -130,9 +131,9 @@ abstract
     preserves-leq-real-ℚ (leq-zero-is-nonnegative-ℚ is-nonneg-q)
 
   reflects-is-nonnegative-real-ℚ :
-    (q : ℚ) → is-nonnegative-ℝ (real-ℚ q) → is-nonnegative-ℚ q
-  reflects-is-nonnegative-real-ℚ q 0≤qℝ =
-    is-nonnegative-leq-zero-ℚ (reflects-leq-real-ℚ zero-ℚ q 0≤qℝ)
+    {q : ℚ} → is-nonnegative-ℝ (real-ℚ q) → is-nonnegative-ℚ q
+  reflects-is-nonnegative-real-ℚ 0≤qℝ =
+    is-nonnegative-leq-zero-ℚ (reflects-leq-real-ℚ 0≤qℝ)
 
 nonnegative-real-ℚ⁰⁺ : ℚ⁰⁺ → ℝ⁰⁺ lzero
 nonnegative-real-ℚ⁰⁺ (q , is-nonneg-q) =
@@ -290,7 +291,7 @@ abstract
     {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) → le-ℝ (real-ℝ⁰⁺ x) y →
     is-nonnegative-ℝ y
   is-nonnegative-le-ℝ⁰⁺ x y x<y =
-    is-nonnegative-leq-ℝ⁰⁺ x y (leq-le-ℝ (real-ℝ⁰⁺ x) y x<y)
+    is-nonnegative-leq-ℝ⁰⁺ x y (leq-le-ℝ x<y)
 ```
 
 ### Nonnegativity is preserved by similarity
@@ -301,7 +302,7 @@ abstract
     {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → is-nonnegative-ℝ x → sim-ℝ x y →
     is-nonnegative-ℝ y
   is-nonnegative-sim-ℝ {x = x} {y = y} 0≤x x~y =
-    preserves-leq-right-sim-ℝ zero-ℝ x y x~y 0≤x
+    preserves-leq-right-sim-ℝ x~y 0≤x
 ```
 
 ### Raising the universe levels of nonnegative real numbers
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index 50018c445b7..3a8df09cc9c 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -14,13 +14,20 @@ open import foundation.disjunction
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtypes
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
+open import real-numbers.absolute-value-real-numbers
+open import real-numbers.addition-real-numbers
 open import real-numbers.apartness-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
@@ -86,3 +93,48 @@ eq-nonzero-ℝ :
   x ＝ y
 eq-nonzero-ℝ _ _ = eq-type-subtype is-nonzero-prop-ℝ
 ```
+
+### Two real numbers are apart if and only if their difference is nonzero
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  where
+
+  abstract
+    is-nonzero-diff-is-apart-ℝ : apart-ℝ x y → is-nonzero-ℝ (x -ℝ y)
+    is-nonzero-diff-is-apart-ℝ x#y =
+      apart-right-sim-ℝ
+        ( x -ℝ y)
+        ( y -ℝ y)
+        ( zero-ℝ)
+        ( right-inverse-law-add-ℝ y)
+        ( preserves-apart-right-add-ℝ (neg-ℝ y) x y x#y)
+
+    is-apart-is-nonzero-diff-ℝ : is-nonzero-ℝ (x -ℝ y) → apart-ℝ x y
+    is-apart-is-nonzero-diff-ℝ x-y#0 =
+      apart-sim-ℝ
+        ( cancel-right-diff-add-ℝ x y)
+        ( sim-eq-ℝ (left-unit-law-add-ℝ y))
+        ( preserves-apart-right-add-ℝ y _ _ x-y#0)
+
+  nonzero-diff-apart-ℝ : apart-ℝ x y → nonzero-ℝ (l1 ⊔ l2)
+  nonzero-diff-apart-ℝ x#y = (x -ℝ y , is-nonzero-diff-is-apart-ℝ x#y)
+```
+
+### If `x < y`, then `y - x` is nonzero
+
+```agda
+nonzero-diff-le-ℝ :
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → nonzero-ℝ (l1 ⊔ l2)
+nonzero-diff-le-ℝ {x = x} {y = y} x<y = nonzero-ℝ⁺ (positive-diff-le-ℝ x<y)
+```
+
+### If `|x| < y`, then `y - x` is nonzero
+
+```agda
+nonzero-diff-le-abs-ℝ :
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ (abs-ℝ x) y → nonzero-ℝ (l1 ⊔ l2)
+nonzero-diff-le-abs-ℝ {x = x} {y = y} |x|<y =
+  nonzero-diff-le-ℝ (concatenate-leq-le-ℝ x (abs-ℝ x) y (leq-abs-ℝ x) |x|<y)
+```
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index d78b0dbdfb7..094ca595e51 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -34,6 +34,7 @@ open import order-theory.large-posets
 
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.large-ring-of-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
@@ -318,7 +319,7 @@ abstract
     {l1 l2 : Level} (n : ℕ) (x : ℝ l1) (y : ℝ l2) → leq-ℝ (abs-ℝ x) (abs-ℝ y) →
     leq-ℝ (abs-ℝ (power-ℝ n x)) (abs-ℝ (power-ℝ n y))
   preserves-leq-abs-power-ℝ 0 _ _ _ =
-    preserves-leq-sim-ℝ _ _ _ _
+    preserves-leq-sim-ℝ
       ( inv-tr
         ( sim-ℝ one-ℝ)
         ( abs-real-ℝ⁺ (raise-ℝ⁺ _ one-ℝ⁺))
@@ -333,9 +334,9 @@ abstract
     in chain-of-inequalities
       abs-ℝ (power-ℝ (succ-ℕ n) x)
       ≤ abs-ℝ (power-ℝ n x *ℝ x)
-        by leq-eq-ℝ _ _ (ap abs-ℝ (power-succ-ℝ n x))
+        by leq-eq-ℝ (ap abs-ℝ (power-succ-ℝ n x))
       ≤ abs-ℝ (power-ℝ n x) *ℝ abs-ℝ x
-        by leq-eq-ℝ _ _ (abs-mul-ℝ _ _)
+        by leq-eq-ℝ (abs-mul-ℝ _ _)
       ≤ abs-ℝ (power-ℝ n y) *ℝ abs-ℝ y
         by
           preserves-leq-mul-ℝ⁰⁺
@@ -346,9 +347,9 @@ abstract
             ( preserves-leq-abs-power-ℝ n x y |x|≤|y|)
             ( |x|≤|y|)
       ≤ abs-ℝ (power-ℝ n y *ℝ y)
-        by leq-eq-ℝ _ _ (inv (abs-mul-ℝ _ _))
+        by leq-eq-ℝ (inv (abs-mul-ℝ _ _))
       ≤ abs-ℝ (power-ℝ (succ-ℕ n) y)
-        by leq-eq-ℝ _ _ (ap abs-ℝ (inv (power-succ-ℝ n y)))
+        by leq-eq-ℝ (ap abs-ℝ (inv (power-succ-ℝ n y)))
 ```
 
 ### If `x` and `y` are nonnegative, and `x ≤ y`, `xⁿ ≤ yⁿ`
@@ -359,7 +360,7 @@ abstract
     {l1 l2 : Level} (n : ℕ) (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) → leq-ℝ⁰⁺ x y →
     leq-ℝ (power-ℝ n (real-ℝ⁰⁺ x)) (power-ℝ n (real-ℝ⁰⁺ y))
   preserves-leq-power-real-ℝ⁰⁺ {l1} {l2} 0 _ _ _ =
-    leq-sim-ℝ _ _
+    leq-sim-ℝ
       ( transitive-sim-ℝ _ one-ℝ _
         ( sim-raise-ℝ l2 one-ℝ)
         ( symmetric-sim-ℝ (sim-raise-ℝ l1 one-ℝ)))
@@ -370,7 +371,7 @@ abstract
       chain-of-inequalities
       power-ℝ (succ-ℕ n) x
       ≤ power-ℝ n x *ℝ x
-        by leq-eq-ℝ _ _ (power-succ-ℝ n x)
+        by leq-eq-ℝ (power-succ-ℝ n x)
       ≤ power-ℝ n y *ℝ y
         by
           preserves-leq-mul-ℝ⁰⁺
@@ -381,7 +382,7 @@ abstract
             ( preserves-leq-power-real-ℝ⁰⁺ n x⁰⁺ y⁰⁺ x≤y)
             ( x≤y)
       ≤ power-ℝ (succ-ℕ n) y
-        by leq-eq-ℝ _ _ (inv (power-succ-ℝ n y))
+        by leq-eq-ℝ (inv (power-succ-ℝ n y))
 ```
 
 ### If `|r| < 1`, `rⁿ` approaches 0
@@ -401,13 +402,12 @@ abstract
       let
         is-pos-ε =
           reflects-is-positive-real-ℚ
-            ( ε)
             ( is-positive-le-ℝ⁰⁺ (nonnegative-abs-ℝ r) (real-ℚ ε) |r|<ε)
         ε⁺ = (ε , is-pos-ε)
       is-zero-limit-sequence-leq-abs-rational-zero-limit-sequence-ℝ
         ( λ n → power-ℝ n r)
         ( (λ n → rational-ℚ⁺ (power-ℚ⁺ n ε⁺)) ,
-          is-zero-limit-power-le-one-ℚ⁺ ε⁺ (reflects-le-real-ℚ ε one-ℚ ε<1ℝ))
+          is-zero-limit-power-le-one-ℚ⁺ ε⁺ (reflects-le-real-ℚ ε<1ℝ))
         ( λ n →
           chain-of-inequalities
             abs-ℝ (power-ℝ n r)
@@ -420,11 +420,11 @@ abstract
                   ( inv-tr
                     ( leq-ℝ (abs-ℝ r))
                     ( abs-real-ℝ⁺ (positive-real-ℚ⁺ ε⁺))
-                    ( leq-le-ℝ _ _ |r|<ε))
+                    ( leq-le-ℝ |r|<ε))
             ≤ power-ℝ n (real-ℚ ε)
-              by leq-eq-ℝ _ _ (abs-real-ℝ⁺ (power-ℝ⁺ n (positive-real-ℚ⁺ ε⁺)))
+              by leq-eq-ℝ (abs-real-ℝ⁺ (power-ℝ⁺ n (positive-real-ℚ⁺ ε⁺)))
             ≤ real-ℚ (power-ℚ n ε)
-              by leq-eq-ℝ _ _ (power-real-ℚ n ε)
+              by leq-eq-ℝ (power-real-ℚ n ε)
             ≤ real-ℚ⁺ (power-ℚ⁺ n ε⁺)
-              by leq-eq-ℝ _ _ (ap real-ℚ (power-rational-ℚ⁺ n ε⁺)))
+              by leq-eq-ℝ (ap real-ℚ (power-rational-ℚ⁺ n ε⁺)))
 ```
diff --git a/src/real-numbers/real-sequences-approximating-zero.lagda.md b/src/real-numbers/real-sequences-approximating-zero.lagda.md
index 4f50b606de9..e36d1f6603c 100644
--- a/src/real-numbers/real-sequences-approximating-zero.lagda.md
+++ b/src/real-numbers/real-sequences-approximating-zero.lagda.md
@@ -37,6 +37,7 @@ open import real-numbers.absolute-value-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.limits-sequences-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
@@ -59,10 +60,7 @@ the
 ```agda
 is-zero-limit-prop-sequence-ℝ : {l : Level} → sequence (ℝ l) → Prop l
 is-zero-limit-prop-sequence-ℝ {l} σ =
-  is-limit-prop-sequence-Metric-Space
-    ( metric-space-ℝ l)
-    ( σ)
-    ( raise-ℝ l zero-ℝ)
+  is-limit-prop-sequence-ℝ σ (raise-ℝ l zero-ℝ)
 
 is-zero-limit-sequence-ℝ : {l : Level} → sequence (ℝ l) → UU l
 is-zero-limit-sequence-ℝ σ = type-Prop (is-zero-limit-prop-sequence-ℝ σ)
@@ -93,24 +91,24 @@ abstract
               dist-ℝ (a n) (raise-ℝ l zero-ℝ)
               ≤ dist-ℝ (a n) zero-ℝ
                 by
-                  leq-sim-ℝ _ _
+                  leq-sim-ℝ
                     ( preserves-dist-right-sim-ℝ
                       ( symmetric-sim-ℝ (sim-raise-ℝ l zero-ℝ)))
               ≤ abs-ℝ (a n)
-                by leq-eq-ℝ _ _ (right-zero-law-dist-ℝ (a n))
+                by leq-eq-ℝ (right-zero-law-dist-ℝ (a n))
               ≤ real-ℚ (b n)
                 by H n
               ≤ real-ℚ (rational-abs-ℚ (b n))
-                by preserves-leq-real-ℚ _ _ (leq-abs-ℚ (b n))
+                by preserves-leq-real-ℚ (leq-abs-ℚ (b n))
               ≤ real-ℚ (rational-dist-ℚ (b n) zero-ℚ)
                 by
-                  leq-eq-ℝ _ _
+                  leq-eq-ℝ
                     ( ap
                       ( real-ℚ ∘ rational-ℚ⁰⁺)
                       ( inv (right-zero-law-dist-ℚ (b n))))
               ≤ real-ℚ⁺ ε
                 by
-                  preserves-leq-real-ℚ _ _
+                  preserves-leq-real-ℚ
                     ( leq-dist-neighborhood-ℚ ε _ _ (is-mod-μ ε n με≤n))))
       ( lim-b=0)
 ```
diff --git a/src/real-numbers/series-real-numbers.lagda.md b/src/real-numbers/series-real-numbers.lagda.md
new file mode 100644
index 00000000000..c123f129447
--- /dev/null
+++ b/src/real-numbers/series-real-numbers.lagda.md
@@ -0,0 +1,42 @@
+# Series of real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.series-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.series-metric-abelian-groups
+
+open import foundation.universe-levels
+
+open import lists.sequences
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.metric-abelian-group-of-real-numbers
+```
+
+</details>
+
+## Idea
+
+A {{#concept "series" Disambiguation="of real numbers" Agda=series-ℝ}} of
+[real numbers](real-numbers.dedekind-real-numbers.md) is an infinite sum
+$$\Sigma_{n=0}^∞ a_n$$, which is evaluated for convergence in the
+[metric abelian group of real numbers](real-numbers.metric-abelian-group-of-real-numbers.md).
+
+## Definition
+
+```agda
+series-ℝ : (l : Level) → UU (lsuc l)
+series-ℝ l = series-Metric-Ab (metric-ab-ℝ l)
+
+series-terms-ℝ : {l : Level} → sequence (ℝ l) → series-ℝ l
+series-terms-ℝ = series-terms-Metric-Ab
+
+terms-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)
+terms-series-ℝ = term-series-Metric-Ab
+```

From eac6aa2768ec150a56dfe0eee64fc72e70c36118 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 21:01:59 -0800
Subject: [PATCH 049/134] Fix link

---
 src/real-numbers/convergent-series-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/convergent-series-real-numbers.lagda.md b/src/real-numbers/convergent-series-real-numbers.lagda.md
index 5b3446be744..0813cae301b 100644
--- a/src/real-numbers/convergent-series-real-numbers.lagda.md
+++ b/src/real-numbers/convergent-series-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import real-numbers.series-real-numbers
 
 A [series](real-numbers.series-real-numbers.md) of
 [real numbers](real-numbers.dedekind-real-numbers.md)
-{{#concepts "converges" Disambiguation="series of real numbers" Agda=is-convergent-series-ℝ}}
+{{#concepts "converges" Disambiguation="series of real numbers" Agda=is-sum-series-ℝ}}
 to `x` if the sequence of its partial sums
 [converges](metric-spaces.limits-of-sequences-metric-spaces.md) to `x` in the
 [standard metric space of real numbers](real-numbers.metric-space-of-real-numbers.md).

From 6fd2d52adcc6c59e54ad1971521f3e9d46dcb18d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 21:02:54 -0800
Subject: [PATCH 050/134] Correct spelling

---
 src/real-numbers/convergent-series-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/convergent-series-real-numbers.lagda.md b/src/real-numbers/convergent-series-real-numbers.lagda.md
index 0813cae301b..9c8b3bd2db6 100644
--- a/src/real-numbers/convergent-series-real-numbers.lagda.md
+++ b/src/real-numbers/convergent-series-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import real-numbers.series-real-numbers
 
 A [series](real-numbers.series-real-numbers.md) of
 [real numbers](real-numbers.dedekind-real-numbers.md)
-{{#concepts "converges" Disambiguation="series of real numbers" Agda=is-sum-series-ℝ}}
+{{#concept "converges" Disambiguation="series of real numbers" Agda=is-sum-series-ℝ}}
 to `x` if the sequence of its partial sums
 [converges](metric-spaces.limits-of-sequences-metric-spaces.md) to `x` in the
 [standard metric space of real numbers](real-numbers.metric-space-of-real-numbers.md).

From 7d10f9c1057d00072c6ba163c924996c54b50f27 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 8 Nov 2025 10:53:12 +0100
Subject: [PATCH 051/134] Update src/real-numbers/powers-real-numbers.lagda.md

---
 src/real-numbers/powers-real-numbers.lagda.md | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index d7812a9177e..497f3b7c043 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -58,6 +58,8 @@ defined by [iteratively](foundation.iterating-functions.md)
 [multiplying](real-numbers.multiplication-real-numbers.md) `x` with itself `n`
 times.
 
+Note that this operation defines`0⁰` to be the empty product, `1`.
+
 ## Definition
 
 ```agda

From 43ad5594a49f02b2dbf8093af81c32fa78099290 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 08:34:23 -0800
Subject: [PATCH 052/134] Fix

---
 src/real-numbers.lagda.md                     |  1 +
 .../geometric-sequences-real-numbers.lagda.md | 10 +---
 ...iplicative-monoid-of-real-numbers.lagda.md |  2 +-
 .../limits-sequences-real-numbers.lagda.md    | 28 +++++++++
 ...nuity-multiplication-real-numbers.lagda.md |  9 +--
 ...continuous-functions-real-numbers.lagda.md | 59 +++++++++++++++++++
 6 files changed, 95 insertions(+), 14 deletions(-)
 create mode 100644 src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md

diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index a33ee5156c3..3b92a04dd4e 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -95,5 +95,6 @@ open import real-numbers.subsets-real-numbers public
 open import real-numbers.suprema-families-real-numbers public
 open import real-numbers.totally-bounded-subsets-real-numbers public
 open import real-numbers.transposition-addition-subtraction-cuts-dedekind-real-numbers public
+open import real-numbers.uniformly-continuous-functions-real-numbers public
 open import real-numbers.upper-dedekind-real-numbers public
 ```
diff --git a/src/real-numbers/geometric-sequences-real-numbers.lagda.md b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
index c70f545f881..2db218fddaf 100644
--- a/src/real-numbers/geometric-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
@@ -44,6 +44,7 @@ open import real-numbers.rational-real-numbers
 open import real-numbers.series-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.uniformly-continuous-functions-real-numbers
 ```
 
 </details>
@@ -231,13 +232,8 @@ module _
               by ap-mul-ℝ refl (right-unit-law-diff-ℝ one-ℝ)
             ＝ a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)
               by right-unit-law-mul-ℝ _)
-          ( uniformly-continuous-map-limit-sequence-Metric-Space
-            ( metric-space-ℝ l)
-            ( metric-space-ℝ l)
-            ( comp-uniformly-continuous-function-Metric-Space
-              ( metric-space-ℝ l)
-              ( metric-space-ℝ l)
-              ( metric-space-ℝ l)
+          ( uniformly-continuous-map-limit-sequence-ℝ
+            ( comp-uniformly-continuous-function-ℝ
               ( uniformly-continuous-right-mul-ℝ
                 ( l)
                 ( a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)))
diff --git a/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md b/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
index 87f9524dc39..8b3e790aaf7 100644
--- a/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
@@ -50,7 +50,7 @@ large-monoid-mul-ℝ =
     ( large-similarity-relation-sim-ℝ)
     ( raise-ℝ)
     ( sim-raise-ℝ)
-    ( preserves-sim-mul-ℝ)
+    ( λ _ _ x~x' _ _ y~y' → preserves-sim-mul-ℝ x~x' y~y')
     ( one-ℝ)
     ( left-unit-law-mul-ℝ)
     ( right-unit-law-mul-ℝ)
diff --git a/src/real-numbers/limits-sequences-real-numbers.lagda.md b/src/real-numbers/limits-sequences-real-numbers.lagda.md
index 664e9e49cca..340bba67d8e 100644
--- a/src/real-numbers/limits-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/limits-sequences-real-numbers.lagda.md
@@ -24,6 +24,7 @@ open import real-numbers.cauchy-sequences-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-addition-real-numbers
 open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.uniformly-continuous-functions-real-numbers
 ```
 
 </details>
@@ -79,3 +80,30 @@ module _
           ( Hu)
           ( Hv))
 ```
+
+### Uniformly continuous functions from `ℝ` to `ℝ` preserve limits
+
+```agda
+module _
+  {l1 l2 : Level}
+  (f : uniformly-continuous-function-ℝ l1 l2)
+  (u : sequence (ℝ l1))
+  (lim : ℝ l1)
+  where
+
+  abstract
+    uniformly-continuous-map-limit-sequence-ℝ :
+      is-limit-sequence-ℝ u lim →
+      is-limit-sequence-ℝ
+        ( map-sequence
+          ( map-uniformly-continuous-function-ℝ f)
+          ( u))
+        ( map-uniformly-continuous-function-ℝ f lim)
+    uniformly-continuous-map-limit-sequence-ℝ =
+      uniformly-continuous-map-limit-sequence-Metric-Space
+        ( metric-space-ℝ l1)
+        ( metric-space-ℝ l2)
+        ( f)
+        ( u)
+        ( lim)
+```
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index ed960956fe8..f8c377f7ce5 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -45,6 +45,7 @@ open import real-numbers.multiplication-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.uniformly-continuous-functions-real-numbers
 ```
 
 </details>
@@ -155,16 +156,12 @@ module _
         ( is-lipschitz-left-mul-ℝ l2 c)
 
   uniformly-continuous-right-mul-ℝ :
-    uniformly-continuous-function-Metric-Space
-      ( metric-space-ℝ l2)
-      ( metric-space-ℝ (l1 ⊔ l2))
+    uniformly-continuous-function-ℝ l2 (l1 ⊔ l2)
   uniformly-continuous-right-mul-ℝ =
     ( mul-ℝ c , is-uniformly-continuous-right-mul-ℝ)
 
   uniformly-continuous-left-mul-ℝ :
-    uniformly-continuous-function-Metric-Space
-      ( metric-space-ℝ l2)
-      ( metric-space-ℝ (l1 ⊔ l2))
+    uniformly-continuous-function-ℝ l2 (l1 ⊔ l2)
   uniformly-continuous-left-mul-ℝ =
     ( mul-ℝ' c , is-uniformly-continuous-left-mul-ℝ)
 ```
diff --git a/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md b/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
new file mode 100644
index 00000000000..c77135801b7
--- /dev/null
+++ b/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
@@ -0,0 +1,59 @@
+# Uniformly continuous functions of real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.uniformly-continuous-functions-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.universe-levels
+
+open import metric-spaces.uniformly-continuous-functions-metric-spaces
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+```
+
+</details>
+
+## Idea
+
+A
+{{#concept "uniformly continuous function" Disambiguation="from ℝ to ℝ" Agda=uniformly-continuous-function-ℝ}}
+from the [real numbers](real-numbers.dedekind-real-numbers.md) to the real
+numbers is a function `f : ℝ → ℝ` such that there
+[exists](foundation.existential-quantification.md) a modulus `μ : ℚ⁺ → ℚ⁺` such
+that for all `x y : ℝ` within a `μ ε`
+[neighborhood](real-numbers.metric-space-of-real-numbers.md) of each other,
+`f x` and `f y` are within an `ε` neighborhood.
+
+## Definition
+
+```agda
+uniformly-continuous-function-ℝ : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
+uniformly-continuous-function-ℝ l1 l2 =
+  uniformly-continuous-function-Metric-Space
+    ( metric-space-ℝ l1)
+    ( metric-space-ℝ l2)
+
+comp-uniformly-continuous-function-ℝ :
+  {l1 l2 l3 : Level} →
+  uniformly-continuous-function-ℝ l2 l3 →
+  uniformly-continuous-function-ℝ l1 l2 →
+  uniformly-continuous-function-ℝ l1 l3
+comp-uniformly-continuous-function-ℝ {l1} {l2} {l3} =
+  comp-uniformly-continuous-function-Metric-Space
+    ( metric-space-ℝ l1)
+    ( metric-space-ℝ l2)
+    ( metric-space-ℝ l3)
+
+map-uniformly-continuous-function-ℝ :
+  {l1 l2 : Level} → uniformly-continuous-function-ℝ l1 l2 → ℝ l1 → ℝ l2
+map-uniformly-continuous-function-ℝ {l1} {l2} =
+  map-uniformly-continuous-function-Metric-Space
+    ( metric-space-ℝ l1)
+    ( metric-space-ℝ l2)
+```

From 76b2d963ebfa5efc5d31378305c24180a6442a62 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 08:47:32 -0800
Subject: [PATCH 053/134] Fix

---
 .../binary-maximum-real-numbers.lagda.md      | 23 -------------------
 .../positive-real-numbers.lagda.md            |  8 -------
 src/real-numbers/powers-real-numbers.lagda.md |  5 ----
 3 files changed, 36 deletions(-)

diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 43e78e69113..69b9224d864 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -320,29 +320,6 @@ abstract
         ( right-leq-left-least-upper-bound-Large-Poset ℝ-Large-Poset x y y≤x))
 ```
 
-### The binary maximum preserves lower neighborhoods
-
-```agda
-abstract
-  left-leq-right-max-ℝ :
-    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → leq-ℝ x y →
-    sim-ℝ (max-ℝ x y) y
-  left-leq-right-max-ℝ {x = x} {y = y} x≤y =
-    sim-sim-leq-ℝ
-      ( sim-is-least-binary-upper-bound-Large-Poset ℝ-Large-Poset x y
-        ( is-least-binary-upper-bound-max-ℝ x y)
-        ( left-leq-right-least-upper-bound-Large-Poset ℝ-Large-Poset x y x≤y))
-
-  right-leq-left-max-ℝ :
-    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → leq-ℝ y x →
-    sim-ℝ (max-ℝ x y) x
-  right-leq-left-max-ℝ {x = x} {y = y} y≤x =
-    sim-sim-leq-ℝ
-      ( sim-is-least-binary-upper-bound-Large-Poset ℝ-Large-Poset x y
-        ( is-least-binary-upper-bound-max-ℝ x y)
-        ( right-leq-left-least-upper-bound-Large-Poset ℝ-Large-Poset x y y≤x))
-```
-
 ### For any `ε : ℚ⁺`, `(max-ℝ x y - ε < x) ∨ (max-ℝ x y - ε < y)`
 
 ```agda
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 05d059e1013..668e1644d95 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -302,11 +302,3 @@ raise-ℝ⁺ : {l1 : Level} (l : Level) → ℝ⁺ l1 → ℝ⁺ (l ⊔ l1)
 raise-ℝ⁺ l (x , 0<x) =
   ( raise-ℝ l x , preserves-le-right-sim-ℝ zero-ℝ x _ (sim-raise-ℝ _ _) 0<x)
 ```
-
-### Raising the universe level of positive real numbers
-
-```agda
-raise-ℝ⁺ : {l1 : Level} (l : Level) → ℝ⁺ l1 → ℝ⁺ (l ⊔ l1)
-raise-ℝ⁺ l (x , 0<x) =
-  ( raise-ℝ l x , preserves-le-right-sim-ℝ zero-ℝ x _ (sim-raise-ℝ _ _) 0<x)
-```
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 58ccc17d1e1..55cb4942444 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -52,11 +52,6 @@ open import real-numbers.real-sequences-approximating-zero
 open import real-numbers.similarity-real-numbers
 open import real-numbers.squares-real-numbers
 open import real-numbers.strict-inequality-real-numbers
-open import real-numbers.positive-real-numbers
-open import real-numbers.raising-universe-levels-real-numbers
-open import real-numbers.rational-real-numbers
-open import real-numbers.similarity-real-numbers
-open import real-numbers.squares-real-numbers
 ```
 
 </details>

From 76bec78f97bfc3c140229c75e9945e01ac3b8ed5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 09:16:38 -0800
Subject: [PATCH 054/134] Progress

---
 ...per-closed-intervals-real-numbers.lagda.md | 36 +++++++++++++++++++
 1 file changed, 36 insertions(+)
 create mode 100644 src/real-numbers/proper-closed-intervals-real-numbers.lagda.md

diff --git a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
new file mode 100644
index 00000000000..6c8f97ed6e4
--- /dev/null
+++ b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
@@ -0,0 +1,36 @@
+# Proper closed intervals in the real numbers
+
+```agda
+module real-numbers.proper-closed-intervals-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.closed-intervals-real-numbers
+open import foundation.universe-levels
+open import foundation.dependent-pair-types
+```
+
+</details>
+
+## Idea
+
+A
+{{#concept "proper closed interval" Disambiguation="in ℝ" Agda=proper-closed-interval-ℝ}}
+in the [real numbers](real-numbers.dedekind-real-numbers.md) is a
+[closed interval](real-numbers.closed-intervals-real-numbers.md) in which the
+lower bound is
+[strictly less than](real-numbers.strict-inequality-real-numbers.md) the upper
+bound.
+
+## Definition
+
+```agda
+proper-closed-interval-ℝ : (l1 l2 : Level) → UU ?
+proper-closed-interval-ℝ =
+  Σ (ℝ l1) (λ x → Σ (ℝ l2) (λ y → le-ℝ x y))
+```

From 507c7fa3302d4e87eb6792772311f10f28145b71 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 09:40:53 -0800
Subject: [PATCH 055/134] Progress

---
 ...per-closed-intervals-real-numbers.lagda.md | 86 ++++++++++++++++++-
 1 file changed, 83 insertions(+), 3 deletions(-)

diff --git a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
index 6c8f97ed6e4..999b3a89adc 100644
--- a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
@@ -9,10 +9,18 @@ module real-numbers.proper-closed-intervals-real-numbers where
 ```agda
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.subsets-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.closed-intervals-real-numbers
+
+open import metric-spaces.metric-spaces
+open import foundation.subtypes
+open import foundation.conjunction
 open import foundation.universe-levels
+open import metric-spaces.complete-metric-spaces
 open import foundation.dependent-pair-types
+open import metric-spaces.closed-subsets-metric-spaces
 ```
 
 </details>
@@ -30,7 +38,79 @@ bound.
 ## Definition
 
 ```agda
-proper-closed-interval-ℝ : (l1 l2 : Level) → UU ?
-proper-closed-interval-ℝ =
-  Σ (ℝ l1) (λ x → Σ (ℝ l2) (λ y → le-ℝ x y))
+proper-closed-interval-ℝ : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
+proper-closed-interval-ℝ l1 l2 = Σ (ℝ l1) (λ x → Σ (ℝ l2) (λ y → le-ℝ x y))
+
+closed-interval-proper-closed-interval-ℝ :
+  {l1 l2 : Level} → proper-closed-interval-ℝ l1 l2 → closed-interval-ℝ l1 l2
+closed-interval-proper-closed-interval-ℝ (a , b , a<b) =
+  ((a , b) , leq-le-ℝ a<b)
+
+lower-bound-proper-closed-interval-ℝ :
+  {l1 l2 : Level} → proper-closed-interval-ℝ l1 l2 → ℝ l1
+lower-bound-proper-closed-interval-ℝ (a , b , a<b) = a
+
+upper-bound-proper-closed-interval-ℝ :
+  {l1 l2 : Level} → proper-closed-interval-ℝ l1 l2 → ℝ l2
+upper-bound-proper-closed-interval-ℝ (a , b , a<b) = b
+
+subtype-proper-closed-interval-ℝ :
+  {l1 l2 : Level} (l : Level) → proper-closed-interval-ℝ l1 l2 →
+  subset-ℝ (l1 ⊔ l2 ⊔ l) l
+subtype-proper-closed-interval-ℝ l (a , b , _) x =
+  leq-prop-ℝ a x ∧ leq-prop-ℝ x b
+
+type-proper-closed-interval-ℝ :
+  {l1 l2 : Level} (l : Level) → proper-closed-interval-ℝ l1 l2 →
+  UU (l1 ⊔ l2 ⊔ lsuc l)
+type-proper-closed-interval-ℝ l [a,b] =
+  type-subtype (subtype-proper-closed-interval-ℝ l [a,b])
+```
+
+## Properties
+
+### The metric space associated with a proper closed interval
+
+```agda
+metric-space-proper-closed-interval-ℝ :
+  {l1 l2 : Level} (l : Level) → proper-closed-interval-ℝ l1 l2 →
+  Metric-Space (l1 ⊔ l2 ⊔ lsuc l) l
+metric-space-proper-closed-interval-ℝ l [a,b] =
+  metric-space-closed-interval-ℝ
+    ( l)
+    ( closed-interval-proper-closed-interval-ℝ [a,b])
+```
+
+### The metric space associated with a proper closed interval is closed
+
+```agda
+abstract
+  is-closed-subset-proper-closed-interval-ℝ :
+    {l1 l2 l3 : Level} ([a,b] : proper-closed-interval-ℝ l1 l2) →
+    is-closed-subset-Metric-Space
+      ( metric-space-ℝ l3)
+      ( subtype-proper-closed-interval-ℝ l3 [a,b])
+  is-closed-subset-proper-closed-interval-ℝ [a,b] =
+    is-closed-subset-closed-interval-ℝ
+      ( closed-interval-proper-closed-interval-ℝ [a,b])
+
+closed-subset-proper-closed-interval-ℝ :
+  {l1 l2 : Level} (l : Level) → proper-closed-interval-ℝ l1 l2 →
+  closed-subset-Metric-Space (l1 ⊔ l2 ⊔ l) (metric-space-ℝ l)
+closed-subset-proper-closed-interval-ℝ l [a,b] =
+  closed-subset-closed-interval-ℝ
+    ( l)
+    ( closed-interval-proper-closed-interval-ℝ [a,b])
+```
+
+### The complete metric space associated with a proper closed interval
+
+```agda
+complete-metric-space-proper-closed-interval-ℝ :
+  {l1 l2 : Level} (l : Level) → proper-closed-interval-ℝ l1 l2 →
+  Complete-Metric-Space (l1 ⊔ l2 ⊔ lsuc l) l
+complete-metric-space-proper-closed-interval-ℝ l [a,b] =
+  complete-metric-space-closed-interval-ℝ
+    ( l)
+    ( closed-interval-proper-closed-interval-ℝ [a,b])
 ```

From 41992ba02ce55e53ebf26676c88939607bbfda57 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 10:59:20 -0800
Subject: [PATCH 056/134] Pull changes

---
 .../apartness-real-numbers.lagda.md           | 90 ++++++++++++++++++-
 1 file changed, 87 insertions(+), 3 deletions(-)

diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 023123e63ef..31770cd4ac0 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -8,6 +8,7 @@ module real-numbers.apartness-real-numbers where
 
 ```agda
 open import foundation.apartness-relations
+open import foundation.binary-transport
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
@@ -21,7 +22,12 @@ open import foundation.propositions
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
+open import real-numbers.absolute-value-real-numbers
+open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -45,6 +51,10 @@ module _
 
   apart-ℝ : UU (l1 ⊔ l2)
   apart-ℝ = type-Prop apart-prop-ℝ
+
+apart-le-ℝ :
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → apart-ℝ x y
+apart-le-ℝ = inl-disjunction
 ```
 
 ## Properties
@@ -61,8 +71,8 @@ antireflexive-apart-ℝ x =
 
 ```agda
 symmetric-apart-ℝ :
-  {l1 l2 : Level} → (x : ℝ l1) (y : ℝ l2) → apart-ℝ x y → apart-ℝ y x
-symmetric-apart-ℝ x y =
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → apart-ℝ x y → apart-ℝ y x
+symmetric-apart-ℝ {x = x} {y = y} =
   elim-disjunction (apart-prop-ℝ y x) inr-disjunction inl-disjunction
 ```
 
@@ -94,7 +104,7 @@ apart-prop-Large-Apartness-Relation large-apartness-relation-ℝ =
   apart-prop-ℝ
 antirefl-Large-Apartness-Relation large-apartness-relation-ℝ =
   antireflexive-apart-ℝ
-symmetric-Large-Apartness-Relation large-apartness-relation-ℝ =
+symmetric-Large-Apartness-Relation large-apartness-relation-ℝ _ _ =
   symmetric-apart-ℝ
 cotransitive-Large-Apartness-Relation large-apartness-relation-ℝ =
   cotransitive-apart-ℝ
@@ -107,3 +117,77 @@ nonequal-apart-ℝ : {l : Level} (x y : ℝ l) → apart-ℝ x y → x ≠ y
 nonequal-apart-ℝ x y =
   nonequal-apart-Large-Apartness-Relation large-apartness-relation-ℝ
 ```
+
+### Apartness is preserved by translation
+
+```agda
+abstract
+  preserves-apart-left-add-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    apart-ℝ y z → apart-ℝ (x +ℝ y) (x +ℝ z)
+  preserves-apart-left-add-ℝ x y z =
+    map-disjunction
+      ( preserves-le-left-add-ℝ x y z)
+      ( preserves-le-left-add-ℝ x z y)
+
+  preserves-apart-right-add-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    apart-ℝ y z → apart-ℝ (y +ℝ x) (z +ℝ x)
+  preserves-apart-right-add-ℝ x y z y#z =
+    binary-tr
+      ( apart-ℝ)
+      ( commutative-add-ℝ x y)
+      ( commutative-add-ℝ x z)
+      ( preserves-apart-left-add-ℝ x y z y#z)
+```
+
+### Apartness is preserved by negation
+
+```agda
+abstract
+  preserves-apart-neg-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → apart-ℝ x y →
+    apart-ℝ (neg-ℝ x) (neg-ℝ y)
+  preserves-apart-neg-ℝ x y =
+    elim-disjunction
+      ( apart-prop-ℝ _ _)
+      ( inr-disjunction ∘ neg-le-ℝ)
+      ( inl-disjunction ∘ neg-le-ℝ)
+```
+
+### Apartness is preserved by similarity
+
+```agda
+abstract
+  apart-right-sim-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    sim-ℝ x y → apart-ℝ z x → apart-ℝ z y
+  apart-right-sim-ℝ z x y x~y =
+    map-disjunction
+      ( preserves-le-right-sim-ℝ z x y x~y)
+      ( preserves-le-left-sim-ℝ z x y x~y)
+
+  apart-left-sim-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    sim-ℝ x y → apart-ℝ x z → apart-ℝ y z
+  apart-left-sim-ℝ z x y x~y =
+    map-disjunction
+      ( preserves-le-left-sim-ℝ z x y x~y)
+      ( preserves-le-right-sim-ℝ z x y x~y)
+
+  apart-sim-ℝ :
+    {l1 l2 l3 l4 : Level} {x : ℝ l1} {x' : ℝ l2} {y : ℝ l3} {y' : ℝ l4} →
+    sim-ℝ x x' → sim-ℝ y y' → apart-ℝ x y → apart-ℝ x' y'
+  apart-sim-ℝ {x = x} {x' = x'} {y = y} {y' = y'} x~x' y~y' x#y =
+    apart-left-sim-ℝ
+      ( y')
+      ( x)
+      ( x')
+      ( x~x')
+      ( apart-right-sim-ℝ
+        ( x)
+        ( y)
+        ( y')
+        ( y~y')
+        ( x#y))
+```

From 80ae32beeb18b45d67d18be83cf78b54a800f8b2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 10:59:56 -0800
Subject: [PATCH 057/134] Progress

---
 ...ddition-positive-rational-numbers.lagda.md |  10 ++
 src/metric-spaces.lagda.md                    |   1 +
 .../apartness-located-metric-spaces.lagda.md  | 170 ++++++++++++++++++
 .../located-metric-spaces.lagda.md            |  36 +++-
 .../metrics-of-metric-spaces.lagda.md         |  41 +++++
 src/real-numbers.lagda.md                     |   1 +
 .../apartness-real-numbers.lagda.md           |  64 +++++++
 ...ated-metric-space-of-real-numbers.lagda.md |  39 ++++
 8 files changed, 360 insertions(+), 2 deletions(-)
 create mode 100644 src/metric-spaces/apartness-located-metric-spaces.lagda.md
 create mode 100644 src/real-numbers/located-metric-space-of-real-numbers.lagda.md

diff --git a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
index 85668704afe..fde4689bf76 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -226,6 +226,16 @@ module _
     eq-add-split-ℚ⁺
 
   abstract
+    le-left-summand-split-ℚ⁺ : le-ℚ⁺ left-summand-split-ℚ⁺ x
+    le-left-summand-split-ℚ⁺ = le-mediant-zero-ℚ⁺ x
+
+    le-right-summand-split-ℚ⁺ : le-ℚ⁺ right-summand-split-ℚ⁺ x
+    le-right-summand-split-ℚ⁺ =
+      tr
+        ( le-ℚ⁺ right-summand-split-ℚ⁺)
+        ( eq-add-split-ℚ⁺)
+        ( le-right-add-ℚ⁺ left-summand-split-ℚ⁺ right-summand-split-ℚ⁺)
+
     le-add-split-ℚ⁺ :
       (p q r s : ℚ) →
       le-ℚ p (q +ℚ rational-ℚ⁺ left-summand-split-ℚ⁺) →
diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index 4dd23a4e057..d08a6ffd1f1 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -56,6 +56,7 @@ metric space, `N d₂ x y` [or](foundation.disjunction.md)
 ```agda
 module metric-spaces where
 
+open import metric-spaces.apartness-located-metric-spaces public
 open import metric-spaces.approximations-located-metric-spaces public
 open import metric-spaces.approximations-metric-spaces public
 open import metric-spaces.bounded-distance-decompositions-of-metric-spaces public
diff --git a/src/metric-spaces/apartness-located-metric-spaces.lagda.md b/src/metric-spaces/apartness-located-metric-spaces.lagda.md
new file mode 100644
index 00000000000..4aeb3403b34
--- /dev/null
+++ b/src/metric-spaces/apartness-located-metric-spaces.lagda.md
@@ -0,0 +1,170 @@
+# Apartness in located metric spaces
+
+```agda
+module metric-spaces.apartness-located-metric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.strict-inequality-positive-rational-numbers
+
+open import foundation.apartness-relations
+open import foundation.binary-relations
+open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.empty-types
+open import foundation.existential-quantification
+open import foundation.negation
+open import foundation.propositional-truncations
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import logic.functoriality-existential-quantification
+
+open import metric-spaces.located-metric-spaces
+```
+
+</details>
+
+## Idea
+
+A [located metric space](metric-spaces.located-metric-spaces.md) `M` induces an
+[apartness relation](foundation.apartness-relations.md) `#` such that `x # y` if
+there [exists](foundation.existential-quantification.md) an `ε : ℚ⁺` such that
+`x` and `y` are [not](foundation.negation.md) in an `ε`-neighborhood of each
+other.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (M : Located-Metric-Space l1 l2)
+  where
+
+  apart-prop-Located-Metric-Space :
+    Relation-Prop l2 (type-Located-Metric-Space M)
+  apart-prop-Located-Metric-Space x y =
+    ∃ ℚ⁺ (λ ε → ¬' (neighborhood-prop-Located-Metric-Space M ε x y))
+
+  apart-Located-Metric-Space : Relation l2 (type-Located-Metric-Space M)
+  apart-Located-Metric-Space =
+    type-Relation-Prop apart-prop-Located-Metric-Space
+```
+
+## Properties
+
+### The apartness relation of a located metric space is antireflexive
+
+```agda
+module _
+  {l1 l2 : Level} (M : Located-Metric-Space l1 l2)
+  where
+
+  abstract
+    is-antireflexive-apart-Located-Metric-Space :
+      is-antireflexive (apart-prop-Located-Metric-Space M)
+    is-antireflexive-apart-Located-Metric-Space x =
+      elim-exists
+        ( empty-Prop)
+        ( λ ε ¬Nεxx → ¬Nεxx (refl-neighborhood-Located-Metric-Space M ε x))
+```
+
+### The apartness relation of a located metric space is symmetric
+
+```agda
+module _
+  {l1 l2 : Level} (M : Located-Metric-Space l1 l2)
+  where
+
+  abstract
+    is-symmetric-apart-Located-Metric-Space :
+      is-symmetric (apart-Located-Metric-Space M)
+    is-symmetric-apart-Located-Metric-Space x y =
+      map-tot-exists
+        ( λ ε ¬Nεxy Nεyx →
+          ¬Nεxy (symmetric-neighborhood-Located-Metric-Space M ε y x Nεyx))
+```
+
+### The apartness relation of a located metric space is cotransitive
+
+```agda
+module _
+  {l1 l2 : Level} (M : Located-Metric-Space l1 l2)
+  where
+
+  abstract
+    is-cotransitive-apart-Located-Metric-Space :
+      is-cotransitive (apart-prop-Located-Metric-Space M)
+    is-cotransitive-apart-Located-Metric-Space x y z x#y =
+      let
+        motive =
+          apart-prop-Located-Metric-Space M x z ∨
+          apart-prop-Located-Metric-Space M y z
+        open do-syntax-trunc-Prop motive
+      in do
+        (dxy , ¬Ndxy) ← x#y
+        let (δ , ε , δ+ε=dxy) = split-ℚ⁺ dxy
+        elim-disjunction
+          ( motive)
+          ( λ ¬Nδ'xz →
+            inl-disjunction (intro-exists (mediant-zero-ℚ⁺ δ) ¬Nδ'xz))
+          ( λ Nδxz →
+            elim-disjunction
+              ( motive)
+              ( λ ¬Nε'yz →
+                inr-disjunction (intro-exists (mediant-zero-ℚ⁺ ε) ¬Nε'yz))
+              ( λ Nεyz →
+                ex-falso
+                  ( ¬Ndxy
+                    ( tr
+                      ( λ d → neighborhood-Located-Metric-Space M d x y)
+                      ( δ+ε=dxy)
+                      ( triangular-neighborhood-Located-Metric-Space M
+                        ( x)
+                        ( z)
+                        ( y)
+                        ( δ)
+                        ( ε)
+                        ( symmetric-neighborhood-Located-Metric-Space M
+                          ( ε)
+                          ( y)
+                          ( z)
+                          ( Nεyz))
+                        ( Nδxz)))))
+              ( is-located-Located-Metric-Space M
+                ( y)
+                ( z)
+                ( mediant-zero-ℚ⁺ ε)
+                ( ε)
+                ( le-mediant-zero-ℚ⁺ ε)))
+          ( is-located-Located-Metric-Space M
+            ( x)
+            ( z)
+            ( mediant-zero-ℚ⁺ δ)
+            ( δ)
+            ( le-mediant-zero-ℚ⁺ δ))
+```
+
+### The apartness relation on located metric spaces is an apartness relation
+
+```agda
+module _
+  {l1 l2 : Level} (M : Located-Metric-Space l1 l2)
+  where
+
+  is-apartness-relation-apart-Located-Metric-Space :
+    is-apartness-relation (apart-prop-Located-Metric-Space M)
+  is-apartness-relation-apart-Located-Metric-Space =
+    ( is-antireflexive-apart-Located-Metric-Space M ,
+      is-symmetric-apart-Located-Metric-Space M ,
+      is-cotransitive-apart-Located-Metric-Space M)
+
+  apartness-relation-Located-Metric-Space :
+    Apartness-Relation l2 (type-Located-Metric-Space M)
+  apartness-relation-Located-Metric-Space =
+    ( apart-prop-Located-Metric-Space M ,
+      is-apartness-relation-apart-Located-Metric-Space)
+```
diff --git a/src/metric-spaces/located-metric-spaces.lagda.md b/src/metric-spaces/located-metric-spaces.lagda.md
index 1fef763229a..60354eb0195 100644
--- a/src/metric-spaces/located-metric-spaces.lagda.md
+++ b/src/metric-spaces/located-metric-spaces.lagda.md
@@ -7,12 +7,14 @@ module metric-spaces.located-metric-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
+open import foundation.binary-relations
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
@@ -101,9 +103,39 @@ module _
   type-Located-Metric-Space =
     type-Metric-Space metric-space-Located-Metric-Space
 
-  is-located-metric-space-Located-Metric-Space :
+  neighborhood-prop-Located-Metric-Space :
+    ℚ⁺ → Relation-Prop l2 type-Located-Metric-Space
+  neighborhood-prop-Located-Metric-Space =
+    neighborhood-prop-Metric-Space metric-space-Located-Metric-Space
+
+  neighborhood-Located-Metric-Space : ℚ⁺ → Relation l2 type-Located-Metric-Space
+  neighborhood-Located-Metric-Space =
+    neighborhood-Metric-Space metric-space-Located-Metric-Space
+
+  refl-neighborhood-Located-Metric-Space :
+    (d : ℚ⁺) (x : type-Located-Metric-Space) →
+    neighborhood-Located-Metric-Space d x x
+  refl-neighborhood-Located-Metric-Space =
+    refl-neighborhood-Metric-Space metric-space-Located-Metric-Space
+
+  symmetric-neighborhood-Located-Metric-Space :
+    (d : ℚ⁺) (x y : type-Located-Metric-Space) →
+    neighborhood-Located-Metric-Space d x y →
+    neighborhood-Located-Metric-Space d y x
+  symmetric-neighborhood-Located-Metric-Space =
+    symmetric-neighborhood-Metric-Space metric-space-Located-Metric-Space
+
+  triangular-neighborhood-Located-Metric-Space :
+    (x y z : type-Located-Metric-Space) (d₁ d₂ : ℚ⁺) →
+    neighborhood-Located-Metric-Space d₂ y z →
+    neighborhood-Located-Metric-Space d₁ x y →
+    neighborhood-Located-Metric-Space (d₁ +ℚ⁺ d₂) x z
+  triangular-neighborhood-Located-Metric-Space =
+    triangular-neighborhood-Metric-Space metric-space-Located-Metric-Space
+
+  is-located-Located-Metric-Space :
     is-located-Metric-Space metric-space-Located-Metric-Space
-  is-located-metric-space-Located-Metric-Space = pr2 X
+  is-located-Located-Metric-Space = pr2 X
 
   set-Located-Metric-Space : Set l1
   set-Located-Metric-Space = set-Metric-Space metric-space-Located-Metric-Space
diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index 641b00f2ba4..c87bef666af 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -11,6 +11,7 @@ open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.equivalences
+open import foundation.functoriality-disjunction
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
@@ -20,10 +21,13 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.equality-of-metric-spaces
+open import metric-spaces.located-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.metrics
 
+open import real-numbers.dedekind-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -206,6 +210,43 @@ module _
   isometric-equiv-metric-is-metric-of-Metric-Space = (id-equiv , is-metric-M-ρ)
 ```
 
+### If `ρ` is a metric for a metric space `M`, then `M` is located
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  (ρ : distance-function l2 (set-Metric-Space M))
+  (is-metric-M-ρ : is-metric-of-Metric-Space M ρ)
+  where
+
+  abstract
+    is-located-is-metric-of-Metric-Space : is-located-Metric-Space M
+    is-located-is-metric-of-Metric-Space x y δ ε δ<ε =
+      map-disjunction
+        ( λ δ<ρxy Nδxy →
+          not-le-leq-ℝ
+            ( real-ℝ⁰⁺ (ρ x y))
+            ( real-ℚ⁺ δ)
+            ( forward-implication (is-metric-M-ρ δ x y) Nδxy)
+            ( le-real-is-in-lower-cut-ℚ
+              ( rational-ℚ⁺ δ)
+              ( real-ℝ⁰⁺ (ρ x y))
+              ( δ<ρxy)))
+        ( λ ρxy<ε →
+          backward-implication
+            ( is-metric-M-ρ ε x y)
+            ( leq-le-ℝ _ _
+              ( le-real-is-in-upper-cut-ℚ
+                ( rational-ℚ⁺ ε)
+                ( real-ℝ⁰⁺ (ρ x y))
+                ( ρxy<ε))))
+        ( is-located-lower-upper-cut-ℝ
+          ( real-ℝ⁰⁺ (ρ x y))
+          ( rational-ℚ⁺ δ)
+          ( rational-ℚ⁺ ε)
+          ( δ<ε))
+```
+
 ### If `ρ` is a metric for a metric space `M` at the appropriate universe level, then `M` is equal to the metric space induced by `ρ`
 
 ```agda
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 3e2872c1e15..51c2d072482 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -37,6 +37,7 @@ open import real-numbers.large-multiplicative-monoid-of-real-numbers public
 open import real-numbers.large-ring-of-real-numbers public
 open import real-numbers.limits-sequences-real-numbers public
 open import real-numbers.lipschitz-continuity-multiplication-real-numbers public
+open import real-numbers.located-metric-space-of-real-numbers public
 open import real-numbers.lower-dedekind-real-numbers public
 open import real-numbers.maximum-finite-families-real-numbers public
 open import real-numbers.maximum-inhabited-finitely-enumerable-subsets-real-numbers public
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 31770cd4ac0..209984acc3d 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -13,8 +13,12 @@ open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.functoriality-disjunction
+open import foundation.dependent-pair-types
+open import foundation.binary-relations
 open import foundation.identity-types
+open import foundation.existential-quantification
 open import foundation.large-apartness-relations
+open import foundation.propositional-truncations
 open import foundation.large-binary-relations
 open import foundation.negated-equality
 open import foundation.negation
@@ -28,7 +32,15 @@ open import real-numbers.dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequalities-addition-real-numbers
+open import logic.functoriality-existential-quantification
+open import metric-spaces.apartness-located-metric-spaces
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.located-metric-space-of-real-numbers
 open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.metric-space-of-real-numbers
 ```
 
 </details>
@@ -191,3 +203,55 @@ abstract
         ( y~y')
         ( x#y))
 ```
+
+### Apartness on the real numbers is equivalent to apartness in the located metric space of real numbers
+
+```agda
+apart-prop-located-metric-space-ℝ : {l : Level} → Relation-Prop l (ℝ l)
+apart-prop-located-metric-space-ℝ {l} =
+  apart-prop-Located-Metric-Space (located-metric-space-ℝ l)
+
+apart-located-metric-space-ℝ : {l : Level} → Relation l (ℝ l)
+apart-located-metric-space-ℝ {l} =
+  apart-Located-Metric-Space (located-metric-space-ℝ l)
+
+module _
+  {l : Level} (x y : ℝ l)
+  where
+
+  abstract opaque
+    apart-located-metric-space-apart-ℝ :
+      apart-ℝ x y → apart-located-metric-space-ℝ x y
+    apart-located-metric-space-apart-ℝ =
+      elim-disjunction
+        ( apart-prop-located-metric-space-ℝ x y)
+        ( λ x<y →
+          map-tot-exists
+            ( λ ε x+ε<y Nεxy →
+              not-leq-le-ℝ
+                ( x +ℝ real-ℚ⁺ ε)
+                ( y)
+                ( x+ε<y)
+                ( right-leq-real-bound-neighborhood-ℝ ε x y Nεxy))
+            ( exists-positive-rational-separation-le-ℝ x<y))
+        ( λ y<x →
+          map-tot-exists
+            ( λ ε y+ε<x Nεxy →
+              not-leq-le-ℝ
+                ( y +ℝ real-ℚ⁺ ε)
+                ( x)
+                ( y+ε<x)
+                ( left-leq-real-bound-neighborhood-ℝ ε x y Nεxy))
+            ( exists-positive-rational-separation-le-ℝ y<x))
+
+    apart-apart-located-metric-space-ℝ :
+      apart-located-metric-space-ℝ x y → apart-ℝ x y
+    apart-apart-located-metric-space-ℝ =
+      let
+        motive = apart-prop-ℝ x y
+        open do-syntax-trunc-Prop motive
+      in do
+        ( ε , ¬Nεxy) ← ?
+        {!   !}
+
+```
diff --git a/src/real-numbers/located-metric-space-of-real-numbers.lagda.md b/src/real-numbers/located-metric-space-of-real-numbers.lagda.md
new file mode 100644
index 00000000000..27314945626
--- /dev/null
+++ b/src/real-numbers/located-metric-space-of-real-numbers.lagda.md
@@ -0,0 +1,39 @@
+# The located metric space of real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.located-metric-space-of-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import metric-spaces.located-metric-spaces
+open import metric-spaces.metrics-of-metric-spaces
+open import real-numbers.distance-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import foundation.universe-levels
+open import foundation.dependent-pair-types
+```
+
+## Idea
+
+The [metric space of real numbers](real-numbers.metric-space-of-real-numbers.md)
+is [located](metric-spaces.located-metric-spaces.md).
+
+## Definition
+
+```agda
+abstract
+  is-located-metric-space-ℝ :
+    (l : Level) → is-located-Metric-Space (metric-space-ℝ l)
+  is-located-metric-space-ℝ l =
+    is-located-is-metric-of-Metric-Space
+      ( metric-space-ℝ l)
+      ( nonnegative-dist-ℝ)
+      ( dist-is-metric-of-metric-space-ℝ l)
+
+located-metric-space-ℝ : (l : Level) → Located-Metric-Space (lsuc l) l
+located-metric-space-ℝ l = (metric-space-ℝ l , is-located-metric-space-ℝ l)
+```

From b0404c0d9896a76c738fe066ff86532dac326b6a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:02:07 -0800
Subject: [PATCH 058/134] Progress

---
 src/real-numbers/absolute-value-real-numbers.lagda.md | 1 +
 src/real-numbers/apartness-real-numbers.lagda.md      | 9 ++++-----
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index b2b49882e2c..42441789952 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -34,6 +34,7 @@ open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 209984acc3d..12c3d58a6ca 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -31,7 +31,6 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
 open import logic.functoriality-existential-quantification
 open import metric-spaces.apartness-located-metric-spaces
 
@@ -163,8 +162,8 @@ abstract
   preserves-apart-neg-ℝ x y =
     elim-disjunction
       ( apart-prop-ℝ _ _)
-      ( inr-disjunction ∘ neg-le-ℝ)
-      ( inl-disjunction ∘ neg-le-ℝ)
+      ( inr-disjunction ∘ neg-le-ℝ _ _)
+      ( inl-disjunction ∘ neg-le-ℝ _ _)
 ```
 
 ### Apartness is preserved by similarity
@@ -246,12 +245,12 @@ module _
 
     apart-apart-located-metric-space-ℝ :
       apart-located-metric-space-ℝ x y → apart-ℝ x y
-    apart-apart-located-metric-space-ℝ =
+    apart-apart-located-metric-space-ℝ x#y =
       let
         motive = apart-prop-ℝ x y
         open do-syntax-trunc-Prop motive
       in do
-        ( ε , ¬Nεxy) ← ?
+        ( ε , ¬Nεxy) ← x#y
         {!   !}
 
 ```

From 53d3a169bc1da5aed5d303866030de86273f9007 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:52:36 -0800
Subject: [PATCH 059/134] Progress

---
 .../metrics-of-metric-spaces.lagda.md         |  15 +--
 .../absolute-value-real-numbers.lagda.md      |  31 ++++-
 .../apartness-real-numbers.lagda.md           | 109 ++++++++++++++----
 ...ated-metric-space-of-real-numbers.lagda.md |   8 +-
 .../nonzero-real-numbers.lagda.md             |  84 +++++++++-----
 .../positive-real-numbers.lagda.md            |  49 ++++++++
 6 files changed, 229 insertions(+), 67 deletions(-)

diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index d61d43e03f8..49b8a73e747 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -25,12 +25,11 @@ open import metric-spaces.located-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.metrics
 
-open import real-numbers.dedekind-real-numbers
-open import real-numbers.nonnegative-real-numbers
-open import real-numbers.rational-real-numbers
 open import real-numbers.addition-nonnegative-real-numbers
+open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
 open import real-numbers.saturation-inequality-nonnegative-real-numbers
 open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.strict-inequalities-addition-real-numbers
@@ -235,22 +234,16 @@ module _
             ( real-ℚ⁺ δ)
             ( forward-implication (is-metric-M-ρ δ x y) Nδxy)
             ( le-real-is-in-lower-cut-ℚ
-              ( rational-ℚ⁺ δ)
               ( real-ℝ⁰⁺ (ρ x y))
               ( δ<ρxy)))
         ( λ ρxy<ε →
           backward-implication
             ( is-metric-M-ρ ε x y)
-            ( leq-le-ℝ _ _
+            ( leq-le-ℝ
               ( le-real-is-in-upper-cut-ℚ
-                ( rational-ℚ⁺ ε)
                 ( real-ℝ⁰⁺ (ρ x y))
                 ( ρxy<ε))))
-        ( is-located-lower-upper-cut-ℝ
-          ( real-ℝ⁰⁺ (ρ x y))
-          ( rational-ℚ⁺ δ)
-          ( rational-ℚ⁺ ε)
-          ( δ<ε))
+        ( is-located-lower-upper-cut-ℝ (real-ℝ⁰⁺ (ρ x y)) δ<ε)
 ```
 
 ### If `ρ` is a metric for a metric space `M` at the appropriate universe level, then `M` is equal to the metric space induced by `ρ`
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index cb70eb81914..c3fa5795e3d 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -28,6 +28,7 @@ open import metric-spaces.short-functions-metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
 open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.isometry-negation-real-numbers
@@ -35,8 +36,8 @@ open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
+open import real-numbers.negative-real-numbers
 open import real-numbers.nonnegative-real-numbers
-open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
@@ -44,6 +45,7 @@ open import real-numbers.saturation-inequality-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.square-roots-nonnegative-real-numbers
 open import real-numbers.squares-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
@@ -134,11 +136,24 @@ abstract opaque
           ( x)
           ( 0≤x)
           ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ 0≤x))))
+```
+
+### The absolute value of a positive real number is itself
 
+```agda
+abstract
   abs-real-ℝ⁺ : {l : Level} (x : ℝ⁺ l) → abs-ℝ (real-ℝ⁺ x) ＝ real-ℝ⁺ x
   abs-real-ℝ⁺ x = abs-real-ℝ⁰⁺ (nonnegative-ℝ⁺ x)
 ```
 
+### The absolute value of a negative real number is its negation
+
+```agda
+abstract
+  abs-real-ℝ⁻ : {l : Level} (x : ℝ⁻ l) → abs-ℝ (real-ℝ⁻ x) ＝ neg-ℝ (real-ℝ⁻ x)
+  abs-real-ℝ⁻ x⁻@(x , _) = inv (abs-neg-ℝ x) ∙ abs-real-ℝ⁺ (neg-ℝ⁻ x⁻)
+```
+
 ### `x` is between `-|x|` and `|x|`
 
 ```agda
@@ -404,3 +419,17 @@ abstract
       ＝ abs-ℝ x *ℝ abs-ℝ y
         by inv (ap-mul-ℝ (eq-abs-sqrt-square-ℝ x) (eq-abs-sqrt-square-ℝ y))
 ```
+
+### For any `ε : ℚ⁺`, `(abs-ℝ x - ε < x) ∨ (abs-ℝ x - ε < -x)`
+
+```agda
+abstract opaque
+  unfolding abs-ℝ
+
+  approximate-below-abs-ℝ :
+    {l : Level} (x : ℝ l) (ε : ℚ⁺) →
+    disjunction-type
+      ( le-ℝ (abs-ℝ x -ℝ real-ℚ⁺ ε) x)
+      ( le-ℝ (abs-ℝ x -ℝ real-ℚ⁺ ε) (neg-ℝ x))
+  approximate-below-abs-ℝ x = approximate-below-max-ℝ x (neg-ℝ x)
+```
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index f7d84a46a63..75d5f030b22 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -7,40 +7,39 @@ module real-numbers.apartness-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.apartness-relations
+open import foundation.binary-relations
 open import foundation.binary-transport
+open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.functoriality-disjunction
-open import foundation.dependent-pair-types
-open import foundation.binary-relations
-open import foundation.identity-types
-open import foundation.existential-quantification
 open import foundation.large-apartness-relations
-open import foundation.propositional-truncations
 open import foundation.large-binary-relations
+open import foundation.logical-equivalences
 open import foundation.negated-equality
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import real-numbers.absolute-value-real-numbers
-open import real-numbers.addition-real-numbers
-open import real-numbers.dedekind-real-numbers
-open import real-numbers.negation-real-numbers
-open import real-numbers.similarity-real-numbers
 open import logic.functoriality-existential-quantification
+
 open import metric-spaces.apartness-located-metric-spaces
 
+open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
-open import real-numbers.rational-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.distance-real-numbers
 open import real-numbers.located-metric-space-of-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.nonzero-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
-open import real-numbers.metric-space-of-real-numbers
 ```
 
 </details>
@@ -204,6 +203,63 @@ abstract
         ( x#y))
 ```
 
+### Two real numbers are apart if and only if their difference is nonzero
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  where
+
+  abstract
+    is-nonzero-diff-is-apart-ℝ : apart-ℝ x y → is-nonzero-ℝ (x -ℝ y)
+    is-nonzero-diff-is-apart-ℝ x#y =
+      apart-right-sim-ℝ
+        ( x -ℝ y)
+        ( y -ℝ y)
+        ( zero-ℝ)
+        ( right-inverse-law-add-ℝ y)
+        ( preserves-apart-right-add-ℝ (neg-ℝ y) x y x#y)
+
+    apart-is-nonzero-diff-ℝ : is-nonzero-ℝ (x -ℝ y) → apart-ℝ x y
+    apart-is-nonzero-diff-ℝ x-y#0 =
+      apart-sim-ℝ
+        ( cancel-right-diff-add-ℝ x y)
+        ( sim-eq-ℝ (left-unit-law-add-ℝ y))
+        ( preserves-apart-right-add-ℝ y _ _ x-y#0)
+
+  nonzero-diff-apart-ℝ : apart-ℝ x y → nonzero-ℝ (l1 ⊔ l2)
+  nonzero-diff-apart-ℝ x#y = (x -ℝ y , is-nonzero-diff-is-apart-ℝ x#y)
+```
+
+### Apartness on the real numbers is equivalent to having a positive distance
+
+```agda
+module _
+  {l1 l2 : Level}
+  (x : ℝ l1)
+  (y : ℝ l2)
+  where
+
+  abstract
+    apart-is-positive-dist-ℝ : is-positive-ℝ (dist-ℝ x y) → apart-ℝ x y
+    apart-is-positive-dist-ℝ 0<|x-y| =
+      apart-is-nonzero-diff-ℝ
+        ( x)
+        ( y)
+        ( is-nonzero-is-positive-abs-ℝ (x -ℝ y) 0<|x-y|)
+
+    is-positive-dist-apart-ℝ : apart-ℝ x y → is-positive-ℝ (dist-ℝ x y)
+    is-positive-dist-apart-ℝ x#y =
+      is-positive-abs-is-nonzero-ℝ
+        ( x -ℝ y)
+        ( is-nonzero-diff-is-apart-ℝ x y x#y)
+
+  apart-iff-is-positive-dist-ℝ :
+    apart-ℝ x y ↔ is-positive-ℝ (dist-ℝ x y)
+  apart-iff-is-positive-dist-ℝ =
+    ( is-positive-dist-apart-ℝ , apart-is-positive-dist-ℝ)
+```
+
 ### Apartness on the real numbers is equivalent to apartness in the located metric space of real numbers
 
 ```agda
@@ -216,10 +272,11 @@ apart-located-metric-space-ℝ {l} =
   apart-Located-Metric-Space (located-metric-space-ℝ l)
 
 module _
-  {l : Level} (x y : ℝ l)
+  {l : Level}
+  (x y : ℝ l)
   where
 
-  abstract opaque
+  abstract
     apart-located-metric-space-apart-ℝ :
       apart-ℝ x y → apart-located-metric-space-ℝ x y
     apart-located-metric-space-apart-ℝ =
@@ -247,11 +304,17 @@ module _
     apart-apart-located-metric-space-ℝ :
       apart-located-metric-space-ℝ x y → apart-ℝ x y
     apart-apart-located-metric-space-ℝ x#y =
-      let
-        motive = apart-prop-ℝ x y
-        open do-syntax-trunc-Prop motive
-      in do
-        ( ε , ¬Nεxy) ← x#y
-        {!   !}
-
+      apart-is-positive-dist-ℝ
+        ( x)
+        ( y)
+        ( is-positive-exists-not-le-positive-rational-ℝ
+          ( dist-ℝ x y)
+          ( map-tot-exists
+            ( λ ε ¬Nεxy |x-y|≤ε → ¬Nεxy (neighborhood-dist-ℝ ε x y |x-y|≤ε))
+            ( x#y)))
+
+  apart-iff-apart-located-metric-space-ℝ :
+    apart-ℝ x y ↔ apart-located-metric-space-ℝ x y
+  apart-iff-apart-located-metric-space-ℝ =
+    ( apart-located-metric-space-apart-ℝ , apart-apart-located-metric-space-ℝ)
 ```
diff --git a/src/real-numbers/located-metric-space-of-real-numbers.lagda.md b/src/real-numbers/located-metric-space-of-real-numbers.lagda.md
index 27314945626..92ca8636c54 100644
--- a/src/real-numbers/located-metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/located-metric-space-of-real-numbers.lagda.md
@@ -9,14 +9,18 @@ module real-numbers.located-metric-space-of-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
 open import metric-spaces.located-metric-spaces
 open import metric-spaces.metrics-of-metric-spaces
+
 open import real-numbers.distance-real-numbers
 open import real-numbers.metric-space-of-real-numbers
-open import foundation.universe-levels
-open import foundation.dependent-pair-types
 ```
 
+</details>
+
 ## Idea
 
 The [metric space of real numbers](real-numbers.metric-space-of-real-numbers.md)
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index 3a8df09cc9c..f2d2fe8590e 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -9,9 +9,11 @@ module real-numbers.nonzero-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.identity-types
+open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
@@ -19,14 +21,15 @@ open import foundation.universe-levels
 
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
-open import real-numbers.apartness-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -45,7 +48,7 @@ equivalently if it is [negative](real-numbers.negative-real-numbers.md)
 
 ```agda
 is-nonzero-prop-ℝ : {l : Level} → ℝ l → Prop l
-is-nonzero-prop-ℝ x = apart-prop-ℝ x zero-ℝ
+is-nonzero-prop-ℝ x = is-negative-prop-ℝ x ∨ is-positive-prop-ℝ x
 
 is-nonzero-ℝ : {l : Level} → ℝ l → UU l
 is-nonzero-ℝ x = type-Prop (is-nonzero-prop-ℝ x)
@@ -94,34 +97,6 @@ eq-nonzero-ℝ :
 eq-nonzero-ℝ _ _ = eq-type-subtype is-nonzero-prop-ℝ
 ```
 
-### Two real numbers are apart if and only if their difference is nonzero
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
-  where
-
-  abstract
-    is-nonzero-diff-is-apart-ℝ : apart-ℝ x y → is-nonzero-ℝ (x -ℝ y)
-    is-nonzero-diff-is-apart-ℝ x#y =
-      apart-right-sim-ℝ
-        ( x -ℝ y)
-        ( y -ℝ y)
-        ( zero-ℝ)
-        ( right-inverse-law-add-ℝ y)
-        ( preserves-apart-right-add-ℝ (neg-ℝ y) x y x#y)
-
-    is-apart-is-nonzero-diff-ℝ : is-nonzero-ℝ (x -ℝ y) → apart-ℝ x y
-    is-apart-is-nonzero-diff-ℝ x-y#0 =
-      apart-sim-ℝ
-        ( cancel-right-diff-add-ℝ x y)
-        ( sim-eq-ℝ (left-unit-law-add-ℝ y))
-        ( preserves-apart-right-add-ℝ y _ _ x-y#0)
-
-  nonzero-diff-apart-ℝ : apart-ℝ x y → nonzero-ℝ (l1 ⊔ l2)
-  nonzero-diff-apart-ℝ x#y = (x -ℝ y , is-nonzero-diff-is-apart-ℝ x#y)
-```
-
 ### If `x < y`, then `y - x` is nonzero
 
 ```agda
@@ -138,3 +113,52 @@ nonzero-diff-le-abs-ℝ :
 nonzero-diff-le-abs-ℝ {x = x} {y = y} |x|<y =
   nonzero-diff-le-ℝ (concatenate-leq-le-ℝ x (abs-ℝ x) y (leq-abs-ℝ x) |x|<y)
 ```
+
+### `x` is nonzero if and only if `|x|` is positive
+
+```agda
+module _
+  {l : Level} (x : ℝ l)
+  where
+
+  abstract opaque
+    unfolding abs-ℝ
+
+    is-positive-abs-is-nonzero-ℝ : is-nonzero-ℝ x → is-positive-ℝ (abs-ℝ x)
+    is-positive-abs-is-nonzero-ℝ =
+      elim-disjunction
+        ( is-positive-prop-ℝ (abs-ℝ x))
+        ( λ is-neg-x →
+          inv-tr
+            ( is-positive-ℝ)
+            ( abs-real-ℝ⁻ (x , is-neg-x))
+            ( neg-is-negative-ℝ x is-neg-x))
+        ( λ is-pos-x →
+          inv-tr
+            ( is-positive-ℝ)
+            ( abs-real-ℝ⁺ (x , is-pos-x))
+            ( is-pos-x))
+
+    is-nonzero-is-positive-abs-ℝ : is-positive-ℝ (abs-ℝ x) → is-nonzero-ℝ x
+    is-nonzero-is-positive-abs-ℝ 0<|x| =
+      let
+        open do-syntax-trunc-Prop (is-nonzero-prop-ℝ x)
+      in do
+        (ε , 0+ε<|x|) ← exists-positive-rational-separation-le-ℝ 0<|x|
+        let
+          0<|x|-ε = le-transpose-left-add-ℝ zero-ℝ (real-ℚ⁺ ε) (abs-ℝ x) 0+ε<|x|
+        elim-disjunction
+          ( is-nonzero-prop-ℝ x)
+          ( λ |x|-ε<x →
+            is-nonzero-is-positive-ℝ (transitive-le-ℝ _ _ _ |x|-ε<x 0<|x|-ε))
+          ( λ |x|-ε<-x →
+            is-nonzero-is-negative-ℝ
+              ( binary-tr
+                ( le-ℝ)
+                ( neg-neg-ℝ x)
+                ( neg-zero-ℝ)
+                ( transitive-le-ℝ _ _ _
+                  ( neg-le-ℝ 0<|x|-ε)
+                  ( neg-le-ℝ |x|-ε<-x))))
+          ( approximate-below-abs-ℝ x ε)
+```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 668e1644d95..87d7fff6a26 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -14,15 +14,19 @@ open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.binary-transport
 open import foundation.conjunction
 open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
+open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
@@ -36,6 +40,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
@@ -135,6 +140,16 @@ eq-ℝ⁺ : {l : Level} (x y : ℝ⁺ l) → (real-ℝ⁺ x ＝ real-ℝ⁺ y) 
 eq-ℝ⁺ _ _ = eq-type-subtype is-positive-prop-ℝ
 ```
 
+### A real number is positive if it is greater than a positive real number
+
+```agda
+abstract
+  is-positive-le-ℝ⁺ :
+    {l1 l2 : Level} (x : ℝ⁺ l1) (y : ℝ l2) → le-ℝ (real-ℝ⁺ x) y →
+    is-positive-ℝ y
+  is-positive-le-ℝ⁺ (x , 0<x) y x<y = transitive-le-ℝ zero-ℝ x y x<y 0<x
+```
+
 ### A real number is positive if and only if zero is in its lower cut
 
 ```agda
@@ -284,6 +299,40 @@ one-ℝ⁺ : ℝ⁺ lzero
 one-ℝ⁺ = positive-real-ℚ⁺ one-ℚ⁺
 ```
 
+### `x` is positive if and only if there exists a positive rational number it is not less than or equal to
+
+```agda
+module _
+  {l : Level} (x : ℝ l)
+  where
+
+  abstract
+    is-positive-exists-not-le-positive-rational-ℝ :
+      exists ℚ⁺ (λ q → ¬' (leq-prop-ℝ x (real-ℚ⁺ q))) →
+      is-positive-ℝ x
+    is-positive-exists-not-le-positive-rational-ℝ ∃q =
+      let open do-syntax-trunc-Prop (is-positive-prop-ℝ x)
+      in do
+        (q⁺@(q , _) , x≰q) ← ∃q
+        let p⁺@(p , _) = mediant-zero-ℚ⁺ q⁺
+        elim-disjunction
+          ( is-positive-prop-ℝ x)
+          ( is-positive-le-ℝ⁺ (positive-real-ℚ⁺ p⁺) x)
+          ( λ x<q → ex-falso (x≰q (leq-le-ℝ x<q)))
+          ( is-located-le-ℝ x p q (le-mediant-zero-ℚ⁺ q⁺))
+
+    exists-not-le-positive-rational-is-positive-ℝ :
+      is-positive-ℝ x →
+      exists ℚ⁺ (λ q → ¬' (leq-prop-ℝ x (real-ℚ⁺ q)))
+    exists-not-le-positive-rational-is-positive-ℝ 0<x =
+      let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → ¬' (leq-prop-ℝ x (real-ℚ⁺ q))))
+      in do
+        (q , 0<q , q<x) ← dense-rational-le-ℝ _ _ 0<x
+        intro-exists
+          ( q , is-positive-le-zero-ℚ (reflects-le-real-ℚ 0<q))
+          ( not-leq-le-ℝ _ _ q<x)
+```
+
 ### Any rational number in the upper cut of a positive real number is positive
 
 ```agda

From 139c5ace7b8ea8352b162266b8d22917e96cc85d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:53:39 -0800
Subject: [PATCH 060/134] Update
 src/real-numbers/dedekind-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/real-numbers/dedekind-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 0c1531e91fc..d846a3556de 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -312,7 +312,7 @@ module _
           ( lower-cut-ℝ x p)
           ( upper-cut-ℝ x q)
           ( pr2 I)
-          ( is-located-lower-upper-cut-ℝ x ( pr1 I)))
+          ( is-located-lower-upper-cut-ℝ x (pr1 I)))
 
   subset-lower-complement-upper-cut-lower-cut-ℝ :
     lower-cut-ℝ x ⊆ lower-complement-upper-cut-ℝ x

From d9aeb7fda4ea62f8502e4183945b386f770f1137 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:55:14 -0800
Subject: [PATCH 061/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../strict-inequalities-addition-real-numbers.lagda.md          | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 6db92caf63a..1c8789dd676 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -293,7 +293,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) (p : ℚ)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ
 
     le-split-add-rational-ℝ :

From defa2134d99b91f3aaf396f1200f6e6c47458ecd Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:55:32 -0800
Subject: [PATCH 062/134] Update
 src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../strict-inequalities-addition-real-numbers.lagda.md          | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 1c8789dd676..9403114e3a4 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -59,7 +59,7 @@ module _
   {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ le-ℝ
 
     preserves-le-right-add-ℝ : le-ℝ x y → le-ℝ (x +ℝ z) (y +ℝ z)

From fa31c521b7092f0e46eea6d3cf8904418d952271 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:55:52 -0800
Subject: [PATCH 063/134] Update
 src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../strict-inequalities-addition-real-numbers.lagda.md          | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 9403114e3a4..24417dab873 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -44,7 +44,7 @@ open import real-numbers.strict-inequality-real-numbers
 
 ## Idea
 
-This file describes lemmas about
+On this page we describe lemmas about
 [strict inequalities](real-numbers.strict-inequality-real-numbers.md) of
 [real numbers](real-numbers.dedekind-real-numbers.md) related to
 [addition](real-numbers.addition-real-numbers.md) and

From 45757e8f51bf8fccacfd913a0fed7853b7eb9726 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:57:09 -0800
Subject: [PATCH 064/134] Update
 src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../saturation-inequality-nonnegative-real-numbers.lagda.md     | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
index 037e54c2f2c..69f3a5fb114 100644
--- a/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
@@ -36,7 +36,7 @@ Despite being a property of
 [inequality of nonnegative real numbers](real-numbers.inequality-nonnegative-real-numbers.md),
 this is much easier to prove via
 [strict inequality](real-numbers.strict-inequality-nonnegative-real-numbers.md),
-so it is moved to its own file to prevent circular dependency.
+so this page is dedicated to just this property to prevent circular dependency.
 
 ## Definition
 

From 70cef9b710f6564e4559bff99dae0a07f25883ca Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:59:33 -0800
Subject: [PATCH 065/134] Rename files

---
 ...trics-of-metric-spaces-are-uniformly-continuous.lagda.md | 2 +-
 src/metric-spaces/metrics-of-metric-spaces.lagda.md         | 2 +-
 src/real-numbers.lagda.md                                   | 6 +++---
 src/real-numbers/absolute-value-real-numbers.lagda.md       | 2 +-
 src/real-numbers/addition-nonnegative-real-numbers.lagda.md | 4 ++--
 src/real-numbers/binary-maximum-real-numbers.lagda.md       | 2 +-
 .../cauchy-completeness-dedekind-real-numbers.lagda.md      | 2 +-
 src/real-numbers/distance-real-numbers.lagda.md             | 2 +-
 ...ualities-addition-and-subtraction-real-numbers.lagda.md} | 2 +-
 src/real-numbers/infima-families-real-numbers.lagda.md      | 2 +-
 .../inhabited-totally-bounded-subsets-real-numbers.lagda.md | 4 ++--
 src/real-numbers/isometry-addition-real-numbers.lagda.md    | 2 +-
 src/real-numbers/isometry-negation-real-numbers.lagda.md    | 2 +-
 ...ipschitz-continuity-multiplication-real-numbers.lagda.md | 2 +-
 .../maximum-finite-families-real-numbers.lagda.md           | 2 +-
 src/real-numbers/metric-space-of-real-numbers.lagda.md      | 4 ++--
 src/real-numbers/nonnegative-real-numbers.lagda.md          | 2 +-
 src/real-numbers/positive-real-numbers.lagda.md             | 2 +-
 .../saturation-inequality-real-numbers.lagda.md             | 2 +-
 ... => short-function-binary-maximum-real-numbers.lagda.md} | 4 ++--
 ...ualities-addition-and-subtraction-real-numbers.lagda.md} | 2 +-
 src/real-numbers/suprema-families-real-numbers.lagda.md     | 2 +-
 ...addition-subtraction-cuts-dedekind-real-numbers.lagda.md | 2 +-
 23 files changed, 29 insertions(+), 29 deletions(-)
 rename src/real-numbers/{inequalities-addition-real-numbers.lagda.md => inequalities-addition-and-subtraction-real-numbers.lagda.md} (98%)
 rename src/real-numbers/{short-binary-maximum-real-numbers.lagda.md => short-function-binary-maximum-real-numbers.lagda.md} (97%)
 rename src/real-numbers/{strict-inequalities-addition-real-numbers.lagda.md => strict-inequalities-addition-and-subtraction-real-numbers.lagda.md} (99%)

diff --git a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
index c33453db283..0eed19b1799 100644
--- a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
@@ -32,7 +32,7 @@ open import order-theory.large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index d113ff5bac4..271f974d53d 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -28,7 +28,7 @@ open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.saturation-inequality-nonnegative-real-numbers
 open import real-numbers.similarity-nonnegative-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 876511f445d..3b84f6c0b0a 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -23,7 +23,7 @@ open import real-numbers.difference-real-numbers public
 open import real-numbers.distance-real-numbers public
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers public
 open import real-numbers.finitely-enumerable-subsets-real-numbers public
-open import real-numbers.inequalities-addition-real-numbers public
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers public
 open import real-numbers.inequality-lower-dedekind-real-numbers public
 open import real-numbers.inequality-nonnegative-real-numbers public
 open import real-numbers.inequality-positive-real-numbers public
@@ -75,13 +75,13 @@ open import real-numbers.real-numbers-from-lower-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
 open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
-open import real-numbers.short-binary-maximum-real-numbers public
+open import real-numbers.short-function-binary-maximum-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-positive-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.square-roots-nonnegative-real-numbers public
 open import real-numbers.squares-real-numbers public
-open import real-numbers.strict-inequalities-addition-real-numbers public
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers public
 open import real-numbers.strict-inequality-nonnegative-real-numbers public
 open import real-numbers.strict-inequality-positive-real-numbers public
 open import real-numbers.strict-inequality-real-numbers public
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index ba2031c0029..4f77b3aad2d 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -28,7 +28,7 @@ open import metric-spaces.short-functions-metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.isometry-negation-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
index 0735a723925..5298362a872 100644
--- a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
@@ -21,12 +21,12 @@ open import foundation.universe-levels
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 69b9224d864..a321c823561 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -28,7 +28,7 @@ open import order-theory.least-upper-bounds-large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index ab073ba3c24..9d4f3567a1d 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -45,7 +45,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.distance-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index eff38740f50..8e430fb8b0a 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -28,7 +28,7 @@ open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.multiplication-real-numbers
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-and-subtraction-real-numbers.lagda.md
similarity index 98%
rename from src/real-numbers/inequalities-addition-real-numbers.lagda.md
rename to src/real-numbers/inequalities-addition-and-subtraction-real-numbers.lagda.md
index b5af4c0787f..61de6c5cfbf 100644
--- a/src/real-numbers/inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/inequalities-addition-and-subtraction-real-numbers.lagda.md
@@ -3,7 +3,7 @@
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module real-numbers.inequalities-addition-real-numbers where
+module real-numbers.inequalities-addition-and-subtraction-real-numbers where
 ```
 
 <details><summary>Imports</summary>
diff --git a/src/real-numbers/infima-families-real-numbers.lagda.md b/src/real-numbers/infima-families-real-numbers.lagda.md
index add144950d6..0b338cbe370 100644
--- a/src/real-numbers/infima-families-real-numbers.lagda.md
+++ b/src/real-numbers/infima-families-real-numbers.lagda.md
@@ -41,7 +41,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
diff --git a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
index 21124540b7c..8a3677def8c 100644
--- a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
@@ -44,7 +44,7 @@ open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.closed-intervals-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.infima-and-suprema-families-real-numbers
@@ -55,7 +55,7 @@ open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index a7d3686d03b..c42c49edba3 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index 7e5f7562a20..e36974427a6 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import metric-spaces.metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index ed960956fe8..1b56e2c2444 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -36,7 +36,7 @@ open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.inhabited-totally-bounded-subsets-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
index 5e757a72e78..34e2ed1f446 100644
--- a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
@@ -43,7 +43,7 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.suprema-families-real-numbers
 
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index d199e3ef54d..7d78c3c4379 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -44,12 +44,12 @@ open import metric-spaces.triangular-rational-neighborhood-relations
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.transposition-addition-subtraction-cuts-dedekind-real-numbers
 ```
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index fcb7e126e0e..d8219df9e7f 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -36,7 +36,7 @@ open import metric-spaces.metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 668e1644d95..e32729d0f77 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -40,7 +40,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/saturation-inequality-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
index 437a1a7a324..fad1d8e16c4 100644
--- a/src/real-numbers/saturation-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
@@ -21,7 +21,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
similarity index 97%
rename from src/real-numbers/short-binary-maximum-real-numbers.lagda.md
rename to src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
index 5dd32621e15..9cca40873ec 100644
--- a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
@@ -3,7 +3,7 @@
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module real-numbers.short-binary-maximum-real-numbers where
+module real-numbers.short-function-binary-maximum-real-numbers where
 ```
 
 <details><summary>Imports</summary>
@@ -23,7 +23,7 @@ open import order-theory.least-upper-bounds-large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.positive-real-numbers
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
similarity index 99%
rename from src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
rename to src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
index 24417dab873..26b7ac1b7da 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
@@ -3,7 +3,7 @@
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module real-numbers.strict-inequalities-addition-real-numbers where
+module real-numbers.strict-inequalities-addition-and-subtraction-real-numbers where
 ```
 
 <details><summary>Imports</summary>
diff --git a/src/real-numbers/suprema-families-real-numbers.lagda.md b/src/real-numbers/suprema-families-real-numbers.lagda.md
index f730c054982..d51a2e2a299 100644
--- a/src/real-numbers/suprema-families-real-numbers.lagda.md
+++ b/src/real-numbers/suprema-families-real-numbers.lagda.md
@@ -42,7 +42,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 ```
diff --git a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
index 91e5c9c0bef..82a733d1392 100644
--- a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 

From a85cdcbaa33e6a6f4c989c05584e592e304701de Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 12:04:12 -0800
Subject: [PATCH 066/134] Fix

---
 .../inequality-positive-rational-numbers.lagda.md               | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
index 5b51936f6a6..1398ef233b7 100644
--- a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
@@ -97,5 +97,5 @@ antisymmetric-leq-ℚ⁺ = antisymmetric-leq-Poset poset-ℚ⁺
 
 ```agda
 leq-eq-ℚ⁺ : {x y : ℚ⁺} → x ＝ y → leq-ℚ⁺ x y
-leq-eq-ℚ⁺ x=y = leq-eq-ℚ _ _ (ap rational-ℚ⁺ x=y)
+leq-eq-ℚ⁺ x=y = leq-eq-ℚ (ap rational-ℚ⁺ x=y)
 ```

From 1d5f0a217083dd72cdb145026a765ae94b94b38a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 12:25:44 -0800
Subject: [PATCH 067/134] Fix

---
 src/real-numbers/absolute-value-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index 4f77b3aad2d..fc75c7ed59d 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -132,7 +132,7 @@ abstract opaque
           ( zero-ℝ)
           ( x)
           ( 0≤x)
-          ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ _ _ 0≤x))))
+          ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ 0≤x))))
 
   abs-real-ℝ⁺ : {l : Level} (x : ℝ⁺ l) → abs-ℝ (real-ℝ⁺ x) ＝ real-ℝ⁺ x
   abs-real-ℝ⁺ x = abs-real-ℝ⁰⁺ (nonnegative-ℝ⁺ x)

From 399e3c37bb884b897b35304ec9133218680f5b6f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 12:35:18 -0800
Subject: [PATCH 068/134] Fix build

---
 .../powers-positive-rational-numbers.lagda.md          |  8 ++++----
 ...ication-positive-and-negative-real-numbers.lagda.md |  2 +-
 src/real-numbers/powers-real-numbers.lagda.md          | 10 +++++-----
 3 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 847d6dbbbaa..05641432b4b 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -294,9 +294,9 @@ abstract
             ( _)
             ( preserves-leq-power-ℚ⁺ k ε one-ℚ⁺ ε≤1)
         ≤ one-ℚ *ℚ rational-power-ℚ⁺ m ε
-          by leq-eq-ℚ _ _ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
+          by leq-eq-ℚ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
         ≤ rational-power-ℚ⁺ m ε
-          by leq-eq-ℚ _ _ (left-unit-law-mul-ℚ _)
+          by leq-eq-ℚ (left-unit-law-mul-ℚ _)
 ```
 
 ### If `ε` is a positive rational number less than 1, `εⁿ` approaches 0
@@ -327,10 +327,10 @@ abstract
                   rational-dist-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε)) zero-ℚ
                   ≤ rational-abs-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε))
                     by
-                    leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
+                    leq-eq-ℚ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ n ε)
                     by
-                    leq-eq-ℚ _ _
+                    leq-eq-ℚ
                       ( ap rational-ℚ⁰⁺
                         ( abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ (power-ℚ⁺ n ε))))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ m ε)
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 7ecc0768e38..c0fea896952 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -44,7 +44,7 @@ abstract
       ( x *ℝ zero-ℝ)
       ( zero-ℝ)
       ( right-zero-law-mul-ℝ x)
-      ( preserves-le-left-mul-ℝ⁺ (x , is-pos-x) y zero-ℝ is-neg-y)
+      ( preserves-le-left-mul-ℝ⁺ (x , is-pos-x) is-neg-y)
 
 mul-positive-negative-ℝ :
   {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁻ l2 → ℝ⁻ (l1 ⊔ l2)
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 497f3b7c043..8038140ac09 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -293,7 +293,7 @@ abstract
     {l1 l2 : Level} (n : ℕ) (x : ℝ l1) (y : ℝ l2) → leq-ℝ (abs-ℝ x) (abs-ℝ y) →
     leq-ℝ (abs-ℝ (power-ℝ n x)) (abs-ℝ (power-ℝ n y))
   preserves-leq-abs-power-ℝ 0 _ _ _ =
-    preserves-leq-sim-ℝ _ _ _ _
+    preserves-leq-sim-ℝ
       ( inv-tr
         ( sim-ℝ one-ℝ)
         ( abs-real-ℝ⁺ (raise-ℝ⁺ _ one-ℝ⁺))
@@ -310,9 +310,9 @@ abstract
     chain-of-inequalities
       abs-ℝ (power-ℝ (succ-ℕ n) x)
       ≤ abs-ℝ (power-ℝ n x *ℝ x)
-        by leq-eq-ℝ _ _ (ap abs-ℝ (power-succ-ℝ n x))
+        by leq-eq-ℝ (ap abs-ℝ (power-succ-ℝ n x))
       ≤ abs-ℝ (power-ℝ n x) *ℝ abs-ℝ x
-        by leq-eq-ℝ _ _ (abs-mul-ℝ _ _)
+        by leq-eq-ℝ (abs-mul-ℝ _ _)
       ≤ abs-ℝ (power-ℝ n y) *ℝ abs-ℝ y
         by
           preserves-leq-mul-ℝ⁰⁺
@@ -323,7 +323,7 @@ abstract
             ( preserves-leq-abs-power-ℝ n x y |x|≤|y|)
             ( |x|≤|y|)
       ≤ abs-ℝ (power-ℝ n y *ℝ y)
-        by leq-eq-ℝ _ _ (inv (abs-mul-ℝ _ _))
+        by leq-eq-ℝ (inv (abs-mul-ℝ _ _))
       ≤ abs-ℝ (power-ℝ (succ-ℕ n) y)
-        by leq-eq-ℝ _ _ (ap abs-ℝ (inv (power-succ-ℝ n y)))
+        by leq-eq-ℝ (ap abs-ℝ (inv (power-succ-ℝ n y)))
 ```

From 0d47a8a24a42f254dee82ad4a061735969bbc8ed Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 14:04:24 -0800
Subject: [PATCH 069/134] Progress

---
 src/metric-spaces.lagda.md                    |   1 +
 ...tion-points-located-metric-spaces.lagda.md | 120 ++++++++++++++++++
 ...sed-subsets-located-metric-spaces.lagda.md |  22 ++++
 3 files changed, 143 insertions(+)
 create mode 100644 src/metric-spaces/accumulation-points-located-metric-spaces.lagda.md

diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index 4dd23a4e057..a4299c33414 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -56,6 +56,7 @@ metric space, `N d₂ x y` [or](foundation.disjunction.md)
 ```agda
 module metric-spaces where
 
+open import metric-spaces.accumulation-points-located-metric-spaces public
 open import metric-spaces.approximations-located-metric-spaces public
 open import metric-spaces.approximations-metric-spaces public
 open import metric-spaces.bounded-distance-decompositions-of-metric-spaces public
diff --git a/src/metric-spaces/accumulation-points-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-located-metric-spaces.lagda.md
new file mode 100644
index 00000000000..dddb7ba6d1c
--- /dev/null
+++ b/src/metric-spaces/accumulation-points-located-metric-spaces.lagda.md
@@ -0,0 +1,120 @@
+# Accumulation points in located metric spaces
+
+```agda
+module metric-spaces.accumulation-points-located-metric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.inequality-natural-numbers
+open import elementary-number-theory.natural-numbers
+
+open import foundation.conjunction
+open import foundation.dependent-pair-types
+open import foundation.existential-quantification
+open import foundation.function-types
+open import foundation.intersections-subtypes
+open import foundation.propositional-truncations
+open import foundation.propositions
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import lists.sequences
+
+open import metric-spaces.apartness-located-metric-spaces
+open import metric-spaces.closed-subsets-located-metric-spaces
+open import metric-spaces.limits-of-sequences-metric-spaces
+open import metric-spaces.located-metric-spaces
+open import metric-spaces.subspaces-metric-spaces
+```
+
+</details>
+
+## Idea
+
+An
+{{#concept "accumulation point" WDID=Q858223 WD="limit point" Disambiguation="of a metric space" Agda=accumulation-point-Metric-Space}}
+of a subset `S` of a
+[located metric space](metric-spaces.located-metric-spaces.md) `X` is a point
+`x : X` such that there exists a [sequence](lists.sequences.md) `aₙ` of points
+in `S` such that `x` is the
+[limit](metric-spaces.limits-of-sequences-metric-spaces.md) of `aₙ`, and for
+every `n`, `aₙ` is [apart](metric-spaces.apartness-located-metric-spaces.md)
+from `x`.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (X : Located-Metric-Space l1 l2)
+  (S : subset-Located-Metric-Space l3 X)
+  where
+
+  is-accumulation-point-prop-subset-Located-Metric-Space :
+    subset-Metric-Space (l1 ⊔ l2 ⊔ l3) (metric-space-Located-Metric-Space X)
+  is-accumulation-point-prop-subset-Located-Metric-Space x =
+    ∃ ( sequence (type-subtype S))
+      ( λ a →
+        Π-Prop ℕ (λ n → apart-prop-Located-Metric-Space X (pr1 (a n)) x) ∧
+        is-limit-prop-sequence-Metric-Space
+          ( metric-space-Located-Metric-Space X)
+          ( pr1 ∘ a)
+          ( x))
+
+  is-accumulation-point-subset-Located-Metric-Space :
+    type-Located-Metric-Space X → UU (l1 ⊔ l2 ⊔ l3)
+  is-accumulation-point-subset-Located-Metric-Space x =
+    type-Prop (is-accumulation-point-prop-subset-Located-Metric-Space x)
+
+  accumulation-point-subset-Located-Metric-Space : UU (l1 ⊔ l2 ⊔ l3)
+  accumulation-point-subset-Located-Metric-Space =
+    type-subtype is-accumulation-point-prop-subset-Located-Metric-Space
+```
+
+## Properties
+
+### A closed subset of a metric space contains all its accumulation points
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (X : Located-Metric-Space l1 l2)
+  (S : closed-subset-Located-Metric-Space l3 X)
+  where
+
+  is-in-closed-subset-is-accumulation-point-Located-Metric-Space :
+    (x : type-Located-Metric-Space X) →
+    is-accumulation-point-subset-Located-Metric-Space
+      ( X)
+      ( subset-closed-subset-Located-Metric-Space X S)
+      ( x) →
+    is-in-closed-subset-Located-Metric-Space X S x
+  is-in-closed-subset-is-accumulation-point-Located-Metric-Space x is-acc-x =
+    is-closed-subset-closed-subset-Located-Metric-Space
+      ( X)
+      ( S)
+      ( x)
+      ( λ ε →
+        let
+          open
+            do-syntax-trunc-Prop
+              ( ∃
+                ( type-Located-Metric-Space X)
+                ( λ y →
+                  neighborhood-prop-Located-Metric-Space X ε x y ∧
+                  subset-closed-subset-Located-Metric-Space X S y))
+        in do
+          (a , a#x , lim-a=x) ← is-acc-x
+          (μ , is-mod-μ) ← lim-a=x
+          let (y , y∈S) = a (μ ε)
+          intro-exists
+            ( y)
+            ( symmetric-neighborhood-Located-Metric-Space X
+                ( ε)
+                ( y)
+                ( x)
+                ( is-mod-μ ε (μ ε) (refl-leq-ℕ (μ ε))) ,
+              y∈S))
+```
diff --git a/src/metric-spaces/closed-subsets-located-metric-spaces.lagda.md b/src/metric-spaces/closed-subsets-located-metric-spaces.lagda.md
index 59906ef2094..887300e2f4d 100644
--- a/src/metric-spaces/closed-subsets-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/closed-subsets-located-metric-spaces.lagda.md
@@ -7,7 +7,9 @@ module metric-spaces.closed-subsets-located-metric-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.dependent-pair-types
 open import foundation.propositions
+open import foundation.subtypes
 open import foundation.universe-levels
 
 open import metric-spaces.closed-subsets-metric-spaces
@@ -48,4 +50,24 @@ closed-subset-Located-Metric-Space :
   (X : Located-Metric-Space l1 l2) → UU (l1 ⊔ l2 ⊔ lsuc l3)
 closed-subset-Located-Metric-Space l3 X =
   closed-subset-Metric-Space l3 (metric-space-Located-Metric-Space X)
+
+subset-closed-subset-Located-Metric-Space :
+  {l1 l2 l3 : Level} (X : Located-Metric-Space l1 l2) →
+  (S : closed-subset-Located-Metric-Space l3 X) →
+  subset-Located-Metric-Space l3 X
+subset-closed-subset-Located-Metric-Space X = pr1
+
+is-closed-subset-closed-subset-Located-Metric-Space :
+  {l1 l2 l3 : Level} (X : Located-Metric-Space l1 l2) →
+  (S : closed-subset-Located-Metric-Space l3 X) →
+  is-closed-subset-Located-Metric-Space
+    ( X)
+    ( subset-closed-subset-Located-Metric-Space X S)
+is-closed-subset-closed-subset-Located-Metric-Space X = pr2
+
+is-in-closed-subset-Located-Metric-Space :
+  {l1 l2 l3 : Level} (X : Located-Metric-Space l1 l2) →
+  (S : closed-subset-Located-Metric-Space l3 X) →
+  type-Located-Metric-Space X → UU l3
+is-in-closed-subset-Located-Metric-Space X S = is-in-subtype (pr1 S)
 ```

From 6a0ce50d94566c78c87e7645115115c2c3c67041 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 16:51:02 -0800
Subject: [PATCH 070/134] Every point in a proper closed interval is an
 accumulation point

---
 src/metric-spaces.lagda.md                    |   2 +-
 ...ts-subsets-located-metric-spaces.lagda.md} |  89 +++++--
 ...uchy-approximations-metric-spaces.lagda.md |  29 +++
 .../located-metric-spaces.lagda.md            |  19 ++
 src/real-numbers.lagda.md                     |   3 +
 ...ation-points-subsets-real-numbers.lagda.md |  54 ++++
 .../apartness-real-numbers.lagda.md           |   6 +-
 .../binary-maximum-real-numbers.lagda.md      |  58 +++++
 .../binary-minimum-real-numbers.lagda.md      |  60 +++++
 .../closed-intervals-real-numbers.lagda.md    |  51 ++++
 .../isometry-addition-real-numbers.lagda.md   |  11 +
 .../isometry-difference-real-numbers.lagda.md |   9 +
 .../isometry-negation-real-numbers.lagda.md   |  19 ++
 .../nonzero-real-numbers.lagda.md             |   2 +-
 ...per-closed-intervals-real-numbers.lagda.md | 241 +++++++++++++++++-
 ...ction-binary-maximum-real-numbers.lagda.md |   9 +
 ...ction-binary-minimum-real-numbers.lagda.md |  72 ++++++
 17 files changed, 701 insertions(+), 33 deletions(-)
 rename src/metric-spaces/{accumulation-points-located-metric-spaces.lagda.md => accumulation-points-subsets-located-metric-spaces.lagda.md} (50%)
 create mode 100644 src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md
 create mode 100644 src/real-numbers/short-function-binary-minimum-real-numbers.lagda.md

diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index e11c6551c9e..bc48c2ee7e7 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -56,7 +56,7 @@ metric space, `N d₂ x y` [or](foundation.disjunction.md)
 ```agda
 module metric-spaces where
 
-open import metric-spaces.accumulation-points-located-metric-spaces public
+open import metric-spaces.accumulation-points-subsets-located-metric-spaces public
 open import metric-spaces.apartness-located-metric-spaces public
 open import metric-spaces.approximations-located-metric-spaces public
 open import metric-spaces.approximations-metric-spaces public
diff --git a/src/metric-spaces/accumulation-points-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
similarity index 50%
rename from src/metric-spaces/accumulation-points-located-metric-spaces.lagda.md
rename to src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
index dddb7ba6d1c..052d283d8ac 100644
--- a/src/metric-spaces/accumulation-points-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
@@ -1,7 +1,7 @@
-# Accumulation points in located metric spaces
+# Accumulation points of subsets of located metric spaces
 
 ```agda
-module metric-spaces.accumulation-points-located-metric-spaces where
+module metric-spaces.accumulation-points-subsets-located-metric-spaces where
 ```
 
 <details><summary>Imports</summary>
@@ -9,6 +9,7 @@ module metric-spaces.accumulation-points-located-metric-spaces where
 ```agda
 open import elementary-number-theory.inequality-natural-numbers
 open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.conjunction
 open import foundation.dependent-pair-types
@@ -23,7 +24,10 @@ open import foundation.universe-levels
 open import lists.sequences
 
 open import metric-spaces.apartness-located-metric-spaces
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-sequences-metric-spaces
 open import metric-spaces.closed-subsets-located-metric-spaces
+open import metric-spaces.limits-of-cauchy-approximations-metric-spaces
 open import metric-spaces.limits-of-sequences-metric-spaces
 open import metric-spaces.located-metric-spaces
 open import metric-spaces.subspaces-metric-spaces
@@ -37,11 +41,11 @@ An
 {{#concept "accumulation point" WDID=Q858223 WD="limit point" Disambiguation="of a metric space" Agda=accumulation-point-Metric-Space}}
 of a subset `S` of a
 [located metric space](metric-spaces.located-metric-spaces.md) `X` is a point
-`x : X` such that there exists a [sequence](lists.sequences.md) `aₙ` of points
-in `S` such that `x` is the
-[limit](metric-spaces.limits-of-sequences-metric-spaces.md) of `aₙ`, and for
-every `n`, `aₙ` is [apart](metric-spaces.apartness-located-metric-spaces.md)
-from `x`.
+`x : X` such that there exists a
+[Cauchy approximation](metric-spaces.cauchy-approximations-metric-spaces.md) `a`
+with [limit](metric-spaces.limits-of-cauchy-approximations-metric-spaces.md) `x`
+such that for every `ε : ℚ⁺`, `a ε` is in `S` and is
+[apart](metric-spaces.apartness-located-metric-spaces.md) from `x`.
 
 ## Definition
 
@@ -52,16 +56,45 @@ module _
   (S : subset-Located-Metric-Space l3 X)
   where
 
+  is-accumulation-to-point-prop-subset-Located-Metric-Space :
+    type-Located-Metric-Space X →
+    subtype
+      ( l2)
+      ( cauchy-approximation-Metric-Space (subspace-Located-Metric-Space X S))
+  is-accumulation-to-point-prop-subset-Located-Metric-Space x a =
+    Π-Prop
+      ( ℚ⁺)
+      ( λ ε →
+        apart-prop-Located-Metric-Space X
+          ( pr1
+            ( map-cauchy-approximation-Metric-Space
+              ( subspace-Located-Metric-Space X S)
+              ( a)
+              ( ε)))
+          ( x)) ∧
+    is-limit-cauchy-approximation-prop-Metric-Space
+      ( metric-space-Located-Metric-Space X)
+      ( map-short-function-cauchy-approximation-Metric-Space
+        ( subspace-Located-Metric-Space X S)
+        ( metric-space-Located-Metric-Space X)
+        ( short-inclusion-subspace-Metric-Space
+          ( metric-space-Located-Metric-Space X)
+          ( S))
+        ( a))
+      ( x)
+
+  is-accumulation-to-point-subset-Located-Metric-Space :
+    type-Located-Metric-Space X →
+    cauchy-approximation-Metric-Space (subspace-Located-Metric-Space X S) →
+    UU l2
+  is-accumulation-to-point-subset-Located-Metric-Space x a =
+    type-Prop (is-accumulation-to-point-prop-subset-Located-Metric-Space x a)
+
   is-accumulation-point-prop-subset-Located-Metric-Space :
     subset-Metric-Space (l1 ⊔ l2 ⊔ l3) (metric-space-Located-Metric-Space X)
   is-accumulation-point-prop-subset-Located-Metric-Space x =
-    ∃ ( sequence (type-subtype S))
-      ( λ a →
-        Π-Prop ℕ (λ n → apart-prop-Located-Metric-Space X (pr1 (a n)) x) ∧
-        is-limit-prop-sequence-Metric-Space
-          ( metric-space-Located-Metric-Space X)
-          ( pr1 ∘ a)
-          ( x))
+    ∃ ( cauchy-approximation-Metric-Space (subspace-Located-Metric-Space X S))
+      ( is-accumulation-to-point-prop-subset-Located-Metric-Space x)
 
   is-accumulation-point-subset-Located-Metric-Space :
     type-Located-Metric-Space X → UU (l1 ⊔ l2 ⊔ l3)
@@ -106,15 +139,31 @@ module _
                   neighborhood-prop-Located-Metric-Space X ε x y ∧
                   subset-closed-subset-Located-Metric-Space X S y))
         in do
-          (a , a#x , lim-a=x) ← is-acc-x
-          (μ , is-mod-μ) ← lim-a=x
-          let (y , y∈S) = a (μ ε)
+          (approx@(a , _) , a#x , lim-a=x) ← is-acc-x
+          let (y , y∈S) = a ε
           intro-exists
             ( y)
             ( symmetric-neighborhood-Located-Metric-Space X
-                ( ε)
-                ( y)
+              ( ε)
+              ( y)
+              ( x)
+              ( saturated-is-limit-cauchy-approximation-Metric-Space
+                ( metric-space-Located-Metric-Space X)
+                ( map-short-function-cauchy-approximation-Metric-Space
+                  ( subspace-Located-Metric-Space
+                    ( X)
+                    ( subset-closed-subset-Located-Metric-Space X S))
+                  ( metric-space-Located-Metric-Space X)
+                  ( short-inclusion-subspace-Metric-Space
+                    ( metric-space-Located-Metric-Space X)
+                    ( subset-closed-subset-Located-Metric-Space X S))
+                  ( approx))
                 ( x)
-                ( is-mod-μ ε (μ ε) (refl-leq-ℕ (μ ε))) ,
+                ( lim-a=x)
+                ( ε)) ,
               y∈S))
 ```
+
+### To show `x` is an accumulation point of a subset `S` of a located metric space, it suffices to exhibit a sequence in `S` apart from `x` with limit `x`
+
+This remains to be shown.
diff --git a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
index 4ba16ab30ae..6757f63e605 100644
--- a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
@@ -7,6 +7,7 @@ module metric-spaces.limits-of-cauchy-approximations-metric-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.dependent-pair-types
@@ -20,6 +21,7 @@ open import foundation.universe-levels
 open import metric-spaces.cauchy-approximations-metric-spaces
 open import metric-spaces.limits-of-cauchy-approximations-pseudometric-spaces
 open import metric-spaces.metric-spaces
+open import metric-spaces.short-functions-metric-spaces
 ```
 
 </details>
@@ -142,6 +144,33 @@ module _
       ( x)
 ```
 
+### The action of short maps on Cauchy approximations preserves limits
+
+```agda
+module _
+  {l1 l2 l1' l2' : Level}
+  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
+  (f : short-function-Metric-Space A B)
+  (a : cauchy-approximation-Metric-Space A)
+  (lim : type-Metric-Space A)
+  where
+
+  abstract
+    map-short-function-is-limit-cauchy-approximation-Metric-Space :
+      is-limit-cauchy-approximation-Metric-Space A a lim →
+      is-limit-cauchy-approximation-Metric-Space
+        ( B)
+        ( map-short-function-cauchy-approximation-Metric-Space A B f a)
+        ( map-short-function-Metric-Space A B f lim)
+    map-short-function-is-limit-cauchy-approximation-Metric-Space is-lim-a ε δ =
+      is-short-map-short-function-Metric-Space A B
+        ( f)
+        ( ε +ℚ⁺ δ)
+        ( map-cauchy-approximation-Metric-Space A a ε)
+        ( lim)
+        ( is-lim-a ε δ)
+```
+
 ## See also
 
 - [Convergent cauchy approximations](metric-spaces.convergent-cauchy-approximations-metric-spaces.md)
diff --git a/src/metric-spaces/located-metric-spaces.lagda.md b/src/metric-spaces/located-metric-spaces.lagda.md
index 0da99bc7346..273acf8c4bb 100644
--- a/src/metric-spaces/located-metric-spaces.lagda.md
+++ b/src/metric-spaces/located-metric-spaces.lagda.md
@@ -223,3 +223,22 @@ module _
     ( real-dist-located-Metric-Space ,
       is-nonnegative-real-dist-located-Metric-Space)
 ```
+
+### Located metric subspaces
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (X : Located-Metric-Space l1 l2)
+  (S : subset-Located-Metric-Space l3 X)
+  where
+
+  subspace-Located-Metric-Space : Metric-Space (l1 ⊔ l3) l2
+  subspace-Located-Metric-Space =
+    subspace-Metric-Space (metric-space-Located-Metric-Space X) S
+
+  located-subspace-Located-Metric-Space : Located-Metric-Space (l1 ⊔ l3) l2
+  located-subspace-Located-Metric-Space =
+    ( subspace-Located-Metric-Space ,
+      λ (x , _) (y , _) → is-located-Located-Metric-Space X x y)
+```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 0eaee0fa71e..0d22541afac 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -7,6 +7,7 @@ module real-numbers where
 
 open import real-numbers.absolute-value-closed-intervals-real-numbers public
 open import real-numbers.absolute-value-real-numbers public
+open import real-numbers.accumulation-points-subsets-real-numbers public
 open import real-numbers.addition-lower-dedekind-real-numbers public
 open import real-numbers.addition-nonnegative-real-numbers public
 open import real-numbers.addition-real-numbers public
@@ -72,6 +73,7 @@ open import real-numbers.nonzero-real-numbers public
 open import real-numbers.positive-and-negative-real-numbers public
 open import real-numbers.positive-real-numbers public
 open import real-numbers.powers-real-numbers public
+open import real-numbers.proper-closed-intervals-real-numbers public
 open import real-numbers.raising-universe-levels-real-numbers public
 open import real-numbers.rational-lower-dedekind-real-numbers public
 open import real-numbers.rational-real-numbers public
@@ -83,6 +85,7 @@ open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
 open import real-numbers.series-real-numbers public
 open import real-numbers.short-function-binary-maximum-real-numbers public
+open import real-numbers.short-function-binary-minimum-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-positive-real-numbers public
 open import real-numbers.similarity-real-numbers public
diff --git a/src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md b/src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md
new file mode 100644
index 00000000000..9cae91b49d2
--- /dev/null
+++ b/src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md
@@ -0,0 +1,54 @@
+# Accumulation points of subsets of the real numbers
+
+```agda
+module real-numbers.accumulation-points-subsets-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.propositions
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import lists.sequences
+
+open import metric-spaces.accumulation-points-subsets-located-metric-spaces
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.located-metric-space-of-real-numbers
+open import real-numbers.subsets-real-numbers
+```
+
+</details>
+
+## Idea
+
+An
+{{#concept "accumulation point" Disambiguation="of a subset of ℝ" Agda=accumulation-point-subset-ℝ}}
+of a [subset](real-numbers.subsets-real-numbers.md) `S` of the
+[real numbers](real-numbers.dedekind-real-numbers.md) is an
+[accumulation point](metric-spaces.accumulation-points-located-metric-spaces.md)
+of `S` in the
+[located metric space of the real numbers](real-numbers.located-metric-space-of-real-numbers.md).
+
+### Accumulation points of subsets of ℝ
+
+```agda
+module _
+  {l1 l2 : Level} (S : subset-ℝ l1 l2)
+  where
+
+  is-accumulation-point-prop-subset-ℝ : subset-ℝ (l1 ⊔ lsuc l2) l2
+  is-accumulation-point-prop-subset-ℝ =
+    is-accumulation-point-prop-subset-Located-Metric-Space
+      ( located-metric-space-ℝ l2)
+      ( S)
+
+  is-accumulation-point-subset-ℝ : ℝ l2 → UU (l1 ⊔ lsuc l2)
+  is-accumulation-point-subset-ℝ x =
+    type-Prop (is-accumulation-point-prop-subset-ℝ x)
+
+  accumulation-point-subset-ℝ : UU (l1 ⊔ lsuc l2)
+  accumulation-point-subset-ℝ = type-subtype is-accumulation-point-prop-subset-ℝ
+```
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 75d5f030b22..a6a3f19daaf 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -38,7 +38,7 @@ open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -66,6 +66,10 @@ module _
 apart-le-ℝ :
   {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → apart-ℝ x y
 apart-le-ℝ = inl-disjunction
+
+apart-le-ℝ' :
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ y x → apart-ℝ x y
+apart-le-ℝ' = inr-disjunction
 ```
 
 ## Properties
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index a321c823561..eda3d58f579 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -9,8 +9,11 @@ module real-numbers.binary-maximum-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
+open import foundation.action-on-identifications-binary-functions
+open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
@@ -108,6 +111,11 @@ module _
         ( upper-real-max-ℝ)
         ( is-disjoint-lower-upper-max-ℝ)
         ( is-located-lower-upper-max-ℝ)
+
+ap-max-ℝ :
+  {l1 l2 : Level} → {x x' : ℝ l1} → x ＝ x' →
+  {y y' : ℝ l2} → y ＝ y' → max-ℝ x y ＝ max-ℝ x' y'
+ap-max-ℝ = ap-binary max-ℝ
 ```
 
 ## Properties
@@ -377,3 +385,53 @@ module _
               ( is-located-lower-upper-cut-ℝ y q<r))
           ( is-located-lower-upper-cut-ℝ x q<r)
 ```
+
+### If `x < z` and `y < z`, then `max-ℝ x y < x`
+
+```agda
+abstract
+  le-max-le-le-ℝ :
+    {l1 l2 l3 : Level} {x : ℝ l1} {y : ℝ l2} {z : ℝ l3} → le-ℝ x z → le-ℝ y z →
+    le-ℝ (max-ℝ x y) z
+  le-max-le-le-ℝ {x = x} {y = y} {z = z} x<z y<z =
+    let open do-syntax-trunc-Prop (le-prop-ℝ (max-ℝ x y) z)
+    in do
+      (p , x<p , p<z) ← dense-rational-le-ℝ x z x<z
+      (q , y<q , q<z) ← dense-rational-le-ℝ y z y<z
+      rec-coproduct
+        ( λ p≤q →
+          concatenate-leq-le-ℝ
+            ( max-ℝ x y)
+            ( real-ℚ q)
+            ( z)
+            ( leq-max-leq-leq-ℝ
+              ( x)
+              ( y)
+              ( real-ℚ q)
+              ( transitive-leq-ℝ
+                ( x)
+                ( real-ℚ p)
+                ( real-ℚ q)
+                ( preserves-leq-real-ℚ p≤q)
+                ( leq-le-ℝ x<p))
+              ( leq-le-ℝ y<q))
+            ( q<z))
+        ( λ q≤p →
+          concatenate-leq-le-ℝ
+            ( max-ℝ x y)
+            ( real-ℚ p)
+            ( z)
+            ( leq-max-leq-leq-ℝ
+              ( x)
+              ( y)
+              ( real-ℚ p)
+              ( leq-le-ℝ x<p)
+              ( transitive-leq-ℝ
+                ( y)
+                ( real-ℚ q)
+                ( real-ℚ p)
+                ( preserves-leq-real-ℚ q≤p)
+                ( leq-le-ℝ y<q)))
+            ( p<z))
+        ( linear-leq-ℚ p q)
+```
diff --git a/src/real-numbers/binary-minimum-real-numbers.lagda.md b/src/real-numbers/binary-minimum-real-numbers.lagda.md
index b0ae4c06407..37f19c055e8 100644
--- a/src/real-numbers/binary-minimum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-minimum-real-numbers.lagda.md
@@ -11,6 +11,8 @@ module real-numbers.binary-minimum-real-numbers where
 ```agda
 open import elementary-number-theory.positive-rational-numbers
 
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.disjunction
@@ -30,6 +32,7 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -54,6 +57,11 @@ module _
   opaque
     min-ℝ : ℝ (l1 ⊔ l2)
     min-ℝ = neg-ℝ (max-ℝ (neg-ℝ x) (neg-ℝ y))
+
+ap-min-ℝ :
+  {l1 l2 : Level} → {x x' : ℝ l1} → x ＝ x' →
+  {y y' : ℝ l2} → y ＝ y' → min-ℝ x y ＝ min-ℝ x' y'
+ap-min-ℝ = ap-binary min-ℝ
 ```
 
 ## Properties
@@ -130,6 +138,18 @@ module _
           ( is-greatest-binary-lower-bound-min-ℝ x y))
 ```
 
+### The binary minimum is commutative
+
+```agda
+abstract opaque
+  unfolding min-ℝ
+
+  commutative-min-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → min-ℝ x y ＝ min-ℝ y x
+  commutative-min-ℝ x y =
+    ap neg-ℝ (commutative-max-ℝ (neg-ℝ x) (neg-ℝ y))
+```
+
 ### The large poset of real numbers has meets
 
 ```agda
@@ -183,3 +203,43 @@ module _
           ( λ max-ε<-y → inr-disjunction (case y max-ε<-y))
           ( approximate-below-max-ℝ (neg-ℝ x) (neg-ℝ y) ε)
 ```
+
+### If `x ≤ y`, the minimum of `x` and `y` is similar to `x`
+
+```agda
+abstract opaque
+  unfolding min-ℝ
+
+  left-leq-right-min-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → leq-ℝ x y →
+    sim-ℝ (min-ℝ x y) x
+  left-leq-right-min-ℝ {x = x} {y = y} x≤y =
+    similarity-reasoning-ℝ
+      min-ℝ x y
+      ~ℝ neg-ℝ (neg-ℝ x)
+        by preserves-sim-neg-ℝ (right-leq-left-max-ℝ (neg-leq-ℝ x≤y))
+      ~ℝ x
+        by sim-eq-ℝ (neg-neg-ℝ x)
+
+  right-leq-left-min-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → leq-ℝ y x →
+    sim-ℝ (min-ℝ x y) y
+  right-leq-left-min-ℝ {y = y} y≤x =
+    tr (λ z → sim-ℝ z y) (commutative-min-ℝ _ _) (left-leq-right-min-ℝ y≤x)
+```
+
+### If `x < y` and `x < z`, then `x < min-ℝ y z`
+
+```agda
+abstract opaque
+  unfolding min-ℝ
+
+  le-min-le-le-ℝ :
+    {l1 l2 l3 : Level} {x : ℝ l1} {y : ℝ l2} {z : ℝ l3} → le-ℝ x y → le-ℝ x z →
+    le-ℝ x (min-ℝ y z)
+  le-min-le-le-ℝ {x = x} {y = y} {z = z} x<y x<z =
+    tr
+      ( λ w → le-ℝ w (min-ℝ y z))
+      ( neg-neg-ℝ x)
+      ( neg-le-ℝ (le-max-le-le-ℝ (neg-le-ℝ x<y) (neg-le-ℝ x<z)))
+```
diff --git a/src/real-numbers/closed-intervals-real-numbers.lagda.md b/src/real-numbers/closed-intervals-real-numbers.lagda.md
index d8460f5bdc8..7ee471f9ab0 100644
--- a/src/real-numbers/closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/closed-intervals-real-numbers.lagda.md
@@ -26,15 +26,20 @@ open import foundation.universe-levels
 open import metric-spaces.closed-subsets-metric-spaces
 open import metric-spaces.complete-metric-spaces
 open import metric-spaces.metric-spaces
+open import metric-spaces.short-functions-metric-spaces
 open import metric-spaces.subspaces-metric-spaces
 
 open import order-theory.closed-intervals-large-posets
 
+open import real-numbers.binary-maximum-real-numbers
+open import real-numbers.binary-minimum-real-numbers
 open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.short-function-binary-maximum-real-numbers
+open import real-numbers.short-function-binary-minimum-real-numbers
 ```
 
 </details>
@@ -68,6 +73,11 @@ subtype-closed-interval-ℝ :
   subtype (l1 ⊔ l2 ⊔ l) (ℝ l)
 subtype-closed-interval-ℝ = subtype-closed-interval-Large-Poset ℝ-Large-Poset
 
+type-closed-interval-ℝ :
+  {l1 l2 : Level} (l : Level) → closed-interval-ℝ l1 l2 → UU (l1 ⊔ l2 ⊔ lsuc l)
+type-closed-interval-ℝ l [a,b] =
+  type-subtype (subtype-closed-interval-ℝ l [a,b])
+
 lower-bound-closed-interval-ℝ : {l1 l2 : Level} → closed-interval-ℝ l1 l2 → ℝ l1
 lower-bound-closed-interval-ℝ =
   lower-bound-closed-interval-Large-Poset ℝ-Large-Poset
@@ -165,3 +175,44 @@ complete-metric-space-unit-interval-ℝ :
 complete-metric-space-unit-interval-ℝ l =
   complete-metric-space-closed-interval-ℝ l unit-closed-interval-ℝ
 ```
+
+### The clamping function
+
+```agda
+clamp-closed-interval-ℝ :
+  {l1 l2 l3 : Level} ([a,b] : closed-interval-ℝ l1 l2) → ℝ l3 →
+  type-closed-interval-ℝ (l1 ⊔ l2 ⊔ l3) [a,b]
+clamp-closed-interval-ℝ ((a , b) , a≤b) x =
+  ( max-ℝ a (min-ℝ b x) ,
+    leq-left-max-ℝ _ _ ,
+    leq-max-leq-leq-ℝ _ _ _ a≤b (leq-left-min-ℝ b x))
+```
+
+### The clamping function is a short function
+
+```agda
+abstract
+  is-short-clamp-closed-interval-ℝ :
+    {l1 l2 l3 : Level} ([a,b] : closed-interval-ℝ l1 l2) →
+    is-short-function-Metric-Space
+      ( metric-space-ℝ l3)
+      ( metric-space-closed-interval-ℝ (l1 ⊔ l2 ⊔ l3) [a,b])
+      ( clamp-closed-interval-ℝ [a,b])
+  is-short-clamp-closed-interval-ℝ [a,b]@((a , b) , a≤b) =
+    is-short-comp-is-short-function-Metric-Space
+      ( metric-space-ℝ _)
+      ( metric-space-ℝ _)
+      ( metric-space-ℝ _)
+      ( max-ℝ a)
+      ( min-ℝ b)
+      ( is-short-function-left-max-ℝ a)
+      ( is-short-function-left-min-ℝ b)
+
+short-clamp-closed-interval-ℝ :
+  {l1 l2 l3 : Level} ([a,b] : closed-interval-ℝ l1 l2) →
+  short-function-Metric-Space
+    ( metric-space-ℝ l3)
+    ( metric-space-closed-interval-ℝ (l1 ⊔ l2 ⊔ l3) [a,b])
+short-clamp-closed-interval-ℝ [a,b] =
+  ( clamp-closed-interval-ℝ [a,b] , is-short-clamp-closed-interval-ℝ [a,b])
+```
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index c42c49edba3..1107315e4d4 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -20,6 +20,7 @@ open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-space-of-isometries-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-functions-metric-spaces
+open import metric-spaces.short-functions-metric-spaces
 open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
@@ -99,6 +100,16 @@ module _
       ( metric-space-ℝ l2)
       ( metric-space-ℝ (l1 ⊔ l2))
   isometry-right-add-ℝ = ( (λ y → add-ℝ y x) , is-isometry-right-add-ℝ)
+
+  short-left-add-ℝ :
+    short-function-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+  short-left-add-ℝ =
+    short-isometry-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+      ( isometry-left-add-ℝ)
 ```
 
 ### Addition is an isometry from `ℝ` to the metric space of isometries `ℝ → ℝ`
diff --git a/src/real-numbers/isometry-difference-real-numbers.lagda.md b/src/real-numbers/isometry-difference-real-numbers.lagda.md
index 73b411df6a4..6b909afc88f 100644
--- a/src/real-numbers/isometry-difference-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-difference-real-numbers.lagda.md
@@ -13,6 +13,7 @@ open import foundation.dependent-pair-types
 open import foundation.universe-levels
 
 open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.short-functions-metric-spaces
 open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
@@ -61,6 +62,14 @@ module _
     isometry-Metric-Space (metric-space-ℝ l2) (metric-space-ℝ (l1 ⊔ l2))
   isometry-diff-ℝ = (diff-ℝ x , is-isometry-diff-ℝ)
 
+  short-diff-ℝ :
+    short-function-Metric-Space (metric-space-ℝ l2) (metric-space-ℝ (l1 ⊔ l2))
+  short-diff-ℝ =
+    short-isometry-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+      ( isometry-diff-ℝ)
+
   uniformly-continuous-diff-ℝ :
     uniformly-continuous-function-Metric-Space
       ( metric-space-ℝ l2)
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index e36974427a6..05c832b67fc 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -21,6 +21,7 @@ open import foundation.universe-levels
 
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-spaces
+open import metric-spaces.short-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
@@ -123,3 +124,21 @@ module _
       ( metric-space-ℝ l1)
   isometry-neg-ℝ = (neg-ℝ , is-isometry-neg-ℝ)
 ```
+
+### Negation on the real numbers is short
+
+```agda
+abstract
+  is-short-neg-ℝ :
+    {l : Level} →
+    is-short-function-Metric-Space
+      ( metric-space-ℝ l)
+      ( metric-space-ℝ l)
+      ( neg-ℝ)
+  is-short-neg-ℝ =
+    is-short-is-isometry-Metric-Space
+      ( metric-space-ℝ _)
+      ( metric-space-ℝ _)
+      ( neg-ℝ)
+      ( is-isometry-neg-ℝ)
+```
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index f2d2fe8590e..d9ae44bf9b0 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -29,7 +29,7 @@ open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
index 999b3a89adc..51360ec483c 100644
--- a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
@@ -7,20 +7,47 @@ module real-numbers.proper-closed-intervals-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import real-numbers.dedekind-real-numbers
-open import real-numbers.strict-inequality-real-numbers
-open import real-numbers.subsets-real-numbers
-open import real-numbers.inequality-real-numbers
-open import real-numbers.metric-space-of-real-numbers
-open import real-numbers.closed-intervals-real-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.unit-fractions-rational-numbers
 
-open import metric-spaces.metric-spaces
-open import foundation.subtypes
+open import foundation.cartesian-product-types
 open import foundation.conjunction
-open import foundation.universe-levels
-open import metric-spaces.complete-metric-spaces
 open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import metric-spaces.cauchy-approximations-metric-spaces
 open import metric-spaces.closed-subsets-metric-spaces
+open import metric-spaces.complete-metric-spaces
+open import metric-spaces.limits-of-cauchy-approximations-metric-spaces
+open import metric-spaces.metric-space-of-rational-numbers
+open import metric-spaces.metric-spaces
+open import metric-spaces.short-functions-metric-spaces
+open import metric-spaces.subspaces-metric-spaces
+
+open import order-theory.large-posets
+
+open import real-numbers.accumulation-points-subsets-real-numbers
+open import real-numbers.addition-real-numbers
+open import real-numbers.apartness-real-numbers
+open import real-numbers.binary-maximum-real-numbers
+open import real-numbers.binary-minimum-real-numbers
+open import real-numbers.closed-intervals-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.isometry-addition-real-numbers
+open import real-numbers.isometry-difference-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.subsets-real-numbers
 ```
 
 </details>
@@ -60,6 +87,10 @@ subtype-proper-closed-interval-ℝ :
 subtype-proper-closed-interval-ℝ l (a , b , _) x =
   leq-prop-ℝ a x ∧ leq-prop-ℝ x b
 
+is-in-proper-closed-interval-ℝ :
+  {l1 l2 l3 : Level} → proper-closed-interval-ℝ l1 l2 → ℝ l3 → UU (l1 ⊔ l2 ⊔ l3)
+is-in-proper-closed-interval-ℝ (a , b , _) x = leq-ℝ a x × leq-ℝ x b
+
 type-proper-closed-interval-ℝ :
   {l1 l2 : Level} (l : Level) → proper-closed-interval-ℝ l1 l2 →
   UU (l1 ⊔ l2 ⊔ lsuc l)
@@ -114,3 +145,193 @@ complete-metric-space-proper-closed-interval-ℝ l [a,b] =
     ( l)
     ( closed-interval-proper-closed-interval-ℝ [a,b])
 ```
+
+### The clamp function
+
+```agda
+clamp-proper-closed-interval-ℝ :
+  {l1 l2 l3 : Level} ([a,b] : proper-closed-interval-ℝ l1 l2) → ℝ l3 →
+  type-proper-closed-interval-ℝ (l1 ⊔ l2 ⊔ l3) [a,b]
+clamp-proper-closed-interval-ℝ [a,b] =
+  clamp-closed-interval-ℝ (closed-interval-proper-closed-interval-ℝ [a,b])
+```
+
+### The clamp function is a short function
+
+```agda
+abstract
+  is-short-clamp-proper-closed-interval-ℝ :
+    {l1 l2 l3 : Level} ([a,b] : proper-closed-interval-ℝ l1 l2) →
+    is-short-function-Metric-Space
+      ( metric-space-ℝ l3)
+      ( metric-space-proper-closed-interval-ℝ (l1 ⊔ l2 ⊔ l3) [a,b])
+      ( clamp-proper-closed-interval-ℝ [a,b])
+  is-short-clamp-proper-closed-interval-ℝ [a,b] =
+    is-short-clamp-closed-interval-ℝ
+      ( closed-interval-proper-closed-interval-ℝ [a,b])
+
+short-clamp-proper-closed-interval-ℝ :
+  {l1 l2 l3 : Level} ([a,b] : proper-closed-interval-ℝ l1 l2) →
+  short-function-Metric-Space
+    ( metric-space-ℝ l3)
+    ( metric-space-proper-closed-interval-ℝ (l1 ⊔ l2 ⊔ l3) [a,b])
+short-clamp-proper-closed-interval-ℝ [a,b] =
+  short-clamp-closed-interval-ℝ (closed-interval-proper-closed-interval-ℝ [a,b])
+```
+
+### Every real number in a proper closed interval is an accumulation point in that proper closed interval
+
+Note that this cannot be made more universe-polymorphic.
+
+```agda
+abstract
+  is-accumulation-point-is-in-proper-closed-interval-ℝ :
+    {l : Level} ([a,b] : proper-closed-interval-ℝ l l) →
+    (x : ℝ l) → is-in-proper-closed-interval-ℝ [a,b] x →
+    is-accumulation-point-subset-ℝ
+      ( subtype-proper-closed-interval-ℝ l [a,b])
+      ( x)
+  is-accumulation-point-is-in-proper-closed-interval-ℝ
+    {l} [a,b]@(a , b , a<b) x (a≤x , x≤b) =
+    let
+      open inequality-reasoning-Large-Poset ℝ-Large-Poset
+      motive =
+        is-accumulation-point-prop-subset-ℝ
+          ( subtype-proper-closed-interval-ℝ _ [a,b])
+          ( x)
+      short-clamp-diff =
+        comp-short-function-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-ℝ lzero)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( comp-short-function-Metric-Space
+            ( metric-space-ℝ lzero)
+            ( metric-space-ℝ l)
+            ( metric-space-proper-closed-interval-ℝ l [a,b])
+            ( short-clamp-proper-closed-interval-ℝ [a,b])
+            ( short-diff-ℝ x))
+          ( short-isometry-Metric-Space
+            ( metric-space-ℚ)
+            ( metric-space-ℝ lzero)
+            ( isometry-metric-space-real-ℚ))
+      short-clamp-add =
+        comp-short-function-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-ℝ lzero)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( comp-short-function-Metric-Space
+            ( metric-space-ℝ lzero)
+            ( metric-space-ℝ l)
+            ( metric-space-proper-closed-interval-ℝ l [a,b])
+            ( short-clamp-proper-closed-interval-ℝ [a,b])
+            ( short-left-add-ℝ x))
+          ( short-isometry-Metric-Space
+            ( metric-space-ℚ)
+            ( metric-space-ℝ lzero)
+            ( isometry-metric-space-real-ℚ))
+      short-inclusion =
+        short-inclusion-subspace-Metric-Space
+          ( metric-space-ℝ l)
+          ( subtype-proper-closed-interval-ℝ l [a,b])
+      approx-clamp-diff =
+        map-short-function-cauchy-approximation-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( short-clamp-diff)
+          ( cauchy-approximation-rational-ℚ⁺)
+      approx-clamp-add =
+        map-short-function-cauchy-approximation-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( short-clamp-add)
+          ( cauchy-approximation-rational-ℚ⁺)
+    in
+      elim-disjunction
+        ( motive)
+        ( λ a<x →
+          intro-exists
+            ( approx-clamp-diff)
+            ( ( λ ε →
+                apart-located-metric-space-apart-ℝ _ _
+                  ( apart-le-ℝ
+                    ( le-max-le-le-ℝ
+                      ( a<x)
+                      ( concatenate-leq-le-ℝ
+                        ( min-ℝ b (x -ℝ real-ℚ⁺ ε))
+                        ( x -ℝ real-ℚ⁺ ε)
+                        ( x)
+                        ( leq-right-min-ℝ _ _)
+                        ( le-diff-real-ℝ⁺ x (positive-real-ℚ⁺ ε)))))) ,
+              tr
+                ( is-limit-cauchy-approximation-Metric-Space
+                  ( metric-space-ℝ l)
+                  ( map-short-function-cauchy-approximation-Metric-Space
+                    ( metric-space-proper-closed-interval-ℝ l [a,b])
+                    ( metric-space-ℝ l)
+                    ( short-inclusion)
+                    ( approx-clamp-diff)))
+                ( equational-reasoning
+                  max-ℝ a (min-ℝ b (x -ℝ zero-ℝ))
+                  ＝ max-ℝ a (min-ℝ b x)
+                    by ap-max-ℝ refl (ap-min-ℝ refl (right-unit-law-diff-ℝ x))
+                  ＝ max-ℝ a x
+                    by
+                      ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
+                  ＝ x
+                    by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
+                ( map-short-function-is-limit-cauchy-approximation-Metric-Space
+                  ( metric-space-ℚ)
+                  ( metric-space-ℝ l)
+                  ( comp-short-function-Metric-Space
+                    ( metric-space-ℚ)
+                    ( metric-space-proper-closed-interval-ℝ l [a,b])
+                    ( metric-space-ℝ l)
+                    ( short-inclusion)
+                    ( short-clamp-diff))
+                  ( cauchy-approximation-rational-ℚ⁺)
+                  ( zero-ℚ)
+                  ( is-zero-limit-rational-ℚ⁺))))
+        ( λ x<b →
+          intro-exists
+            ( approx-clamp-add)
+            ( ( λ ε →
+                apart-located-metric-space-apart-ℝ _ _
+                  ( apart-le-ℝ'
+                    ( concatenate-le-leq-ℝ
+                      ( x)
+                      ( min-ℝ b (x +ℝ real-ℚ⁺ ε))
+                      ( max-ℝ a (min-ℝ b (x +ℝ real-ℚ⁺ ε)))
+                      ( le-min-le-le-ℝ
+                        ( x<b)
+                        ( le-left-add-real-ℝ⁺ x (positive-real-ℚ⁺ ε)))
+                      ( leq-right-max-ℝ _ _)))) ,
+              tr
+                ( is-limit-cauchy-approximation-Metric-Space
+                  ( metric-space-ℝ l)
+                  ( map-short-function-cauchy-approximation-Metric-Space
+                    ( metric-space-proper-closed-interval-ℝ l [a,b])
+                    ( metric-space-ℝ l)
+                    ( short-inclusion)
+                    ( approx-clamp-add)))
+                ( equational-reasoning
+                  max-ℝ a (min-ℝ b (x +ℝ zero-ℝ))
+                  ＝ max-ℝ a (min-ℝ b x)
+                    by ap-max-ℝ refl (ap-min-ℝ refl (right-unit-law-add-ℝ x))
+                  ＝ max-ℝ a x
+                    by ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
+                  ＝ x
+                    by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
+                ( map-short-function-is-limit-cauchy-approximation-Metric-Space
+                  ( metric-space-ℚ)
+                  ( metric-space-ℝ l)
+                  ( comp-short-function-Metric-Space
+                    ( metric-space-ℚ)
+                    ( metric-space-proper-closed-interval-ℝ l [a,b])
+                    ( metric-space-ℝ l)
+                    ( short-inclusion)
+                    ( short-clamp-add))
+                  ( cauchy-approximation-rational-ℚ⁺)
+                  ( zero-ℚ)
+                  ( is-zero-limit-rational-ℚ⁺))))
+        ( cotransitive-le-ℝ a b x a<b)
+```
diff --git a/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
index 9cca40873ec..d0716d0b192 100644
--- a/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
@@ -17,6 +17,7 @@ open import foundation.universe-levels
 
 open import metric-spaces.metric-space-of-short-functions-metric-spaces
 open import metric-spaces.short-functions-metric-spaces
+open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import order-theory.least-upper-bounds-large-posets
 
@@ -29,6 +30,7 @@ open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.uniformly-continuous-functions-real-numbers
 ```
 
 </details>
@@ -128,6 +130,13 @@ module _
       ( metric-space-ℝ (l1 ⊔ l2))
   short-left-max-ℝ =
     (max-ℝ x , is-short-function-left-max-ℝ)
+
+  uniformly-continuous-left-max-ℝ : uniformly-continuous-function-ℝ l2 (l1 ⊔ l2)
+  uniformly-continuous-left-max-ℝ =
+    uniformly-continuous-short-function-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+      ( short-left-max-ℝ)
 ```
 
 ### The binary maximum is a short function from `ℝ` to the metric space of short functions `ℝ → ℝ`
diff --git a/src/real-numbers/short-function-binary-minimum-real-numbers.lagda.md b/src/real-numbers/short-function-binary-minimum-real-numbers.lagda.md
new file mode 100644
index 00000000000..5ad24658411
--- /dev/null
+++ b/src/real-numbers/short-function-binary-minimum-real-numbers.lagda.md
@@ -0,0 +1,72 @@
+# The binary minimum of real numbers is a short function
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.short-function-binary-minimum-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.short-functions-metric-spaces
+
+open import real-numbers.binary-maximum-real-numbers
+open import real-numbers.binary-minimum-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.isometry-negation-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.short-function-binary-maximum-real-numbers
+```
+
+</details>
+
+## Idea
+
+For any `a : ℝ`, the
+[binary minimum](real-numbers.binary-minimum-real-numbers.md) with `a` is a
+[short function](metric-spaces.short-functions-metric-spaces.md) `ℝ → ℝ` for the
+[standard real metric structure](real-numbers.metric-space-of-real-numbers.md).
+
+## Proof
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1)
+  where
+
+  abstract opaque
+    unfolding min-ℝ
+
+    is-short-function-left-min-ℝ :
+      is-short-function-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( min-ℝ x)
+    is-short-function-left-min-ℝ =
+      is-short-comp-is-short-function-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( neg-ℝ)
+        ( λ y → max-ℝ (neg-ℝ x) (neg-ℝ y))
+        ( is-short-neg-ℝ)
+        ( is-short-comp-is-short-function-Metric-Space
+          ( metric-space-ℝ l2)
+          ( metric-space-ℝ l2)
+          ( metric-space-ℝ (l1 ⊔ l2))
+          ( max-ℝ (neg-ℝ x))
+          ( neg-ℝ)
+          ( is-short-function-left-max-ℝ (neg-ℝ x))
+          ( is-short-neg-ℝ))
+
+  short-left-min-ℝ :
+    short-function-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+  short-left-min-ℝ = (min-ℝ x , is-short-function-left-min-ℝ)
+```

From 3ed6bf1e9481dac7b218a72e8a22729756437859 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 17:38:19 -0800
Subject: [PATCH 071/134] Sequential accumulation points

---
 ...nts-subsets-located-metric-spaces.lagda.md | 67 ++++++++++++++++++-
 .../cauchy-sequences-metric-spaces.lagda.md   | 48 ++++++++++++-
 2 files changed, 111 insertions(+), 4 deletions(-)

diff --git a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
index 052d283d8ac..4e7ac4d9973 100644
--- a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
@@ -23,6 +23,8 @@ open import foundation.universe-levels
 
 open import lists.sequences
 
+open import logic.functoriality-existential-quantification
+
 open import metric-spaces.apartness-located-metric-spaces
 open import metric-spaces.cauchy-approximations-metric-spaces
 open import metric-spaces.cauchy-sequences-metric-spaces
@@ -38,7 +40,7 @@ open import metric-spaces.subspaces-metric-spaces
 ## Idea
 
 An
-{{#concept "accumulation point" WDID=Q858223 WD="limit point" Disambiguation="of a metric space" Agda=accumulation-point-Metric-Space}}
+{{#concept "accumulation point" WDID=Q858223 WD="limit point" Disambiguation="of a metric space" Agda=accumulation-point-subset-Located-Metric-Space}}
 of a subset `S` of a
 [located metric space](metric-spaces.located-metric-spaces.md) `X` is a point
 `x : X` such that there exists a
@@ -164,6 +166,65 @@ module _
               y∈S))
 ```
 
-### To show `x` is an accumulation point of a subset `S` of a located metric space, it suffices to exhibit a sequence in `S` apart from `x` with limit `x`
+### The property of being a sequential accumulation point
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (X : Located-Metric-Space l1 l2)
+  (S : subset-Located-Metric-Space l3 X)
+  (x : type-Located-Metric-Space X)
+  where
+
+  is-sequential-accumulation-point-prop-subset-Located-Metric-Space :
+    Prop (l1 ⊔ l2 ⊔ l3)
+  is-sequential-accumulation-point-prop-subset-Located-Metric-Space =
+    ∃ ( sequence (type-subtype S))
+      ( λ a →
+        Π-Prop ℕ (λ n → apart-prop-Located-Metric-Space X (pr1 (a n)) x) ∧
+        is-limit-prop-sequence-Metric-Space
+          ( metric-space-Located-Metric-Space X)
+          ( pr1 ∘ a)
+          ( x))
+
+  is-sequential-accumulation-point-subset-Located-Metric-Space :
+    UU (l1 ⊔ l2 ⊔ l3)
+  is-sequential-accumulation-point-subset-Located-Metric-Space =
+    type-Prop is-sequential-accumulation-point-prop-subset-Located-Metric-Space
+```
+
+### If `x` is an accumulation point of `S`, it is a sequential accumulation point of `S`
 
-This remains to be shown.
+The converse has yet to be proved.
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (X : Located-Metric-Space l1 l2)
+  (S : subset-Located-Metric-Space l3 X)
+  (x : type-Located-Metric-Space X)
+  where
+
+  abstract
+    is-sequential-accumulation-point-is-accumulation-point-subset-Located-Metric-Space :
+      is-accumulation-point-subset-Located-Metric-Space X S x →
+      is-sequential-accumulation-point-subset-Located-Metric-Space X S x
+    is-sequential-accumulation-point-is-accumulation-point-subset-Located-Metric-Space =
+      map-exists
+        ( _)
+        ( map-cauchy-sequence-cauchy-approximation-Metric-Space
+          ( subspace-Located-Metric-Space X S))
+        ( λ a (a#x , lim-a=x) →
+          ( ( λ n → a#x _) ,
+            is-limit-cauchy-sequence-cauchy-approximation-Metric-Space
+              ( metric-space-Located-Metric-Space X)
+              ( map-short-function-cauchy-approximation-Metric-Space
+                ( subspace-Located-Metric-Space X S)
+                ( metric-space-Located-Metric-Space X)
+                ( short-inclusion-subspace-Metric-Space
+                  ( metric-space-Located-Metric-Space X)
+                  ( S))
+                ( a))
+              ( x)
+              ( lim-a=x)))
+```
diff --git a/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md b/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
index 932310526dc..4fe9086ee64 100644
--- a/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
@@ -19,6 +19,7 @@ open import elementary-number-theory.multiplicative-group-of-positive-rational-n
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.strict-inequality-natural-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 open import elementary-number-theory.unit-fractions-rational-numbers
@@ -27,6 +28,8 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
+open import foundation.existential-quantification
+open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositional-truncations
@@ -399,7 +402,7 @@ module _
     map-cauchy-approximation-Metric-Space
       ( M)
       ( x)
-      ( positive-reciprocal-rational-ℕ⁺ (succ-nonzero-ℕ' n))
+      ( positive-reciprocal-rational-succ-ℕ n)
 
   modulus-of-convergence-cauchy-sequence-cauchy-approximation-Metric-Space :
     ℚ⁺ → ℕ
@@ -484,6 +487,49 @@ module _
     is-cauchy-sequence-cauchy-sequence-cauchy-approximation-Metric-Space
 ```
 
+### If a Cauchy approximation has a limit, its corresponding Cauchy sequence has the same limit
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  (x : cauchy-approximation-Metric-Space M)
+  (lim : type-Metric-Space M)
+  (is-lim : is-limit-cauchy-approximation-Metric-Space M x lim)
+  where
+
+  abstract
+    is-limit-cauchy-sequence-cauchy-approximation-Metric-Space :
+      is-limit-cauchy-sequence-Metric-Space
+        ( M)
+        ( cauchy-sequence-cauchy-approximation-Metric-Space M x)
+        ( lim)
+    is-limit-cauchy-sequence-cauchy-approximation-Metric-Space =
+      is-limit-bound-modulus-sequence-Metric-Space M _ _
+        ( λ ε →
+          let
+            (n⁺ , 1/n⁺<ε) = smaller-reciprocal-ℚ⁺ ε
+          in
+            ( nat-nonzero-ℕ n⁺ ,
+              λ m n≤m →
+                monotonic-neighborhood-Metric-Space M
+                  ( map-cauchy-sequence-cauchy-approximation-Metric-Space M x m)
+                  ( lim)
+                  ( positive-reciprocal-rational-succ-ℕ m)
+                  ( ε)
+                  ( transitive-le-ℚ _ (reciprocal-rational-ℕ⁺ n⁺) _
+                    ( 1/n⁺<ε)
+                    ( inv-le-ℚ⁺
+                      ( positive-rational-ℕ⁺ n⁺)
+                      ( positive-rational-ℕ⁺ (succ-nonzero-ℕ' m))
+                      ( preserves-le-rational-ℕ
+                        ( le-succ-leq-ℕ _ _ n≤m))))
+                  ( saturated-is-limit-cauchy-approximation-Metric-Space M
+                    ( x)
+                    ( lim)
+                    ( is-lim)
+                    ( positive-reciprocal-rational-succ-ℕ m))))
+```
+
 ### If the Cauchy sequence associated with a Cauchy approximation has a limit modulus at `l`, its associated approximation converges to `l`
 
 ```agda

From 74b1601180a4c93e913be91295b381d08434bfac Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 17:56:46 -0800
Subject: [PATCH 072/134] Prove equivalence between sequential and
 approximation versions

---
 ...ddition-positive-rational-numbers.lagda.md |  9 ++-
 ...nts-subsets-located-metric-spaces.lagda.md | 75 +++++++++++++++++++
 .../located-metric-spaces.lagda.md            |  8 ++
 3 files changed, 88 insertions(+), 4 deletions(-)

diff --git a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
index e4961479540..fd1909ddece 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -453,11 +453,12 @@ module _
           ( left-summand-split-ℚ⁺ p)
           ( right-summand-split-ℚ⁺ p))
 
-    bound-double-le-ℚ⁺ :
-      Σ ℚ⁺ (λ q → le-ℚ⁺ (q +ℚ⁺ q) p)
-    bound-double-le-ℚ⁺ =
-      ( modulus-le-double-le-ℚ⁺ , le-double-le-modulus-le-double-le-ℚ⁺)
+  bound-double-le-ℚ⁺ :
+    Σ ℚ⁺ (λ q → le-ℚ⁺ (q +ℚ⁺ q) p)
+  bound-double-le-ℚ⁺ =
+    ( modulus-le-double-le-ℚ⁺ , le-double-le-modulus-le-double-le-ℚ⁺)
 
+  abstract
     double-le-ℚ⁺ : exists ℚ⁺ (λ q → le-prop-ℚ⁺ (q +ℚ⁺ q) p)
     double-le-ℚ⁺ = unit-trunc-Prop bound-double-le-ℚ⁺
 ```
diff --git a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
index 4e7ac4d9973..ea018b42ca2 100644
--- a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
@@ -7,15 +7,18 @@ module metric-spaces.accumulation-points-subsets-located-metric-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.inequality-natural-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.intersections-subtypes
+open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
@@ -228,3 +231,75 @@ module _
               ( x)
               ( lim-a=x)))
 ```
+
+### If `x` is a sequential accumulation point of `S`, it is an accumulation point of `S`
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (X : Located-Metric-Space l1 l2)
+  (S : subset-Located-Metric-Space l3 X)
+  (x : type-Located-Metric-Space X)
+  where
+
+  abstract
+    is-accumulation-point-is-sequential-accumulation-point-subset-Located-Metric-Space :
+      is-sequential-accumulation-point-subset-Located-Metric-Space X S x →
+      is-accumulation-point-subset-Located-Metric-Space X S x
+    is-accumulation-point-is-sequential-accumulation-point-subset-Located-Metric-Space
+      is-seq-acc-x =
+      let
+        open
+          do-syntax-trunc-Prop
+            ( is-accumulation-point-prop-subset-Located-Metric-Space X S x)
+      in do
+        (σ , σ#x , lim-σ=x) ← is-seq-acc-x
+        μ@(mod-μ , is-mod-μ) ← lim-σ=x
+        intro-exists
+          ( cauchy-approximation-cauchy-sequence-Metric-Space
+            ( subspace-Located-Metric-Space X S)
+            ( σ ,
+              is-cauchy-has-limit-modulus-sequence-Metric-Space
+                ( metric-space-Located-Metric-Space X)
+                ( pr1 ∘ σ)
+                ( x)
+                ( μ)))
+          ( ( λ ε → σ#x _) ,
+            ( λ ε δ →
+              let ε' = modulus-le-double-le-ℚ⁺ ε
+              in
+                monotonic-neighborhood-Located-Metric-Space
+                  ( X)
+                  ( pr1 (σ (mod-μ ε')))
+                  ( x)
+                  ( ε')
+                  ( ε +ℚ⁺ δ)
+                  ( transitive-le-ℚ _ _ _
+                    ( le-left-add-ℚ⁺ ε δ)
+                    ( le-modulus-le-double-le-ℚ⁺ ε))
+                  ( is-mod-μ ε' (mod-μ ε') (refl-leq-ℕ (mod-μ ε')))))
+```
+
+### Being an accumulation point is equivalent to being a sequential accumulation point
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (X : Located-Metric-Space l1 l2)
+  (S : subset-Located-Metric-Space l3 X)
+  (x : type-Located-Metric-Space X)
+  where
+
+  is-accumulation-point-iff-is-sequential-accumulation-point-subset-Located-Metric-Space :
+    is-accumulation-point-subset-Located-Metric-Space X S x ↔
+    is-sequential-accumulation-point-subset-Located-Metric-Space X S x
+  is-accumulation-point-iff-is-sequential-accumulation-point-subset-Located-Metric-Space =
+    ( is-sequential-accumulation-point-is-accumulation-point-subset-Located-Metric-Space
+        ( X)
+        ( S)
+        ( x) ,
+      is-accumulation-point-is-sequential-accumulation-point-subset-Located-Metric-Space
+        ( X)
+        ( S)
+        ( x))
+```
diff --git a/src/metric-spaces/located-metric-spaces.lagda.md b/src/metric-spaces/located-metric-spaces.lagda.md
index 273acf8c4bb..5c1f581c70e 100644
--- a/src/metric-spaces/located-metric-spaces.lagda.md
+++ b/src/metric-spaces/located-metric-spaces.lagda.md
@@ -133,6 +133,14 @@ module _
   triangular-neighborhood-Located-Metric-Space =
     triangular-neighborhood-Metric-Space metric-space-Located-Metric-Space
 
+  monotonic-neighborhood-Located-Metric-Space :
+    (x y : type-Located-Metric-Space) (d₁ d₂ : ℚ⁺) →
+    le-ℚ⁺ d₁ d₂ →
+    neighborhood-Located-Metric-Space d₁ x y →
+    neighborhood-Located-Metric-Space d₂ x y
+  monotonic-neighborhood-Located-Metric-Space =
+    monotonic-neighborhood-Metric-Space metric-space-Located-Metric-Space
+
   is-located-Located-Metric-Space :
     is-located-Metric-Space metric-space-Located-Metric-Space
   is-located-Located-Metric-Space = pr2 X

From 2bda7c6c76ac7a3214a22235e452e99200e55154 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 17:59:26 -0800
Subject: [PATCH 073/134] Prove equivalence between sequential and
 approximation versions

---
 ...nts-subsets-located-metric-spaces.lagda.md | 22 ++++++++++++++-----
 1 file changed, 16 insertions(+), 6 deletions(-)

diff --git a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
index ea018b42ca2..475db6744c7 100644
--- a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
@@ -179,16 +179,26 @@ module _
   (x : type-Located-Metric-Space X)
   where
 
+  is-sequence-accumulating-to-point-prop-subset-Located-Metric-Space :
+    subtype l2 (sequence (type-subtype S))
+  is-sequence-accumulating-to-point-prop-subset-Located-Metric-Space a =
+    Π-Prop ℕ (λ n → apart-prop-Located-Metric-Space X (pr1 (a n)) x) ∧
+    is-limit-prop-sequence-Metric-Space
+      ( metric-space-Located-Metric-Space X)
+      ( pr1 ∘ a)
+      ( x)
+
+  is-sequence-accumulating-to-point-subset-Located-Metric-Space :
+    sequence (type-subtype S) → UU l2
+  is-sequence-accumulating-to-point-subset-Located-Metric-Space =
+    is-in-subtype
+      ( is-sequence-accumulating-to-point-prop-subset-Located-Metric-Space)
+
   is-sequential-accumulation-point-prop-subset-Located-Metric-Space :
     Prop (l1 ⊔ l2 ⊔ l3)
   is-sequential-accumulation-point-prop-subset-Located-Metric-Space =
     ∃ ( sequence (type-subtype S))
-      ( λ a →
-        Π-Prop ℕ (λ n → apart-prop-Located-Metric-Space X (pr1 (a n)) x) ∧
-        is-limit-prop-sequence-Metric-Space
-          ( metric-space-Located-Metric-Space X)
-          ( pr1 ∘ a)
-          ( x))
+      ( is-sequence-accumulating-to-point-prop-subset-Located-Metric-Space)
 
   is-sequential-accumulation-point-subset-Located-Metric-Space :
     UU (l1 ⊔ l2 ⊔ l3)

From 719aff360c9abc37468500052dd79f8611dff7ea Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 19:40:53 -0800
Subject: [PATCH 074/134] Fix link

---
 .../accumulation-points-subsets-real-numbers.lagda.md           | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md b/src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md
index 9cae91b49d2..31cee9ca467 100644
--- a/src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/accumulation-points-subsets-real-numbers.lagda.md
@@ -28,7 +28,7 @@ An
 {{#concept "accumulation point" Disambiguation="of a subset of ℝ" Agda=accumulation-point-subset-ℝ}}
 of a [subset](real-numbers.subsets-real-numbers.md) `S` of the
 [real numbers](real-numbers.dedekind-real-numbers.md) is an
-[accumulation point](metric-spaces.accumulation-points-located-metric-spaces.md)
+[accumulation point](metric-spaces.accumulation-points-subsets-located-metric-spaces.md)
 of `S` in the
 [located metric space of the real numbers](real-numbers.located-metric-space-of-real-numbers.md).
 

From 8569df8582c3d35496e39a140b7ed34d00d965cd Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 9 Nov 2025 13:06:21 +0100
Subject: [PATCH 075/134] chore: optimize imports `real-numbers`

---
 src/real-numbers/absolute-value-real-numbers.lagda.md |  1 -
 .../addition-lower-dedekind-real-numbers.lagda.md     |  5 -----
 .../addition-nonnegative-real-numbers.lagda.md        |  1 -
 src/real-numbers/addition-real-numbers.lagda.md       |  1 -
 .../addition-upper-dedekind-real-numbers.lagda.md     |  2 --
 src/real-numbers/apartness-real-numbers.lagda.md      |  3 ---
 .../arithmetically-located-dedekind-cuts.lagda.md     |  2 --
 src/real-numbers/binary-maximum-real-numbers.lagda.md |  2 --
 ...cauchy-completeness-dedekind-real-numbers.lagda.md |  4 ----
 .../cauchy-sequences-real-numbers.lagda.md            |  4 ----
 .../closed-intervals-real-numbers.lagda.md            |  4 ----
 src/real-numbers/dedekind-real-numbers.lagda.md       |  9 ---------
 src/real-numbers/difference-real-numbers.lagda.md     |  1 -
 src/real-numbers/distance-real-numbers.lagda.md       |  2 --
 ...ng-closed-rational-intervals-real-numbers.lagda.md |  1 -
 .../inequality-nonnegative-real-numbers.lagda.md      |  1 -
 .../inequality-positive-real-numbers.lagda.md         |  1 -
 .../inequality-upper-dedekind-real-numbers.lagda.md   |  1 -
 .../infima-and-suprema-families-real-numbers.lagda.md |  2 --
 ...-finitely-enumerable-subsets-real-numbers.lagda.md |  1 -
 ...ited-totally-bounded-subsets-real-numbers.lagda.md |  1 -
 .../isometry-addition-real-numbers.lagda.md           |  5 -----
 .../isometry-negation-real-numbers.lagda.md           |  1 -
 .../large-additive-group-of-real-numbers.lagda.md     |  2 --
 ...rge-multiplicative-monoid-of-real-numbers.lagda.md |  2 --
 src/real-numbers/large-ring-of-real-numbers.lagda.md  |  4 ----
 .../limits-sequences-real-numbers.lagda.md            |  2 --
 ...tz-continuity-multiplication-real-numbers.lagda.md |  7 -------
 src/real-numbers/lower-dedekind-real-numbers.lagda.md |  1 -
 .../maximum-finite-families-real-numbers.lagda.md     |  3 ---
 .../maximum-upper-dedekind-real-numbers.lagda.md      |  6 ------
 ...-finitely-enumerable-subsets-real-numbers.lagda.md |  3 ---
 .../minimum-lower-dedekind-real-numbers.lagda.md      |  1 -
 .../minimum-upper-dedekind-real-numbers.lagda.md      |  2 --
 .../multiplication-nonnegative-real-numbers.lagda.md  |  1 -
 ...cation-positive-and-negative-real-numbers.lagda.md |  1 -
 .../multiplication-positive-real-numbers.lagda.md     |  1 -
 ...iplicative-inverses-negative-real-numbers.lagda.md |  1 -
 ...tiplicative-inverses-nonzero-real-numbers.lagda.md |  4 ----
 ...iplicative-inverses-positive-real-numbers.lagda.md |  3 ---
 src/real-numbers/negative-real-numbers.lagda.md       |  1 -
 src/real-numbers/nonnegative-real-numbers.lagda.md    |  2 --
 .../positive-and-negative-real-numbers.lagda.md       |  3 ---
 src/real-numbers/positive-real-numbers.lagda.md       |  9 ---------
 src/real-numbers/powers-real-numbers.lagda.md         |  1 -
 .../raising-universe-levels-real-numbers.lagda.md     |  3 ---
 src/real-numbers/rational-real-numbers.lagda.md       | 11 -----------
 ...-numbers-from-lower-dedekind-real-numbers.lagda.md |  3 ---
 ...-numbers-from-upper-dedekind-real-numbers.lagda.md |  3 ---
 .../similarity-nonnegative-real-numbers.lagda.md      |  7 -------
 src/real-numbers/similarity-real-numbers.lagda.md     |  9 ---------
 src/real-numbers/squares-real-numbers.lagda.md        |  7 -------
 ...ies-addition-and-subtraction-real-numbers.lagda.md |  2 --
 ...trict-inequality-nonnegative-real-numbers.lagda.md |  1 -
 .../strict-inequality-positive-real-numbers.lagda.md  |  1 -
 .../strict-inequality-real-numbers.lagda.md           |  1 -
 src/real-numbers/subsets-real-numbers.lagda.md        |  3 ---
 .../totally-bounded-subsets-real-numbers.lagda.md     |  3 ---
 ...on-subtraction-cuts-dedekind-real-numbers.lagda.md |  1 -
 src/real-numbers/upper-dedekind-real-numbers.lagda.md |  1 -
 60 files changed, 171 deletions(-)

diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index fc75c7ed59d..1956f141160 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -19,7 +19,6 @@ open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
index bd1e58abfe1..6b2493249ea 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -21,27 +21,22 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
-open import foundation.sets
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
 open import group-theory.commutative-monoids
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
 open import group-theory.monoids
 open import group-theory.semigroups
 
-open import logic.functoriality-existential-quantification
-
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 ```
diff --git a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
index 5298362a872..6c29699df8a 100644
--- a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
@@ -28,7 +28,6 @@ open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index 356d51654fd..8f67041ac9a 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -11,7 +11,6 @@ module real-numbers.addition-real-numbers where
 ```agda
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
-open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
diff --git a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
index 616124b9515..7ada6a30a27 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -19,7 +19,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
@@ -30,7 +29,6 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
 open import group-theory.commutative-monoids
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 023123e63ef..242133133ba 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -7,18 +7,15 @@ module real-numbers.apartness-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.apartness-relations
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.functoriality-disjunction
-open import foundation.identity-types
 open import foundation.large-apartness-relations
 open import foundation.large-binary-relations
 open import foundation.negated-equality
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 1592876b213..d2abf4e12a8 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -27,7 +27,6 @@ open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
-open import foundation.binary-transport
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.coproduct-types
@@ -37,7 +36,6 @@ open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index a321c823561..d4da7d69c05 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -25,10 +25,8 @@ open import order-theory.join-semilattices
 open import order-theory.large-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
 
-open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index 9d4f3567a1d..6e102ad464b 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -21,7 +21,6 @@ open import foundation.action-on-identifications-functions
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
-open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.functoriality-disjunction
@@ -34,19 +33,16 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-sequences-complete-metric-spaces
 open import metric-spaces.complete-metric-spaces
 open import metric-spaces.convergent-cauchy-approximations-metric-spaces
 open import metric-spaces.limits-of-cauchy-approximations-metric-spaces
 open import metric-spaces.metric-spaces
 
-open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.distance-real-numbers
 open import real-numbers.inequalities-addition-and-subtraction-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
diff --git a/src/real-numbers/cauchy-sequences-real-numbers.lagda.md b/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
index cac5f34c62a..68762013862 100644
--- a/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
@@ -11,14 +11,10 @@ module real-numbers.cauchy-sequences-real-numbers where
 ```agda
 open import foundation.universe-levels
 
-open import lists.sequences
-
 open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.cauchy-sequences-complete-metric-spaces
 open import metric-spaces.cauchy-sequences-metric-spaces
-open import metric-spaces.convergent-sequences-metric-spaces
 
-open import real-numbers.addition-real-numbers
 open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-addition-real-numbers
diff --git a/src/real-numbers/closed-intervals-real-numbers.lagda.md b/src/real-numbers/closed-intervals-real-numbers.lagda.md
index d8460f5bdc8..09d4fada9e8 100644
--- a/src/real-numbers/closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/closed-intervals-real-numbers.lagda.md
@@ -11,9 +11,6 @@ open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
-open import elementary-number-theory.rational-numbers
-open import elementary-number-theory.strict-inequality-positive-rational-numbers
-open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.logical-equivalences
@@ -26,7 +23,6 @@ open import foundation.universe-levels
 open import metric-spaces.closed-subsets-metric-spaces
 open import metric-spaces.complete-metric-spaces
 open import metric-spaces.metric-spaces
-open import metric-spaces.subspaces-metric-spaces
 
 open import order-theory.closed-intervals-large-posets
 
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index d846a3556de..22ece701910 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -15,8 +15,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
-open import foundation.complements-subtypes
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
@@ -24,28 +22,21 @@ open import foundation.disjoint-subtypes
 open import foundation.disjunction
 open import foundation.embeddings
 open import foundation.empty-types
-open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
-open import foundation.inhabited-types
 open import foundation.logical-equivalences
 open import foundation.negation
-open import foundation.powersets
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.similarity-subtypes
 open import foundation.subtypes
 open import foundation.transport-along-identifications
-open import foundation.truncated-types
 open import foundation.universal-quantification
 open import foundation.universe-levels
 
-open import foundation-core.truncation-levels
-
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.lower-dedekind-real-numbers
diff --git a/src/real-numbers/difference-real-numbers.lagda.md b/src/real-numbers/difference-real-numbers.lagda.md
index e3a1cd036e7..0d13f8b33f0 100644
--- a/src/real-numbers/difference-real-numbers.lagda.md
+++ b/src/real-numbers/difference-real-numbers.lagda.md
@@ -14,7 +14,6 @@ open import elementary-number-theory.rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
-open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 8e430fb8b0a..4bc37c1536a 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -9,11 +9,9 @@ module real-numbers.distance-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
diff --git a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
index 77d3ab8c6b5..977e33b2245 100644
--- a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
@@ -10,7 +10,6 @@ module real-numbers.enclosing-closed-rational-intervals-real-numbers where
 open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.interior-closed-intervals-rational-numbers
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.conjunction
 open import foundation.dependent-pair-types
diff --git a/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
index fa71aea44a1..dc71c190c0a 100644
--- a/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
diff --git a/src/real-numbers/inequality-positive-real-numbers.lagda.md b/src/real-numbers/inequality-positive-real-numbers.lagda.md
index 2ef1d3a686c..15343cc094b 100644
--- a/src/real-numbers/inequality-positive-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-positive-real-numbers.lagda.md
@@ -12,7 +12,6 @@ module real-numbers.inequality-positive-real-numbers where
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.positive-real-numbers
 ```
diff --git a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
index 2b8a3de8896..48dc2d04d4d 100644
--- a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
@@ -18,7 +18,6 @@ open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.unit-type
diff --git a/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md b/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md
index f889ec1a311..6a0b26fdf4d 100644
--- a/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md
+++ b/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md
@@ -8,7 +8,6 @@ module real-numbers.infima-and-suprema-families-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.existential-quantification
 open import foundation.propositional-truncations
 open import foundation.subtypes
 open import foundation.universe-levels
@@ -17,7 +16,6 @@ open import real-numbers.closed-intervals-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.infima-families-real-numbers
-open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
 ```
diff --git a/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
index 945abdb69cb..a5eb36703a0 100644
--- a/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
@@ -20,7 +20,6 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.subsets-real-numbers
 
 open import univalent-combinatorics.finitely-enumerable-subtypes
-open import univalent-combinatorics.finitely-enumerable-types
 open import univalent-combinatorics.inhabited-finitely-enumerable-subtypes
 ```
 
diff --git a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
index 8a3677def8c..3b65a9823d9 100644
--- a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
@@ -32,7 +32,6 @@ open import metric-spaces.approximations-metric-spaces
 open import metric-spaces.inhabited-totally-bounded-subspaces-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.nets-metric-spaces
-open import metric-spaces.subspaces-metric-spaces
 open import metric-spaces.totally-bounded-metric-spaces
 
 open import order-theory.upper-bounds-large-posets
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index c42c49edba3..3b1ca989435 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -11,25 +11,20 @@ module real-numbers.isometry-addition-real-numbers where
 ```agda
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
-open import foundation.function-types
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-space-of-isometries-metric-spaces
-open import metric-spaces.metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-functions-metric-spaces
-open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequalities-addition-and-subtraction-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index e36974427a6..3c2b4494329 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -20,7 +20,6 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/large-additive-group-of-real-numbers.lagda.md b/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
index 77808a473ab..8e876af34b7 100644
--- a/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
@@ -9,10 +9,8 @@ module real-numbers.large-additive-group-of-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.universe-levels
diff --git a/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md b/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
index 87f9524dc39..b5d2a73c59d 100644
--- a/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
@@ -11,9 +11,7 @@ module real-numbers.large-multiplicative-monoid-of-real-numbers where
 ```agda
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
 open import group-theory.commutative-monoids
-open import group-theory.large-abelian-groups
 open import group-theory.large-commutative-monoids
 open import group-theory.large-monoids
 open import group-theory.large-semigroups
diff --git a/src/real-numbers/large-ring-of-real-numbers.lagda.md b/src/real-numbers/large-ring-of-real-numbers.lagda.md
index 5786d7c56cb..98fd1e8782a 100644
--- a/src/real-numbers/large-ring-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-ring-of-real-numbers.lagda.md
@@ -13,16 +13,12 @@ open import commutative-algebra.commutative-rings
 open import commutative-algebra.homomorphisms-commutative-rings
 open import commutative-algebra.large-commutative-rings
 
-open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.ring-of-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.universe-levels
 
-open import group-theory.large-monoids
-
 open import real-numbers.large-additive-group-of-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
diff --git a/src/real-numbers/limits-sequences-real-numbers.lagda.md b/src/real-numbers/limits-sequences-real-numbers.lagda.md
index f40aab460f5..f7ab6051833 100644
--- a/src/real-numbers/limits-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/limits-sequences-real-numbers.lagda.md
@@ -10,7 +10,6 @@ module real-numbers.limits-sequences-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.propositional-truncations
 open import foundation.universe-levels
 
 open import lists.sequences
@@ -19,7 +18,6 @@ open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.limits-of-sequences-metric-spaces
 
 open import real-numbers.addition-real-numbers
-open import real-numbers.cauchy-sequences-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-addition-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index 1b56e2c2444..b44914841b4 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -12,14 +12,10 @@ module real-numbers.lipschitz-continuity-multiplication-real-numbers where
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
-open import foundation.function-extensionality
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositional-truncations
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.cartesian-products-metric-spaces
@@ -28,14 +24,11 @@ open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import order-theory.large-posets
 
-open import real-numbers.absolute-value-closed-intervals-real-numbers
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
-open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
-open import real-numbers.enclosing-closed-rational-intervals-real-numbers
 open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.inhabited-totally-bounded-subsets-real-numbers
diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 6cfa5efab42..0372e6da377 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -24,7 +24,6 @@ open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.sets
 open import foundation.similarity-subtypes
diff --git a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
index 34e2ed1f446..543950300ed 100644
--- a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
@@ -9,7 +9,6 @@ module real-numbers.maximum-finite-families-real-numbers where
 ```agda
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.natural-numbers
-open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
@@ -30,7 +29,6 @@ open import lists.finite-sequences
 
 open import logic.functoriality-existential-quantification
 
-open import order-theory.join-semilattices
 open import order-theory.joins-finite-families-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
 open import order-theory.upper-bounds-large-posets
@@ -40,7 +38,6 @@ open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
diff --git a/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md
index b4eb768f8bd..8819476f7c8 100644
--- a/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md
@@ -16,15 +16,9 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
-open import foundation.disjunction
 open import foundation.existential-quantification
-open import foundation.function-types
-open import foundation.functoriality-cartesian-product-types
-open import foundation.inhabited-types
 open import foundation.intersections-subtypes
 open import foundation.logical-equivalences
-open import foundation.powersets
-open import foundation.propositional-truncations
 open import foundation.subtypes
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md b/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
index 5909da62ca6..57f9d1bf809 100644
--- a/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
@@ -21,9 +21,6 @@ open import real-numbers.infima-families-real-numbers
 open import real-numbers.inhabited-finitely-enumerable-subsets-real-numbers
 open import real-numbers.maximum-inhabited-finitely-enumerable-subsets-real-numbers
 open import real-numbers.negation-real-numbers
-open import real-numbers.subsets-real-numbers
-
-open import univalent-combinatorics.finitely-enumerable-subtypes
 ```
 
 </details>
diff --git a/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md
index 860e3cee2ad..e0c6900799d 100644
--- a/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md
@@ -26,7 +26,6 @@ open import logic.functoriality-existential-quantification
 
 open import order-theory.greatest-lower-bounds-large-posets
 open import order-theory.large-meet-semilattices
-open import order-theory.lower-bounds-large-posets
 
 open import real-numbers.inequality-lower-dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
diff --git a/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md
index 590cb5a2e97..6ea8a8c76e7 100644
--- a/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md
@@ -21,7 +21,6 @@ open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-disjunction
 open import foundation.inhabited-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.subtypes
 open import foundation.unions-subtypes
@@ -30,7 +29,6 @@ open import foundation.universe-levels
 open import logic.functoriality-existential-quantification
 
 open import order-theory.greatest-lower-bounds-large-posets
-open import order-theory.large-inflattices
 open import order-theory.lower-bounds-large-posets
 
 open import real-numbers.inequality-upper-dedekind-real-numbers
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index a1bb60ef8e5..b4af60d1c53 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -26,7 +26,6 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index c0fea896952..9e9e66cc377 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -8,7 +8,6 @@ module real-numbers.multiplication-positive-and-negative-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index 7972741f76c..c151f0ecd28 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -9,7 +9,6 @@ module real-numbers.multiplication-positive-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.minimum-positive-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
diff --git a/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md
index 8acc9d09e71..564230fd9b5 100644
--- a/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md
@@ -17,7 +17,6 @@ open import foundation.universe-levels
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.rational-real-numbers
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index f91c1d2bfcf..ad287640f8b 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -19,23 +19,19 @@ open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
-open import foundation.unit-type
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-negative-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 490dd190442..47b08b6c2a9 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -12,9 +12,7 @@ module real-numbers.multiplicative-inverses-positive-real-numbers where
 open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
-open import elementary-number-theory.maximum-positive-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
-open import elementary-number-theory.minimum-positive-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
@@ -39,7 +37,6 @@ open import foundation.empty-types
 open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.function-types
-open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
diff --git a/src/real-numbers/negative-real-numbers.lagda.md b/src/real-numbers/negative-real-numbers.lagda.md
index fd87ef7b8c7..75b3944778a 100644
--- a/src/real-numbers/negative-real-numbers.lagda.md
+++ b/src/real-numbers/negative-real-numbers.lagda.md
@@ -10,7 +10,6 @@ module real-numbers.negative-real-numbers where
 
 ```agda
 open import elementary-number-theory.negative-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index d8219df9e7f..c7b56ff0af2 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -31,8 +31,6 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import metric-spaces.metric-spaces
-
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index 850496dd467..d1d0920e34d 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -10,13 +10,10 @@ module real-numbers.positive-and-negative-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.identity-types
-open import foundation.propositions
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.nonnegative-real-numbers
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index e32729d0f77..9b8c6101ca9 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -9,31 +9,22 @@ module real-numbers.positive-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-rational-numbers
-open import elementary-number-theory.additive-group-of-rational-numbers
-open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.binary-transport
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
-open import group-theory.groups
-
 open import real-numbers.addition-real-numbers
-open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 8038140ac09..55ff07d63df 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import elementary-number-theory.ring-of-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
-open import foundation.disjunction
 open import foundation.identity-types
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
diff --git a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
index 2782e63dab5..c3f1b1d4179 100644
--- a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
+++ b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
@@ -28,9 +28,6 @@ open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
-open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.metric-space-of-rational-numbers
-
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.similarity-real-numbers
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index df01919a624..285729c4a4d 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -9,7 +9,6 @@ module real-numbers.rational-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -17,20 +16,14 @@ open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.disjunction
-open import foundation.embeddings
 open import foundation.empty-types
 open import foundation.equivalences
-open import foundation.existential-quantification
 open import foundation.function-types
-open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.negation
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.retractions
 open import foundation.sections
@@ -38,15 +31,11 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
index 227fa5c7afc..1f1c549e9a1 100644
--- a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.real-numbers-from-lower-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-rational-numbers
-open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
@@ -27,7 +25,6 @@ open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
-open import foundation.propositions
 open import foundation.subtypes
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
index 80c6cacfa97..d1d478c1a34 100644
--- a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.real-numbers-from-upper-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.difference-rational-numbers
-open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
@@ -25,7 +23,6 @@ open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
-open import foundation.propositions
 open import foundation.subtypes
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
index 3686eb0a01d..6b03b09e6f9 100644
--- a/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
@@ -7,19 +7,12 @@ module real-numbers.similarity-nonnegative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.positive-rational-numbers
-
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
-open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/similarity-real-numbers.lagda.md b/src/real-numbers/similarity-real-numbers.lagda.md
index 5664b3a49bc..2c959f7d469 100644
--- a/src/real-numbers/similarity-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-real-numbers.lagda.md
@@ -7,26 +7,17 @@ module real-numbers.similarity-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.strict-inequality-rational-numbers
-
-open import foundation.complements-subtypes
 open import foundation.dependent-pair-types
-open import foundation.disjunction
-open import foundation.empty-types
-open import foundation.function-types
 open import foundation.identity-types
 open import foundation.large-equivalence-relations
 open import foundation.large-similarity-relations
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.similarity-subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import order-theory.large-posets
 open import order-theory.large-preorders
-open import order-theory.similarity-of-elements-large-posets
 
 open import real-numbers.dedekind-real-numbers
 ```
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index e9995706b55..86d106d8c6a 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -9,30 +9,23 @@ module real-numbers.squares-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
 open import elementary-number-theory.multiplication-negative-rational-numbers
-open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
-open import elementary-number-theory.nonpositive-rational-numbers
-open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.squares-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
-open import foundation.empty-types
 open import foundation.existential-quantification
-open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.transport-along-identifications
diff --git a/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
index 26b7ac1b7da..20fe2fbe234 100644
--- a/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
@@ -21,7 +21,6 @@ open import foundation.binary-transport
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
-open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
@@ -34,7 +33,6 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
diff --git a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
index 07f144ab5d9..af7f51e4931 100644
--- a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
diff --git a/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md b/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md
index a0b7199f32b..e24b886aea1 100644
--- a/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md
@@ -12,7 +12,6 @@ module real-numbers.strict-inequality-positive-real-numbers where
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index 96296b0ca74..d632b09a547 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -26,7 +26,6 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.universe-levels
diff --git a/src/real-numbers/subsets-real-numbers.lagda.md b/src/real-numbers/subsets-real-numbers.lagda.md
index b83bc66f442..c42f82f45eb 100644
--- a/src/real-numbers/subsets-real-numbers.lagda.md
+++ b/src/real-numbers/subsets-real-numbers.lagda.md
@@ -20,11 +20,8 @@ open import foundation.involutions
 open import foundation.logical-equivalences
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import metric-spaces.equality-of-metric-spaces
 open import metric-spaces.images-isometries-metric-spaces
 open import metric-spaces.isometries-metric-spaces
diff --git a/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md
index 3f060650bf2..02cfcd06dbb 100644
--- a/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.totally-bounded-subsets-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.positive-rational-numbers
-
 open import foundation.dependent-pair-types
 open import foundation.propositions
 open import foundation.subtypes
@@ -21,7 +19,6 @@ open import metric-spaces.metric-spaces
 open import metric-spaces.subspaces-metric-spaces
 open import metric-spaces.totally-bounded-metric-spaces
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-negation-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.subsets-real-numbers
diff --git a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
index 82a733d1392..7c96ab1c56e 100644
--- a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
@@ -23,7 +23,6 @@ open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index 133ba99ea07..b9ed0136c8e 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -24,7 +24,6 @@ open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.sets
 open import foundation.similarity-subtypes

From f11b6e2ecc5abdd4eae20346a460807ea293f811 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 9 Nov 2025 13:09:19 +0100
Subject: [PATCH 076/134] fix

---
 .../multiplicative-inverses-nonzero-real-numbers.lagda.md     | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index ad287640f8b..51ab87da430 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -132,8 +132,8 @@ module _
 ### If a real number has a multiplicative inverse, it is nonzero
 
 ```agda
-opaque
-  unfolding leq-ℝ mul-ℝ real-ℚ sim-ℝ
+abstract opaque
+  unfolding mul-ℝ real-ℚ sim-ℝ
 
   is-nonzero-has-right-inverse-mul-ℝ :
     {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →

From 2c0bdbcfc81c2e2f20941257b3fbfdec7d3dbdac Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 9 Nov 2025 13:30:32 +0100
Subject: [PATCH 077/134] chore: optimize imports rational numbers

---
 .../addition-rational-numbers.lagda.md                       | 1 -
 .../difference-rational-numbers.lagda.md                     | 1 -
 .../distance-rational-numbers.lagda.md                       | 2 --
 .../geometric-sequences-rational-numbers.lagda.md            | 4 ----
 .../inequality-positive-rational-numbers.lagda.md            | 1 -
 .../metric-additive-group-of-rational-numbers.lagda.md       | 2 --
 .../multiplicative-group-of-rational-numbers.lagda.md        | 2 --
 .../negative-rational-numbers.lagda.md                       | 2 --
 .../positive-and-negative-rational-numbers.lagda.md          | 2 --
 .../positive-rational-numbers.lagda.md                       | 4 ----
 .../powers-nonnegative-rational-numbers.lagda.md             | 2 --
 .../powers-positive-rational-numbers.lagda.md                | 1 -
 .../powers-rational-numbers.lagda.md                         | 2 --
 .../square-roots-positive-rational-numbers.lagda.md          | 1 -
 .../squares-rational-numbers.lagda.md                        | 2 --
 .../strict-inequality-rational-numbers.lagda.md              | 5 -----
 src/elementary-number-theory/triangular-numbers.lagda.md     | 4 ----
 17 files changed, 38 deletions(-)

diff --git a/src/elementary-number-theory/addition-rational-numbers.lagda.md b/src/elementary-number-theory/addition-rational-numbers.lagda.md
index 4e3854dddfc..72024841c02 100644
--- a/src/elementary-number-theory/addition-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-rational-numbers.lagda.md
@@ -23,7 +23,6 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
-open import foundation.injective-maps
 open import foundation.interchange-law
 open import foundation.retractions
 open import foundation.sections
diff --git a/src/elementary-number-theory/difference-rational-numbers.lagda.md b/src/elementary-number-theory/difference-rational-numbers.lagda.md
index e4830f7cff5..cc782a4c7cd 100644
--- a/src/elementary-number-theory/difference-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/difference-rational-numbers.lagda.md
@@ -17,7 +17,6 @@ open import elementary-number-theory.rational-numbers
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.identity-types
-open import foundation.interchange-law
 ```
 
 </details>
diff --git a/src/elementary-number-theory/distance-rational-numbers.lagda.md b/src/elementary-number-theory/distance-rational-numbers.lagda.md
index 3926d248dc4..ccc7b5f8b30 100644
--- a/src/elementary-number-theory/distance-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-rational-numbers.lagda.md
@@ -21,13 +21,11 @@ open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
-open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
-open import foundation.binary-transport
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.function-types
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 21695e9d15a..07004973a88 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -29,9 +29,7 @@ open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.ring-of-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
-open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.negated-equality
@@ -45,8 +43,6 @@ open import lists.sequences
 open import metric-spaces.limits-of-sequences-metric-spaces
 open import metric-spaces.metric-space-of-rational-numbers
 open import metric-spaces.uniformly-continuous-functions-metric-spaces
-
-open import order-theory.strictly-increasing-sequences-strictly-preordered-sets
 ```
 
 </details>
diff --git a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
index 1398ef233b7..bd42b31bd9b 100644
--- a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
@@ -13,7 +13,6 @@ open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
-open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
index ba29b988788..c692847c8c8 100644
--- a/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
@@ -9,9 +9,7 @@ module elementary-number-theory.metric-additive-group-of-rational-numbers where
 ```agda
 open import analysis.metric-abelian-groups
 
-open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index 30f54cea5a3..9b527c5f536 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -16,7 +16,6 @@ open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
-open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -27,7 +26,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
-open import foundation.empty-types
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.negated-equality
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 7a41b1a752e..15c00ecac48 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -17,13 +17,11 @@ open import elementary-number-theory.negative-integer-fractions
 open import elementary-number-theory.negative-integers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-integers
-open import elementary-number-theory.positive-integer-fractions
 open import elementary-number-theory.positive-integers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index 021ef28f500..09a711a99e7 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -11,13 +11,11 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
-open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-integers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.binary-transport
 open import foundation.cartesian-product-types
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index f146b5f15f9..eda656f8218 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -15,17 +15,13 @@ open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integer-fractions
 open import elementary-number-theory.integers
-open import elementary-number-theory.negative-integers
 open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.nonzero-rational-numbers
-open import elementary-number-theory.positive-and-negative-integers
 open import elementary-number-theory.positive-integer-fractions
 open import elementary-number-theory.positive-integers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.cartesian-product-types
-open import foundation.coproduct-types
 open import foundation.decidable-propositions
 open import foundation.decidable-subtypes
 open import foundation.dependent-pair-types
diff --git a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
index 535de721529..b07d9baf6df 100644
--- a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
@@ -17,7 +17,6 @@ open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-nonnegative-rational-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
-open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-nonnegative-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -27,7 +26,6 @@ open import foundation.empty-types
 open import foundation.identity-types
 
 open import group-theory.powers-of-elements-commutative-monoids
-open import group-theory.powers-of-elements-monoids
 ```
 
 </details>
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 05641432b4b..1e1dbbcd56b 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -25,7 +25,6 @@ open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
-open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index ea1efba98d3..51e308477d0 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -14,9 +14,7 @@ open import elementary-number-theory.inequality-natural-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.multiplication-natural-numbers
-open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
-open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
 open import elementary-number-theory.natural-numbers
diff --git a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
index 7687c6d3d4b..c6621bc79b1 100644
--- a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
@@ -14,7 +14,6 @@ open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
-open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minimum-positive-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index ada7062ba55..41b169790a3 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -18,7 +18,6 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
-open import elementary-number-theory.multiplication-nonpositive-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
@@ -31,7 +30,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
-open import foundation.decidable-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.function-types
diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
index ddaa7ba6166..e237a00ca06 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -13,10 +13,7 @@ open import elementary-number-theory.addition-integer-fractions
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.cross-multiplication-difference-integer-fractions
-open import elementary-number-theory.difference-integers
 open import elementary-number-theory.difference-rational-numbers
-open import elementary-number-theory.inequality-integer-fractions
-open import elementary-number-theory.inequality-integers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integer-fractions
 open import elementary-number-theory.integers
@@ -26,7 +23,6 @@ open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-integers
 open import elementary-number-theory.nonpositive-integers
 open import elementary-number-theory.positive-and-negative-integers
-open import elementary-number-theory.positive-integers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.reduced-integer-fractions
 open import elementary-number-theory.strict-inequality-integer-fractions
@@ -36,7 +32,6 @@ open import elementary-number-theory.strict-inequality-natural-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.decidable-propositions
diff --git a/src/elementary-number-theory/triangular-numbers.lagda.md b/src/elementary-number-theory/triangular-numbers.lagda.md
index 451206ca038..23ae32125c6 100644
--- a/src/elementary-number-theory/triangular-numbers.lagda.md
+++ b/src/elementary-number-theory/triangular-numbers.lagda.md
@@ -35,20 +35,16 @@ open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
 
 open import group-theory.abelian-groups
 open import group-theory.groups
 
-open import lists.sequences
-
 open import metric-spaces.limits-of-sequences-metric-spaces
 open import metric-spaces.metric-space-of-rational-numbers
 open import metric-spaces.rational-sequences-approximating-zero
 open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import ring-theory.partial-sums-sequences-semirings
-open import ring-theory.sums-of-finite-sequences-of-elements-semirings
 ```
 
 </details>

From 908914f81d8a73278c3fec0517560c245dd7ffc9 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Nov 2025 11:38:01 -0800
Subject: [PATCH 078/134] Fixes

---
 ...metric-sequences-rational-numbers.lagda.md | 62 +------------------
 .../apartness-real-numbers.lagda.md           |  2 +-
 2 files changed, 2 insertions(+), 62 deletions(-)

diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index e93c63c655b..3773379808f 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -30,6 +30,7 @@ open import elementary-number-theory.ring-of-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.binary-transport
+open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.negated-equality
@@ -176,67 +177,6 @@ module _
             ( is-zero-limit-power-le-one-abs-ℚ r |r|<1))
 ```
 
-### If `|r| < 1`, the sum of the standard geometric sequence `n ↦ arⁿ` is `a/(1-r)`
-
-```agda
-module _
-  (a r : ℚ)
-  where
-
-  standard-geometric-series-ℚ : series-Metric-Ab metric-ab-add-ℚ
-  standard-geometric-series-ℚ =
-    series-terms-Metric-Ab (seq-standard-geometric-sequence-ℚ a r)
-
-  abstract
-    sum-standard-geometric-sequence-ℚ :
-      (|r|<1 : le-ℚ (rational-abs-ℚ r) one-ℚ) →
-      is-sum-series-Metric-Ab
-        ( standard-geometric-series-ℚ)
-        ( a *ℚ rational-inv-ℚˣ (invertible-diff-le-abs-ℚ r one-ℚ⁺ |r|<1))
-    sum-standard-geometric-sequence-ℚ |r|<1 =
-      let
-        r≠1 =
-          nonequal-map
-            ( rational-abs-ℚ)
-            ( inv-tr
-              ( rational-abs-ℚ r ≠_)
-              ( rational-abs-rational-ℚ⁺ one-ℚ⁺)
-              ( nonequal-le-ℚ |r|<1))
-      in
-        binary-tr
-          ( is-limit-sequence-Metric-Space metric-space-ℚ)
-          ( inv
-            ( eq-htpy (compute-sum-standard-geometric-fin-sequence-ℚ a r r≠1)))
-          ( equational-reasoning
-            a *ℚ
-            ( (one-ℚ -ℚ zero-ℚ) *ℚ
-              rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1))
-            ＝
-              a *ℚ
-              ( one-ℚ *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1))
-              by ap-mul-ℚ refl (ap-mul-ℚ (right-zero-law-diff-ℚ one-ℚ) refl)
-            ＝ a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
-              by ap-mul-ℚ refl (left-unit-law-mul-ℚ _))
-          ( uniformly-continuous-map-limit-sequence-Metric-Space
-            ( metric-space-ℚ)
-            ( metric-space-ℚ)
-            ( comp-uniformly-continuous-function-Metric-Space
-              ( metric-space-ℚ)
-              ( metric-space-ℚ)
-              ( metric-space-ℚ)
-              ( uniformly-continuous-left-mul-ℚ a)
-              ( comp-uniformly-continuous-function-Metric-Space
-                ( metric-space-ℚ)
-                ( metric-space-ℚ)
-                ( metric-space-ℚ)
-                ( uniformly-continuous-right-mul-ℚ
-                  ( rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)))
-                ( uniformly-continuous-diff-ℚ one-ℚ)))
-            ( λ n → power-ℚ n r)
-            ( zero-ℚ)
-            ( is-zero-limit-power-le-one-abs-ℚ r |r|<1))
-```
-
 ## External links
 
 - [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 5067df3652d..a362297a3d6 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -25,7 +25,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 

From 7542dd2ea7af8204392bd8c7259fc6fdf5924b81 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Nov 2025 11:46:34 -0800
Subject: [PATCH 079/134] Fix build

---
 .../geometric-sequences-rational-numbers.lagda.md                | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 45d0873b490..3773379808f 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -30,6 +30,7 @@ open import elementary-number-theory.ring-of-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.binary-transport
+open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.negated-equality

From fc7be890e196e7b67b097ac6b485f97162ccf4a8 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Nov 2025 15:43:37 -0800
Subject: [PATCH 080/134] Multiplicative inverses of nonzero complex numbers

---
 src/complex-numbers.lagda.md                  |   5 +
 .../apartness-complex-numbers.lagda.md        |  16 +-
 src/complex-numbers/complex-numbers.lagda.md  |  17 ++-
 .../conjugation-complex-numbers.lagda.md      |  42 ++++++
 .../magnitude-complex-numbers.lagda.md        |  95 ++++++++++++
 .../multiplication-complex-numbers.lagda.md   |  50 +++++++
 ...-inverses-nonzero-complex-numbers.lagda.md |  85 +++++++++++
 .../nonzero-complex-numbers.lagda.md          |  68 ++++++++-
 .../similarity-complex-numbers.lagda.md       |  51 +++++++
 .../addition-rational-numbers.lagda.md        |   7 +
 src/real-numbers.lagda.md                     |   1 +
 ...addition-nonnegative-real-numbers.lagda.md |  21 +++
 .../addition-positive-real-numbers.lagda.md   | 141 ++++++++++++++++++
 .../binary-maximum-real-numbers.lagda.md      |   1 +
 ...ompleteness-dedekind-real-numbers.lagda.md |   1 +
 .../nonzero-real-numbers.lagda.md             |  41 +++++
 .../positive-real-numbers.lagda.md            |  38 -----
 ...re-roots-nonnegative-real-numbers.lagda.md |  22 +++
 .../squares-real-numbers.lagda.md             |  21 +++
 19 files changed, 669 insertions(+), 54 deletions(-)
 create mode 100644 src/complex-numbers/conjugation-complex-numbers.lagda.md
 create mode 100644 src/complex-numbers/magnitude-complex-numbers.lagda.md
 create mode 100644 src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
 create mode 100644 src/real-numbers/addition-positive-real-numbers.lagda.md

diff --git a/src/complex-numbers.lagda.md b/src/complex-numbers.lagda.md
index 70b0af5ab4d..694c4263882 100644
--- a/src/complex-numbers.lagda.md
+++ b/src/complex-numbers.lagda.md
@@ -5,10 +5,15 @@ module complex-numbers where
 
 open import complex-numbers.addition-complex-numbers public
 open import complex-numbers.additive-group-of-complex-numbers public
+open import complex-numbers.apartness-complex-numbers public
 open import complex-numbers.complex-numbers public
+open import complex-numbers.conjugation-complex-numbers public
 open import complex-numbers.eisenstein-integers public
 open import complex-numbers.gaussian-integers public
+open import complex-numbers.magnitude-complex-numbers public
 open import complex-numbers.multiplication-complex-numbers public
+open import complex-numbers.multiplicative-inverses-nonzero-complex-numbers public
+open import complex-numbers.nonzero-complex-numbers public
 open import complex-numbers.ring-of-complex-numbers public
 open import complex-numbers.similarity-complex-numbers public
 ```
diff --git a/src/complex-numbers/apartness-complex-numbers.lagda.md b/src/complex-numbers/apartness-complex-numbers.lagda.md
index 762d6da4092..9a0aac2539b 100644
--- a/src/complex-numbers/apartness-complex-numbers.lagda.md
+++ b/src/complex-numbers/apartness-complex-numbers.lagda.md
@@ -7,17 +7,19 @@ module complex-numbers.apartness-complex-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.negation
 open import complex-numbers.complex-numbers
-open import foundation.function-types
-open import foundation.large-apartness-relations
+
 open import foundation.dependent-pair-types
-open import foundation.functoriality-disjunction
-open import real-numbers.apartness-real-numbers
-open import foundation.empty-types
 open import foundation.disjunction
+open import foundation.empty-types
+open import foundation.function-types
+open import foundation.functoriality-disjunction
+open import foundation.large-apartness-relations
+open import foundation.negation
 open import foundation.propositions
 open import foundation.universe-levels
+
+open import real-numbers.apartness-real-numbers
 ```
 
 </details>
@@ -66,7 +68,7 @@ abstract
   symmetric-apart-ℂ :
     {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2) → apart-ℂ z w → apart-ℂ w z
   symmetric-apart-ℂ (a , b) (c , d) =
-    map-disjunction (symmetric-apart-ℝ a c) (symmetric-apart-ℝ b d)
+    map-disjunction symmetric-apart-ℝ symmetric-apart-ℝ
 ```
 
 ### Apartness is cotransitive
diff --git a/src/complex-numbers/complex-numbers.lagda.md b/src/complex-numbers/complex-numbers.lagda.md
index ec8571f0de9..7465731aca5 100644
--- a/src/complex-numbers/complex-numbers.lagda.md
+++ b/src/complex-numbers/complex-numbers.lagda.md
@@ -38,6 +38,8 @@ are numbers of the form `a + bi`, where `a` and `b` are
 ℂ : (l : Level) → UU (lsuc l)
 ℂ l = ℝ l × ℝ l
 
+pattern _+iℂ_ x y = (x , y)
+
 re-ℂ : {l : Level} → ℂ l → ℝ l
 re-ℂ = pr1
 
@@ -79,13 +81,6 @@ complex-ℤ[i] : ℤ[i] → ℂ lzero
 complex-ℤ[i] (a , b) = (real-ℤ a , real-ℤ b)
 ```
 
-### The conjugate of a complex number
-
-```agda
-conjugate-ℂ : {l : Level} → ℂ l → ℂ l
-conjugate-ℂ (a , b) = (a , neg-ℝ b)
-```
-
 ### Important complex numbers
 
 ```agda
@@ -108,3 +103,11 @@ i-ℂ = (zero-ℝ , one-ℝ)
 neg-ℂ : {l : Level} → ℂ l → ℂ l
 neg-ℂ (a , b) = (neg-ℝ a , neg-ℝ b)
 ```
+
+### `complex-ℝ one-ℝ` is equal to `one-ℂ`
+
+```agda
+abstract
+  eq-complex-one-ℝ : complex-ℝ one-ℝ ＝ one-ℂ
+  eq-complex-one-ℝ = eq-ℂ refl (inv (eq-raise-ℝ zero-ℝ))
+```
diff --git a/src/complex-numbers/conjugation-complex-numbers.lagda.md b/src/complex-numbers/conjugation-complex-numbers.lagda.md
new file mode 100644
index 00000000000..8f3eb54ba19
--- /dev/null
+++ b/src/complex-numbers/conjugation-complex-numbers.lagda.md
@@ -0,0 +1,42 @@
+# Conjugation of complex numbers
+
+```agda
+module complex-numbers.conjugation-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.complex-numbers
+
+open import foundation.identity-types
+open import foundation.universe-levels
+
+open import real-numbers.negation-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "conjugate" WDID=Q381040 WD="complex conjugate" Disambiguation="of a complex number" Agda=conjugate-ℂ}}
+of a [complex number](complex-numbers.complex-numbers.md) `a + bi` is `a - bi`.
+
+## Definition
+
+```agda
+conjugate-ℂ : {l : Level} → ℂ l → ℂ l
+conjugate-ℂ (a +iℂ b) = a +iℂ neg-ℝ b
+```
+
+## Properties
+
+### Conjugation is an involution
+
+```agda
+abstract
+  is-involution-conjugate-ℂ :
+    {l : Level} (z : ℂ l) → conjugate-ℂ (conjugate-ℂ z) ＝ z
+  is-involution-conjugate-ℂ (a +iℂ b) = eq-ℂ refl (neg-neg-ℝ b)
+```
diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
new file mode 100644
index 00000000000..d46272ba5de
--- /dev/null
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -0,0 +1,95 @@
+# Magnitude of complex numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module complex-numbers.magnitude-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.complex-numbers
+open import complex-numbers.conjugation-complex-numbers
+open import complex-numbers.multiplication-complex-numbers
+
+open import foundation.identity-types
+open import foundation.universe-levels
+
+open import real-numbers.addition-nonnegative-real-numbers
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.multiplication-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.rational-real-numbers
+-- sopen import real-numbers.similarity-real-numbers
+open import real-numbers.square-roots-nonnegative-real-numbers
+open import real-numbers.squares-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "magnitude" WD="magnitude of a complex number" WDID=Q3317982 Agda=magnitude-ℂ}}
+of a [complex number](complex-numbers.complex-numbers.md) `a + bi` is defined as
+$$\sqrt{a^2 + b^2}}$$.
+
+## Definition
+
+```agda
+nonnegative-squared-magnitude-ℂ : {l : Level} → ℂ l → ℝ⁰⁺ l
+nonnegative-squared-magnitude-ℂ (a +iℂ b) =
+  nonnegative-square-ℝ a +ℝ⁰⁺ nonnegative-square-ℝ b
+
+squared-magnitude-ℂ : {l : Level} → ℂ l → ℝ l
+squared-magnitude-ℂ z = real-ℝ⁰⁺ (nonnegative-squared-magnitude-ℂ z)
+
+∥_∥²ℂ : {l : Level} → ℂ l → ℝ l
+∥ z ∥²ℂ = squared-magnitude-ℂ z
+
+nonnegative-magnitude-ℂ : {l : Level} → ℂ l → ℝ⁰⁺ l
+nonnegative-magnitude-ℂ z = sqrt-ℝ⁰⁺ (nonnegative-squared-magnitude-ℂ z)
+
+magnitude-ℂ : {l : Level} → ℂ l → ℝ l
+magnitude-ℂ z = real-ℝ⁰⁺ (nonnegative-magnitude-ℂ z)
+
+∥_∥ℂ : {l : Level} → ℂ l → ℝ l
+∥ z ∥ℂ = magnitude-ℂ z
+```
+
+## Properties
+
+### The square of the magnitude of `z` is the product of `z` and the conjugate of `z`
+
+```agda
+abstract
+  eq-squared-magnitude-mul-conjugate-ℂ :
+    {l : Level} (z : ℂ l) →
+    z *ℂ conjugate-ℂ z ＝ complex-ℝ (squared-magnitude-ℂ z)
+  eq-squared-magnitude-mul-conjugate-ℂ (a +iℂ b) =
+    eq-ℂ
+      ( equational-reasoning
+        square-ℝ a -ℝ (b *ℝ neg-ℝ b)
+        ＝ square-ℝ a -ℝ (neg-ℝ (square-ℝ b))
+          by ap-diff-ℝ refl (right-negative-law-mul-ℝ _ _)
+        ＝ square-ℝ a +ℝ square-ℝ b
+          by ap-add-ℝ refl (neg-neg-ℝ _))
+      ( eq-sim-ℝ
+        ( similarity-reasoning-ℝ
+          a *ℝ neg-ℝ b +ℝ b *ℝ a
+          ~ℝ neg-ℝ (a *ℝ b) +ℝ a *ℝ b
+            by
+              sim-eq-ℝ
+                ( ap-add-ℝ
+                  ( right-negative-law-mul-ℝ a b)
+                  ( commutative-mul-ℝ b a))
+          ~ℝ zero-ℝ
+            by left-inverse-law-add-ℝ (a *ℝ b)
+          ~ℝ raise-ℝ _ zero-ℝ
+            by sim-raise-ℝ _ _))
+```
diff --git a/src/complex-numbers/multiplication-complex-numbers.lagda.md b/src/complex-numbers/multiplication-complex-numbers.lagda.md
index 12598cc434b..7e33daf3dc5 100644
--- a/src/complex-numbers/multiplication-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplication-complex-numbers.lagda.md
@@ -13,6 +13,7 @@ open import complex-numbers.similarity-complex-numbers
 
 open import elementary-number-theory.rational-numbers
 
+open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
@@ -20,9 +21,11 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
 ```
@@ -45,6 +48,11 @@ mul-ℂ (a , b) (c , d) = (a *ℝ c -ℝ b *ℝ d , a *ℝ d +ℝ b *ℝ c)
 infixl 40 _*ℂ_
 _*ℂ_ : {l1 l2 : Level} → ℂ l1 → ℂ l2 → ℂ (l1 ⊔ l2)
 _*ℂ_ = mul-ℂ
+
+ap-mul-ℂ :
+  {l1 l2 : Level} {z z' : ℂ l1} → z ＝ z' → {w w' : ℂ l2} → w ＝ w' →
+  mul-ℂ z w ＝ mul-ℂ z' w'
+ap-mul-ℂ = ap-binary mul-ℂ
 ```
 
 ## Properties
@@ -255,3 +263,45 @@ opaque
         ＝ zero-ℝ
           by left-unit-law-add-ℝ zero-ℝ)
 ```
+
+### The canonical embedding of real numbers in the complex numbers preserves multiplication
+
+```agda
+abstract
+  mul-complex-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) →
+    complex-ℝ x *ℂ complex-ℝ y ＝ complex-ℝ (x *ℝ y)
+  mul-complex-ℝ {l1} {l2} x y =
+    eq-ℂ
+      ( equational-reasoning
+        x *ℝ y -ℝ raise-ℝ l1 zero-ℝ *ℝ raise-ℝ l2 zero-ℝ
+        ＝ x *ℝ y -ℝ zero-ℝ *ℝ zero-ℝ
+          by
+            eq-sim-ℝ
+              ( preserves-sim-diff-ℝ
+                ( refl-sim-ℝ (x *ℝ y))
+                ( symmetric-sim-ℝ
+                  ( preserves-sim-mul-ℝ (sim-raise-ℝ _ _) (sim-raise-ℝ _ _))))
+        ＝ x *ℝ y -ℝ zero-ℝ
+          by ap-diff-ℝ refl (eq-sim-ℝ (right-zero-law-mul-ℝ _))
+        ＝ x *ℝ y
+          by right-unit-law-diff-ℝ (x *ℝ y))
+      ( eq-sim-ℝ
+        ( similarity-reasoning-ℝ
+          x *ℝ raise-ℝ l2 zero-ℝ +ℝ raise-ℝ l1 zero-ℝ *ℝ y
+          ~ℝ x *ℝ zero-ℝ +ℝ zero-ℝ *ℝ y
+            by
+              symmetric-sim-ℝ
+                ( preserves-sim-add-ℝ
+                  ( preserves-sim-left-mul-ℝ _ _ _ (sim-raise-ℝ l2 zero-ℝ))
+                  ( preserves-sim-right-mul-ℝ _ _ _ (sim-raise-ℝ l1 zero-ℝ)))
+          ~ℝ zero-ℝ +ℝ zero-ℝ
+            by
+              preserves-sim-add-ℝ
+                ( right-zero-law-mul-ℝ x)
+                ( left-zero-law-mul-ℝ y)
+          ~ℝ zero-ℝ
+            by sim-eq-ℝ (left-unit-law-add-ℝ zero-ℝ)
+          ~ℝ raise-ℝ (l1 ⊔ l2) zero-ℝ
+            by sim-raise-ℝ (l1 ⊔ l2) zero-ℝ))
+```
diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
new file mode 100644
index 00000000000..320c7b26cdf
--- /dev/null
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -0,0 +1,85 @@
+# Multiplicative inverses of complex numbers
+
+```agda
+module complex-numbers.multiplicative-inverses-nonzero-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.complex-numbers
+-- open import complex-numbers.conjugation-complex-numbers
+open import complex-numbers.magnitude-complex-numbers
+open import complex-numbers.multiplication-complex-numbers
+open import complex-numbers.nonzero-complex-numbers
+open import complex-numbers.similarity-complex-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.identity-types
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.multiplication-real-numbers
+open import real-numbers.multiplicative-inverses-positive-real-numbers
+open import real-numbers.rational-real-numbers
+```
+
+</details>
+
+## Idea
+
+[Nonzero](complex-numbers.nonzero-complex-numbers.md)
+[complex numbers](complex-numbers.complex-numbers.md) have inverses under
+[multiplication](complex-numbers.multiplication-complex-numbers.md).
+
+## Definition
+
+```agda
+complex-inv-nonzero-ℂ : {l : Level} (z : nonzero-ℂ l) → ℂ l
+complex-inv-nonzero-ℂ (z , z≠0) =
+  conjugate-ℂ z *ℂ
+  complex-ℝ (real-inv-ℝ⁺ (positive-squared-magnitude-nonzero-ℂ (z , z≠0)))
+```
+
+## Properties
+
+### Inverse laws of multiplication
+
+```agda
+abstract
+  left-inverse-law-mul-ℂ : {l : Level} (z : nonzero-ℂ l) →
+    sim-ℂ (complex-nonzero-ℂ z *ℂ complex-inv-nonzero-ℂ z) one-ℂ
+  left-inverse-law-mul-ℂ (z@(a +iℂ b) , z≠0) =
+    similarity-reasoning-ℂ
+      z *ℂ
+      ( conjugate-ℂ z *ℂ
+        complex-ℝ
+          ( real-inv-ℝ⁺ (positive-squared-magnitude-nonzero-ℂ (z , z≠0))))
+      ~ℂ
+        ( z *ℂ conjugate-ℂ z) *ℂ
+        ( complex-ℝ
+          ( real-inv-ℝ⁺ (positive-squared-magnitude-nonzero-ℂ (z , z≠0))))
+        by sim-eq-ℂ (inv (associative-mul-ℂ _ _ _))
+      ~ℂ
+        complex-ℝ (squared-magnitude-ℂ z) *ℂ
+        complex-ℝ
+          ( real-inv-ℝ⁺ (positive-squared-magnitude-nonzero-ℂ (z , z≠0)))
+        by sim-eq-ℂ (ap-mul-ℂ (eq-squared-magnitude-mul-conjugate-ℂ z) refl)
+      ~ℂ
+        complex-ℝ
+          ( squared-magnitude-ℂ z *ℝ
+            real-inv-ℝ⁺ (positive-squared-magnitude-nonzero-ℂ (z , z≠0)))
+        by sim-eq-ℂ (mul-complex-ℝ _ _)
+      ~ℂ complex-ℝ one-ℝ
+        by
+          preserves-sim-complex-ℝ
+            ( right-inverse-law-mul-ℝ⁺
+              ( positive-squared-magnitude-nonzero-ℂ (z , z≠0)))
+      ~ℂ one-ℂ
+        by sim-eq-ℂ eq-complex-one-ℝ
+
+  right-inverse-law-mul-ℂ : {l : Level} (z : nonzero-ℂ l) →
+    sim-ℂ (complex-inv-nonzero-ℂ z *ℂ complex-nonzero-ℂ z) one-ℂ
+  right-inverse-law-mul-ℂ z =
+    tr (λ w → sim-ℂ w one-ℂ) (commutative-mul-ℂ _ _) (left-inverse-law-mul-ℂ z)
+```
diff --git a/src/complex-numbers/nonzero-complex-numbers.lagda.md b/src/complex-numbers/nonzero-complex-numbers.lagda.md
index ac02560e7cb..a10fec58972 100644
--- a/src/complex-numbers/nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/nonzero-complex-numbers.lagda.md
@@ -7,10 +7,27 @@ module complex-numbers.nonzero-complex-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import complex-numbers.apartness-complex-numbers
 open import complex-numbers.complex-numbers
-open import foundation.universe-levels
+open import complex-numbers.magnitude-complex-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.disjunction
 open import foundation.propositions
-open import complex-numbers.apartness-complex-numbers
+open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-nonnegative-real-numbers
+open import real-numbers.addition-positive-real-numbers
+open import real-numbers.addition-real-numbers
+open import real-numbers.nonzero-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.squares-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
@@ -32,4 +49,51 @@ is-nonzero-prop-ℂ z = apart-prop-ℂ z zero-ℂ
 
 is-nonzero-ℂ : {l : Level} → ℂ l → UU l
 is-nonzero-ℂ z = type-Prop (is-nonzero-prop-ℂ z)
+
+nonzero-ℂ : (l : Level) → UU (lsuc l)
+nonzero-ℂ l = type-subtype (is-nonzero-prop-ℂ {l})
+
+complex-nonzero-ℂ : {l : Level} → nonzero-ℂ l → ℂ l
+complex-nonzero-ℂ = pr1
+```
+
+## Properties
+
+### A complex number is nonzero if and only if its squared magnitude is positive
+
+```agda
+abstract
+  is-positive-squared-magnitude-is-nonzero-ℂ :
+    {l : Level} (z : ℂ l) → is-nonzero-ℂ z → is-positive-ℝ ∥ z ∥²ℂ
+  is-positive-squared-magnitude-is-nonzero-ℂ (a +iℂ b) =
+    elim-disjunction
+      ( is-positive-prop-ℝ ∥ a +iℂ b ∥²ℂ)
+      ( λ a≠0 →
+        concatenate-le-leq-ℝ
+          ( zero-ℝ)
+          ( square-ℝ a)
+          ( square-ℝ a +ℝ square-ℝ b)
+          ( is-positive-square-is-nonzero-ℝ a a≠0)
+          ( leq-left-add-real-ℝ⁰⁺ _ (nonnegative-square-ℝ b)))
+      ( λ b≠0 →
+        concatenate-le-leq-ℝ
+          ( zero-ℝ)
+          ( square-ℝ b)
+          ( square-ℝ a +ℝ square-ℝ b)
+          ( is-positive-square-is-nonzero-ℝ b b≠0)
+          ( leq-right-add-real-ℝ⁰⁺ _ (nonnegative-square-ℝ a)))
+
+  is-nonzero-is-positive-squared-magnitude-ℂ :
+    {l : Level} (z : ℂ l) → is-positive-ℝ ∥ z ∥²ℂ → is-nonzero-ℂ z
+  is-nonzero-is-positive-squared-magnitude-ℂ (a +iℂ b) 0<a²+b² =
+    elim-disjunction
+      ( is-nonzero-prop-ℂ (a +iℂ b))
+      ( λ 0<a² → inl-disjunction (is-nonzero-square-is-positive-ℝ a 0<a²))
+      ( λ 0<b² → inr-disjunction (is-nonzero-square-is-positive-ℝ b 0<b²))
+      ( is-positive-either-is-positive-add-ℝ (square-ℝ a) (square-ℝ b) 0<a²+b²)
+
+positive-squared-magnitude-nonzero-ℂ :
+  {l : Level} (z : nonzero-ℂ l) → ℝ⁺ l
+positive-squared-magnitude-nonzero-ℂ (z , z≠0) =
+  ( squared-magnitude-ℂ z , is-positive-squared-magnitude-is-nonzero-ℂ z z≠0)
 ```
diff --git a/src/complex-numbers/similarity-complex-numbers.lagda.md b/src/complex-numbers/similarity-complex-numbers.lagda.md
index 2874755ca84..0f64ff89c9d 100644
--- a/src/complex-numbers/similarity-complex-numbers.lagda.md
+++ b/src/complex-numbers/similarity-complex-numbers.lagda.md
@@ -15,6 +15,9 @@ open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
 
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
 ```
 
@@ -45,6 +48,9 @@ sim-ℂ a+bi c+di = type-Prop (sim-prop-ℂ a+bi c+di)
 abstract
   refl-sim-ℂ : {l : Level} (z : ℂ l) → sim-ℂ z z
   refl-sim-ℂ (a , b) = (refl-sim-ℝ a , refl-sim-ℝ b)
+
+  sim-eq-ℂ : {l : Level} {z w : ℂ l} → z ＝ w → sim-ℂ z w
+  sim-eq-ℂ {z = z} refl = refl-sim-ℂ z
 ```
 
 ### Similarity is symmetric
@@ -74,3 +80,48 @@ abstract
   eq-sim-ℂ : {l : Level} {x y : ℂ l} → sim-ℂ x y → x ＝ y
   eq-sim-ℂ (a~c , b~d) = eq-ℂ (eq-sim-ℝ a~c) (eq-sim-ℝ b~d)
 ```
+
+### The canonical embedding of real numbers in the complex numbers preserves similarity
+
+```agda
+abstract
+  preserves-sim-complex-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → sim-ℝ x y →
+    sim-ℂ (complex-ℝ x) (complex-ℝ y)
+  preserves-sim-complex-ℝ {l1} {l2} x~y =
+    ( x~y ,
+      transitive-sim-ℝ
+        ( raise-ℝ l1 zero-ℝ)
+        ( zero-ℝ)
+        ( raise-ℝ l2 zero-ℝ)
+        ( sim-raise-ℝ l2 zero-ℝ)
+        ( symmetric-sim-ℝ (sim-raise-ℝ l1 zero-ℝ)))
+```
+
+### Similarity reasoning
+
+Similarities between complex numbers can be constructed by similarity reasoning
+in the following way:
+
+```text
+similarity-reasoning-ℂ
+  x ~ℂ y by sim-1
+    ~ℂ z by sim-2
+```
+
+```agda
+infixl 1 similarity-reasoning-ℂ_
+infixl 0 step-similarity-reasoning-ℂ
+
+abstract
+  similarity-reasoning-ℂ_ :
+    {l : Level} → (x : ℂ l) → sim-ℂ x x
+  similarity-reasoning-ℂ x = refl-sim-ℂ x
+
+  step-similarity-reasoning-ℂ :
+    {l1 l2 : Level} {x : ℂ l1} {y : ℂ l2} →
+    sim-ℂ x y → {l3 : Level} → (u : ℂ l3) → sim-ℂ y u → sim-ℂ x u
+  step-similarity-reasoning-ℂ {x = x} {y = y} p u q = transitive-sim-ℂ x y u q p
+
+  syntax step-similarity-reasoning-ℂ p u q = p ~ℂ u by q
+```
diff --git a/src/elementary-number-theory/addition-rational-numbers.lagda.md b/src/elementary-number-theory/addition-rational-numbers.lagda.md
index 72024841c02..76aba4829fe 100644
--- a/src/elementary-number-theory/addition-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-rational-numbers.lagda.md
@@ -185,6 +185,13 @@ abstract
       ＝ (p +ℚ q) +ℚ neg-ℚ q by inv (associative-add-ℚ _ _ _)
       ＝ zero-ℚ +ℚ neg-ℚ q by ap (_+ℚ neg-ℚ q) p+q=0
       ＝ neg-ℚ q by left-unit-law-add-ℚ _
+
+  unique-right-neg-ℚ : (p q : ℚ) → p +ℚ q ＝ zero-ℚ → q ＝ neg-ℚ p
+  unique-right-neg-ℚ p q p+q=0 =
+    equational-reasoning
+      q
+      ＝ neg-ℚ (neg-ℚ q) by inv (neg-neg-ℚ q)
+      ＝ neg-ℚ p by ap neg-ℚ (inv (unique-left-neg-ℚ p q p+q=0))
 ```
 
 ### The negatives of rational numbers distribute over addition
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 0d22541afac..783ef57c830 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -10,6 +10,7 @@ open import real-numbers.absolute-value-real-numbers public
 open import real-numbers.accumulation-points-subsets-real-numbers public
 open import real-numbers.addition-lower-dedekind-real-numbers public
 open import real-numbers.addition-nonnegative-real-numbers public
+open import real-numbers.addition-positive-real-numbers public
 open import real-numbers.addition-real-numbers public
 open import real-numbers.addition-upper-dedekind-real-numbers public
 open import real-numbers.apartness-real-numbers public
diff --git a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
index 6c29699df8a..22600ad7dc1 100644
--- a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
@@ -12,9 +12,12 @@ module real-numbers.addition-nonnegative-real-numbers where
 open import elementary-number-theory.addition-nonnegative-rational-numbers
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
+open import foundation.coproduct-types
 open import foundation.dependent-pair-types
+open import foundation.disjunction
 open import foundation.identity-types
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -133,3 +136,21 @@ module _
       le-ℝ⁰⁺ x y → le-ℝ⁰⁺ z w → le-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
     preserves-le-add-ℝ⁰⁺ = preserves-le-add-ℝ
 ```
+
+### Addition with a nonnegative real number is an inflationary map
+
+```agda
+abstract
+  leq-left-add-real-ℝ⁰⁺ :
+    {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁰⁺ l2) → leq-ℝ x (x +ℝ real-ℝ⁰⁺ d)
+  leq-left-add-real-ℝ⁰⁺ x d⁺@(d , pos-d) =
+    tr
+      ( λ y → leq-ℝ y (x +ℝ d))
+      ( right-unit-law-add-ℝ x)
+      ( preserves-leq-left-add-ℝ x zero-ℝ d pos-d)
+
+leq-right-add-real-ℝ⁰⁺ :
+  {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁰⁺ l2) → leq-ℝ x (real-ℝ⁰⁺ d +ℝ x)
+leq-right-add-real-ℝ⁰⁺ x d =
+  tr (leq-ℝ x) (commutative-add-ℝ x (real-ℝ⁰⁺ d)) (leq-left-add-real-ℝ⁰⁺ x d)
+```
diff --git a/src/real-numbers/addition-positive-real-numbers.lagda.md b/src/real-numbers/addition-positive-real-numbers.lagda.md
new file mode 100644
index 00000000000..2874f6e23c2
--- /dev/null
+++ b/src/real-numbers/addition-positive-real-numbers.lagda.md
@@ -0,0 +1,141 @@
+# Addition of positive real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.addition-positive-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
+
+open import foundation.coproduct-types
+open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.propositional-truncations
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+The [positive](real-numbers.positive-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are closed under
+[addition](real-numbers.addition-real-numbers.md).
+
+## Definition
+
+```agda
+abstract
+  is-positive-add-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → is-positive-ℝ x → is-positive-ℝ y →
+    is-positive-ℝ (x +ℝ y)
+  is-positive-add-ℝ {x = x} {y = y} 0<x 0<y =
+    transitive-le-ℝ
+      ( zero-ℝ)
+      ( x)
+      ( x +ℝ y)
+      ( tr
+        ( λ z → le-ℝ z (x +ℝ y))
+        ( right-unit-law-add-ℝ x)
+        ( preserves-le-left-add-ℝ x zero-ℝ y 0<y))
+      ( 0<x)
+
+add-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → ℝ⁺ (l1 ⊔ l2)
+add-ℝ⁺ (x , is-pos-x) (y , is-pos-y) =
+  ( x +ℝ y , is-positive-add-ℝ is-pos-x is-pos-y)
+
+infixl 35 _+ℝ⁺_
+
+_+ℝ⁺_ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → ℝ⁺ (l1 ⊔ l2)
+_+ℝ⁺_ = add-ℝ⁺
+```
+
+### Addition with a positive real number is a strictly inflationary map
+
+```agda
+abstract opaque
+  unfolding add-ℝ le-ℝ
+
+  le-left-add-real-ℝ⁺ :
+    {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁺ l2) → le-ℝ x (x +ℝ real-ℝ⁺ d)
+  le-left-add-real-ℝ⁺ x d⁺@(d , pos-d) =
+    tr
+      ( λ y → le-ℝ y (x +ℝ d))
+      ( right-unit-law-add-ℝ x)
+      ( preserves-le-left-add-ℝ x zero-ℝ d pos-d)
+
+le-right-add-real-ℝ⁺ :
+  {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁺ l2) → le-ℝ x (real-ℝ⁺ d +ℝ x)
+le-right-add-real-ℝ⁺ x d =
+  tr (le-ℝ x) (commutative-add-ℝ x (real-ℝ⁺ d)) (le-left-add-real-ℝ⁺ x d)
+```
+
+### Subtraction by a positive real number is a strictly deflationary map
+
+```agda
+abstract
+  le-diff-real-ℝ⁺ :
+    {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁺ l2) → le-ℝ (x -ℝ real-ℝ⁺ d) x
+  le-diff-real-ℝ⁺ x d⁺@(d , _) =
+    preserves-le-right-sim-ℝ
+      ( x -ℝ d)
+      ( (x -ℝ d) +ℝ d)
+      ( x)
+      ( tr
+        ( λ y → sim-ℝ y x)
+        ( right-swap-add-ℝ x d (neg-ℝ d))
+        ( cancel-right-add-diff-ℝ x d))
+      ( le-left-add-real-ℝ⁺ (x -ℝ d) d⁺)
+```
+
+### If the sum of two real numbers is positive, one of them is positive
+
+```agda
+abstract opaque
+  unfolding add-ℝ
+
+  is-positive-either-is-positive-add-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → is-positive-ℝ (x +ℝ y) →
+    disjunction-type (is-positive-ℝ x) (is-positive-ℝ y)
+  is-positive-either-is-positive-add-ℝ x y 0<x+y =
+    let
+      open do-syntax-trunc-Prop (is-positive-prop-ℝ x ∨ is-positive-prop-ℝ y)
+    in do
+      ((p , q) , p<x , q<y , 0=p+q) ← zero-in-lower-cut-ℝ⁺ (x +ℝ y , 0<x+y)
+      rec-coproduct
+        ( λ is-neg-p →
+          inr-disjunction
+            ( is-positive-exists-ℚ⁺-in-lower-cut-ℝ
+              ( y)
+              ( intro-exists
+                ( neg-ℚ⁻ (p , is-neg-p))
+                ( tr
+                  ( is-in-lower-cut-ℝ y)
+                  ( unique-right-neg-ℚ p q (inv 0=p+q))
+                  ( q<y)))))
+        ( λ is-nonneg-p →
+          inl-disjunction
+            ( is-positive-zero-in-lower-cut-ℝ
+              ( x)
+              ( leq-lower-cut-ℝ x (leq-zero-is-nonnegative-ℚ is-nonneg-p) p<x)))
+        ( decide-is-negative-is-nonnegative-ℚ p)
+```
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 17dfd98f51b..e4526793d3d 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -28,6 +28,7 @@ open import order-theory.join-semilattices
 open import order-theory.large-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
 
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-real-numbers
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index 6e102ad464b..ce1647b2d5d 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -38,6 +38,7 @@ open import metric-spaces.convergent-cauchy-approximations-metric-spaces
 open import metric-spaces.limits-of-cauchy-approximations-metric-spaces
 open import metric-spaces.metric-spaces
 
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index d9ae44bf9b0..3827fcaebe8 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -9,6 +9,7 @@ module real-numbers.nonzero-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.disjunction
@@ -25,10 +26,13 @@ open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
+open import real-numbers.nonnegative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.square-roots-nonnegative-real-numbers
+open import real-numbers.squares-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -162,3 +166,40 @@ module _
                   ( neg-le-ℝ |x|-ε<-x))))
           ( approximate-below-abs-ℝ x ε)
 ```
+
+### `x` is nonzero if and only if `x²` is positive
+
+```agda
+module _
+  {l : Level} (x : ℝ l)
+  where
+
+  abstract
+    is-positive-square-is-nonzero-ℝ :
+      is-nonzero-ℝ x → is-positive-ℝ (square-ℝ x)
+    is-positive-square-is-nonzero-ℝ =
+      elim-disjunction
+        ( is-positive-prop-ℝ (square-ℝ x))
+        ( λ is-neg-x → is-positive-square-ℝ⁻ (x , is-neg-x))
+        ( λ is-pos-x → is-positive-square-ℝ⁺ (x , is-pos-x))
+
+    is-nonzero-square-is-positive-ℝ :
+      is-positive-ℝ (square-ℝ x) → is-nonzero-ℝ x
+    is-nonzero-square-is-positive-ℝ 0<x² =
+      is-nonzero-is-positive-abs-ℝ
+        ( x)
+        ( tr
+          ( is-positive-ℝ)
+          ( equational-reasoning
+            real-sqrt-ℝ⁰⁺ (nonnegative-ℝ⁺ (square-ℝ x , 0<x²))
+            ＝ real-sqrt-ℝ⁰⁺ (nonnegative-square-ℝ x)
+              by
+                ap
+                  ( real-sqrt-ℝ⁰⁺)
+                  ( eq-ℝ⁰⁺
+                    ( nonnegative-ℝ⁺ (square-ℝ x , 0<x²))
+                    ( nonnegative-square-ℝ x)
+                    ( refl))
+            ＝ abs-ℝ x by inv (eq-abs-sqrt-square-ℝ x))
+          ( is-positive-sqrt-ℝ⁺ (square-ℝ x , 0<x²)))
+```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index ab55377d252..e0b81e741e2 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -202,44 +202,6 @@ exists-ℚ⁺-in-lower-cut-ℝ⁺ :
 exists-ℚ⁺-in-lower-cut-ℝ⁺ = ind-Σ exists-ℚ⁺-in-lower-cut-is-positive-ℝ
 ```
 
-### Addition with a positive real number is a strictly inflationary map
-
-```agda
-abstract opaque
-  unfolding add-ℝ le-ℝ
-
-  le-left-add-real-ℝ⁺ :
-    {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁺ l2) → le-ℝ x (x +ℝ real-ℝ⁺ d)
-  le-left-add-real-ℝ⁺ x d⁺@(d , pos-d) =
-    tr
-      ( λ y → le-ℝ y (x +ℝ d))
-      ( right-unit-law-add-ℝ x)
-      ( preserves-le-left-add-ℝ x zero-ℝ d pos-d)
-
-le-right-add-real-ℝ⁺ :
-  {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁺ l2) → le-ℝ x (real-ℝ⁺ d +ℝ x)
-le-right-add-real-ℝ⁺ x d =
-  tr (le-ℝ x) (commutative-add-ℝ x (real-ℝ⁺ d)) (le-left-add-real-ℝ⁺ x d)
-```
-
-### Subtraction by a positive real number is a strictly deflationary map
-
-```agda
-abstract
-  le-diff-real-ℝ⁺ :
-    {l1 l2 : Level} → (x : ℝ l1) (d : ℝ⁺ l2) → le-ℝ (x -ℝ real-ℝ⁺ d) x
-  le-diff-real-ℝ⁺ x d⁺@(d , _) =
-    preserves-le-right-sim-ℝ
-      ( x -ℝ d)
-      ( (x -ℝ d) +ℝ d)
-      ( x)
-      ( tr
-        ( λ y → sim-ℝ y x)
-        ( right-swap-add-ℝ x d (neg-ℝ d))
-        ( cancel-right-add-diff-ℝ x d))
-      ( le-left-add-real-ℝ⁺ (x -ℝ d) d⁺)
-```
-
 ### `x < y` if and only if `y - x` is positive
 
 ```agda
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index d73dc98a0f4..d7c62f65161 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -53,10 +53,14 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
+open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.squares-real-numbers
+open import real-numbers.strict-inequality-nonnegative-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
@@ -646,3 +650,21 @@ abstract
                         ( is-section-square-ℝ⁰⁺ x)
                         ( is-section-square-ℝ⁰⁺ y)))))))
 ```
+
+### The square root of a positive real number is positive
+
+```agda
+abstract opaque
+  unfolding real-sqrt-ℝ⁰⁺
+
+  is-positive-sqrt-ℝ⁺ :
+    {l : Level} (x : ℝ⁺ l) → is-positive-ℝ (real-sqrt-ℝ⁰⁺ (nonnegative-ℝ⁺ x))
+  is-positive-sqrt-ℝ⁺ x⁺@(x , _) =
+    is-positive-zero-in-lower-cut-ℝ
+      ( real-sqrt-ℝ⁰⁺ (nonnegative-ℝ⁺ x⁺))
+      ( λ _ →
+        inv-tr
+          ( is-in-lower-cut-ℝ x)
+          ( left-zero-law-mul-ℚ zero-ℚ)
+          ( zero-in-lower-cut-ℝ⁺ x⁺))
+```
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 86d106d8c6a..4f195703e9e 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -20,6 +20,7 @@ open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.square-roots-positive-rational-numbers
 open import elementary-number-theory.squares-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
@@ -33,6 +34,7 @@ open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
@@ -218,6 +220,25 @@ abstract
       ( preserves-le-right-mul-ℝ⁺ (y , is-positive-le-ℝ⁰⁺ x⁰⁺ y x<y) x<y)
 ```
 
+### For nonnegative real numbers, squaring reflects inequality
+
+```agda
+abstract
+  reflects-leq-square-ℝ⁰⁺ :
+    {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) →
+    leq-ℝ⁰⁺ (square-ℝ⁰⁺ x) (square-ℝ⁰⁺ y) → leq-ℝ⁰⁺ x y
+  reflects-leq-square-ℝ⁰⁺ x⁰⁺@(x , _) y⁰⁺@(y , _) x²≤y² =
+    leq-not-le-ℝ
+      ( y)
+      ( x)
+      ( λ y<x →
+        not-leq-le-ℝ
+          ( square-ℝ y)
+          ( square-ℝ x)
+          ( preserves-le-square-ℝ⁰⁺ y⁰⁺ x⁰⁺ y<x)
+          ( x²≤y²))
+```
+
 ### If a nonnegative rational `q` is in the lower cut of `x`, `q²` is in the lower cut of `x²`
 
 ```agda

From 89adb085bdc3b4249bdef72daf13db3f90999fda Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Nov 2025 17:32:43 -0800
Subject: [PATCH 081/134] Magnitudes multiply

---
 .../magnitude-complex-numbers.lagda.md        | 140 +++++++++++++++++-
 ...-inverses-nonzero-complex-numbers.lagda.md |   2 +-
 src/real-numbers.lagda.md                     |   1 +
 ...ltiplication-nonzero-real-numbers.lagda.md |  68 +++++++++
 ...ositive-and-negative-real-numbers.lagda.md |  22 ++-
 .../multiplication-real-numbers.lagda.md      |  30 ++++
 .../rational-real-numbers.lagda.md            |   8 +
 .../squares-real-numbers.lagda.md             |  50 +++++++
 8 files changed, 318 insertions(+), 3 deletions(-)
 create mode 100644 src/real-numbers/multiplication-nonzero-real-numbers.lagda.md

diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index d46272ba5de..22a709c6efb 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -13,6 +13,7 @@ open import complex-numbers.complex-numbers
 open import complex-numbers.conjugation-complex-numbers
 open import complex-numbers.multiplication-complex-numbers
 
+open import foundation.action-on-identifications-functions
 open import foundation.identity-types
 open import foundation.universe-levels
 
@@ -20,12 +21,13 @@ open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
--- sopen import real-numbers.similarity-real-numbers
+open import real-numbers.similarity-real-numbers
 open import real-numbers.square-roots-nonnegative-real-numbers
 open import real-numbers.squares-real-numbers
 ```
@@ -93,3 +95,139 @@ abstract
           ~ℝ raise-ℝ _ zero-ℝ
             by sim-raise-ℝ _ _))
 ```
+
+### The square of the magnitude of `zw` is the product of the squares of the magnitudes of `z` and `w`
+
+```agda
+abstract
+  distributive-squared-magnitude-mul-ℂ :
+    {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2) →
+    squared-magnitude-ℂ (z *ℂ w) ＝
+    squared-magnitude-ℂ z *ℝ squared-magnitude-ℂ w
+  distributive-squared-magnitude-mul-ℂ (a +iℂ b) (c +iℂ d) =
+    equational-reasoning
+      square-ℝ (a *ℝ c -ℝ b *ℝ d) +ℝ square-ℝ (a *ℝ d +ℝ b *ℝ c)
+      ＝
+        ( square-ℝ (a *ℝ c) +ℝ
+          neg-ℝ (real-ℕ 2 *ℝ ((a *ℝ c) *ℝ (b *ℝ d))) +ℝ
+          square-ℝ (b *ℝ d)) +ℝ
+        ( square-ℝ (a *ℝ d) +ℝ
+          real-ℕ 2 *ℝ ((a *ℝ d) *ℝ (b *ℝ c)) +ℝ
+          square-ℝ (b *ℝ c))
+        by
+          ap-add-ℝ
+            ( square-diff-ℝ _ _)
+            ( square-add-ℝ _ _)
+      ＝
+        ( square-ℝ (a *ℝ c) +ℝ square-ℝ (b *ℝ d) +ℝ
+          neg-ℝ (real-ℕ 2 *ℝ (a *ℝ c *ℝ (b *ℝ d)))) +ℝ
+        ( square-ℝ (a *ℝ d) +ℝ square-ℝ (b *ℝ c) +ℝ
+          real-ℕ 2 *ℝ ((a *ℝ d) *ℝ (b *ℝ c)))
+        by
+          ap-add-ℝ
+            ( right-swap-add-ℝ _ _ _)
+            ( right-swap-add-ℝ _ _ _)
+      ＝
+        ( ( square-ℝ (a *ℝ c) +ℝ square-ℝ (b *ℝ d)) +ℝ
+          ( square-ℝ (a *ℝ d) +ℝ square-ℝ (b *ℝ c))) +ℝ
+        ( neg-ℝ (real-ℕ 2 *ℝ ((a *ℝ c) *ℝ (b *ℝ d))) +ℝ
+          real-ℕ 2 *ℝ ((a *ℝ d) *ℝ (b *ℝ c)))
+        by interchange-law-add-add-ℝ _ _ _ _
+      ＝
+        ( ( square-ℝ (a *ℝ c) +ℝ square-ℝ (b *ℝ d)) +ℝ
+          ( square-ℝ (a *ℝ d) +ℝ square-ℝ (b *ℝ c))) +ℝ
+        ( neg-ℝ (real-ℕ 2 *ℝ (a *ℝ c *ℝ (b *ℝ d))) +ℝ
+          real-ℕ 2 *ℝ (a *ℝ d *ℝ (c *ℝ b)))
+        by
+          ap-add-ℝ
+            ( refl)
+            ( ap-add-ℝ
+              ( refl)
+              ( ap-mul-ℝ refl (ap-mul-ℝ refl (commutative-mul-ℝ b c))))
+      ＝
+        ( ( square-ℝ (a *ℝ c) +ℝ square-ℝ (b *ℝ d)) +ℝ
+          ( square-ℝ (a *ℝ d) +ℝ square-ℝ (b *ℝ c))) +ℝ
+        ( neg-ℝ (real-ℕ 2 *ℝ (a *ℝ c *ℝ (b *ℝ d))) +ℝ
+          real-ℕ 2 *ℝ (a *ℝ c *ℝ (d *ℝ b)))
+        by
+          ap-add-ℝ
+            ( refl)
+            ( ap-add-ℝ refl (ap-mul-ℝ refl (interchange-law-mul-mul-ℝ _ _ _ _)))
+      ＝
+        ( ( square-ℝ (a *ℝ c) +ℝ square-ℝ (b *ℝ d)) +ℝ
+          ( square-ℝ (a *ℝ d) +ℝ square-ℝ (b *ℝ c))) +ℝ
+        ( neg-ℝ (real-ℕ 2 *ℝ (a *ℝ c *ℝ (b *ℝ d))) +ℝ
+          real-ℕ 2 *ℝ (a *ℝ c *ℝ (b *ℝ d)))
+        by
+          ap-add-ℝ
+            ( refl)
+            ( ap-add-ℝ
+              ( refl)
+              ( ap-mul-ℝ refl (ap-mul-ℝ refl (commutative-mul-ℝ d b))))
+      ＝
+        ( ( square-ℝ (a *ℝ c) +ℝ square-ℝ (b *ℝ d)) +ℝ
+          ( square-ℝ (a *ℝ d) +ℝ square-ℝ (b *ℝ c))) +ℝ
+        zero-ℝ
+        by eq-sim-ℝ (preserves-sim-left-add-ℝ _ _ _ (left-inverse-law-add-ℝ _))
+      ＝
+        ( square-ℝ (a *ℝ c) +ℝ square-ℝ (b *ℝ d)) +ℝ
+        ( square-ℝ (a *ℝ d) +ℝ square-ℝ (b *ℝ c))
+        by right-unit-law-add-ℝ _
+      ＝
+        ( square-ℝ (a *ℝ c) +ℝ square-ℝ (a *ℝ d)) +ℝ
+        ( square-ℝ (b *ℝ d) +ℝ square-ℝ (b *ℝ c))
+        by interchange-law-add-add-ℝ _ _ _ _
+      ＝
+        ( square-ℝ a *ℝ square-ℝ c +ℝ square-ℝ a *ℝ square-ℝ d) +ℝ
+        ( square-ℝ b *ℝ square-ℝ d +ℝ square-ℝ b *ℝ square-ℝ c)
+        by
+          ap-add-ℝ
+            ( ap-add-ℝ
+              ( distributive-square-mul-ℝ a c)
+              ( distributive-square-mul-ℝ a d))
+            ( ap-add-ℝ
+              ( distributive-square-mul-ℝ b d)
+              ( distributive-square-mul-ℝ b c))
+      ＝
+        square-ℝ a *ℝ (square-ℝ c +ℝ square-ℝ d) +ℝ
+        square-ℝ b *ℝ (square-ℝ d +ℝ square-ℝ c)
+        by
+          inv
+            ( ap-add-ℝ
+              ( left-distributive-mul-add-ℝ _ _ _)
+              ( left-distributive-mul-add-ℝ _ _ _))
+      ＝
+        square-ℝ a *ℝ (square-ℝ c +ℝ square-ℝ d) +ℝ
+        square-ℝ b *ℝ (square-ℝ c +ℝ square-ℝ d)
+        by ap-add-ℝ refl (ap-mul-ℝ refl (commutative-add-ℝ _ _))
+      ＝
+        (square-ℝ a +ℝ square-ℝ b) *ℝ (square-ℝ c +ℝ square-ℝ d)
+        by
+          inv
+            ( right-distributive-mul-add-ℝ
+              ( square-ℝ a)
+              ( square-ℝ b)
+              ( square-ℝ c +ℝ square-ℝ d))
+```
+
+### The magnitude of `zw` is the product of the magnitudes of `z` and `w`
+
+```agda
+abstract
+  distributive-magnitude-mul-ℂ :
+    {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2) →
+    magnitude-ℂ (z *ℂ w) ＝ magnitude-ℂ z *ℝ magnitude-ℂ w
+  distributive-magnitude-mul-ℂ z w =
+    equational-reasoning
+      real-sqrt-ℝ⁰⁺ (nonnegative-squared-magnitude-ℂ (z *ℂ w))
+      ＝
+        real-sqrt-ℝ⁰⁺
+          ( nonnegative-squared-magnitude-ℂ z *ℝ⁰⁺
+            nonnegative-squared-magnitude-ℂ w)
+        by
+          ap
+            ( real-sqrt-ℝ⁰⁺)
+            ( eq-ℝ⁰⁺ _ _ (distributive-squared-magnitude-mul-ℂ z w))
+      ＝ magnitude-ℂ z *ℝ magnitude-ℂ w
+        by ap real-ℝ⁰⁺ (distributive-sqrt-mul-ℝ⁰⁺ _ _)
+```
diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 320c7b26cdf..71dd292ad7f 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -8,7 +8,7 @@ module complex-numbers.multiplicative-inverses-nonzero-complex-numbers where
 
 ```agda
 open import complex-numbers.complex-numbers
--- open import complex-numbers.conjugation-complex-numbers
+open import complex-numbers.conjugation-complex-numbers
 open import complex-numbers.magnitude-complex-numbers
 open import complex-numbers.multiplication-complex-numbers
 open import complex-numbers.nonzero-complex-numbers
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 783ef57c830..a3ef18a8455 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -60,6 +60,7 @@ open import real-numbers.minimum-lower-dedekind-real-numbers public
 open import real-numbers.minimum-upper-dedekind-real-numbers public
 open import real-numbers.multiplication-negative-real-numbers public
 open import real-numbers.multiplication-nonnegative-real-numbers public
+open import real-numbers.multiplication-nonzero-real-numbers public
 open import real-numbers.multiplication-positive-and-negative-real-numbers public
 open import real-numbers.multiplication-positive-real-numbers public
 open import real-numbers.multiplication-real-numbers public
diff --git a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
new file mode 100644
index 00000000000..5f6c8696300
--- /dev/null
+++ b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
@@ -0,0 +1,68 @@
+# Multiplication of nonzero real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.multiplication-nonzero-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.function-types
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.multiplication-negative-real-numbers
+open import real-numbers.multiplication-positive-and-negative-real-numbers
+open import real-numbers.multiplication-positive-real-numbers
+open import real-numbers.multiplication-real-numbers
+open import real-numbers.multiplicative-inverses-nonzero-real-numbers
+open import real-numbers.nonzero-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "product" Disambiguation="of pairs of nonzero real numbers" Agda=mul-nonzero-ℝ}}
+of two [nonzero](real-numbers.nonzero-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) is their
+[product](real-numbers.multiplication-real-numbers.md) as real numbers, and is
+itself nonzero.
+
+## Definition
+
+```agda
+abstract
+  is-nonzero-mul-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → is-nonzero-ℝ x → is-nonzero-ℝ y →
+    is-nonzero-ℝ (x *ℝ y)
+  is-nonzero-mul-ℝ {x = x} {y = y} x#0 y#0 =
+    let
+      motive = is-nonzero-prop-ℝ (x *ℝ y)
+    in
+      elim-disjunction
+        ( motive)
+        ( λ x<0 →
+          elim-disjunction
+            ( motive)
+            ( inr-disjunction ∘ is-positive-mul-is-negative-ℝ x<0)
+            ( inl-disjunction ∘ is-negative-mul-negative-positive-ℝ x<0)
+            ( y#0))
+        ( λ 0<x →
+          elim-disjunction
+            ( motive)
+            ( inl-disjunction ∘ is-negative-mul-positive-negative-ℝ 0<x)
+            ( inr-disjunction ∘ is-positive-mul-ℝ 0<x)
+            ( y#0))
+        ( x#0)
+
+mul-nonzero-ℝ :
+  {l1 l2 : Level} → nonzero-ℝ l1 → nonzero-ℝ l2 → nonzero-ℝ (l1 ⊔ l2)
+mul-nonzero-ℝ (x , x#0) (y , y#0) =
+  ( x *ℝ y , is-nonzero-mul-ℝ x#0 y#0)
+```
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 9e9e66cc377..353a564925a 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -8,6 +8,7 @@ module real-numbers.multiplication-positive-and-negative-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
@@ -30,7 +31,7 @@ their product too.
 
 ## Lemmas
 
-### The product of a positive and a negative rational number is negative
+### The product of a positive and a negative real number is negative
 
 ```agda
 abstract
@@ -50,3 +51,22 @@ mul-positive-negative-ℝ :
 mul-positive-negative-ℝ (x , is-pos-x) (y , is-neg-y) =
   ( x *ℝ y , is-negative-mul-positive-negative-ℝ is-pos-x is-neg-y)
 ```
+
+### The product of a negative and a positive real number is negative
+
+```agda
+abstract
+  is-negative-mul-negative-positive-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → is-negative-ℝ x → is-positive-ℝ y →
+    is-negative-ℝ (x *ℝ y)
+  is-negative-mul-negative-positive-ℝ {x = x} {y = y} x<0 0<y =
+    tr
+      ( is-negative-ℝ)
+      ( commutative-mul-ℝ y x)
+      ( is-negative-mul-positive-negative-ℝ 0<y x<0)
+
+mul-negative-positive-ℝ :
+  {l1 l2 : Level} → ℝ⁻ l1 → ℝ⁺ l2 → ℝ⁻ (l1 ⊔ l2)
+mul-negative-positive-ℝ (x , is-neg-x) (y , is-pos-y) =
+  ( x *ℝ y , is-negative-mul-negative-positive-ℝ is-neg-x is-pos-y)
+```
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index e228164d2d9..a4d70e4bbd3 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -11,12 +11,15 @@ module real-numbers.multiplication-real-numbers where
 ```agda
 open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-closed-intervals-rational-numbers
+open import elementary-number-theory.addition-integers
+open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.inequality-natural-numbers
 open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.integers
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers
 open import elementary-number-theory.maximum-natural-numbers
 open import elementary-number-theory.maximum-nonnegative-rational-numbers
@@ -58,6 +61,7 @@ open import foundation.universe-levels
 
 open import group-theory.abelian-groups
 open import group-theory.groups
+open import group-theory.multiples-of-elements-abelian-groups
 
 open import logic.functoriality-existential-quantification
 
@@ -1262,3 +1266,29 @@ abstract
       ＝ x *ℝ z +ℝ neg-ℝ y *ℝ z by right-distributive-mul-add-ℝ _ _ _
       ＝ x *ℝ z -ℝ y *ℝ z by ap (x *ℝ z +ℝ_) (left-negative-law-mul-ℝ y z)
 ```
+
+### Multiplication by a natural number is repeated addition
+
+```agda
+abstract
+  left-mul-real-ℕ :
+    {l : Level} (n : ℕ) (x : ℝ l) →
+    real-ℕ n *ℝ x ＝ multiple-Ab (ab-add-ℝ l) n x
+  left-mul-real-ℕ 0 x =
+    eq-sim-ℝ
+      ( transitive-sim-ℝ _ _ _ (sim-raise-ℝ _ zero-ℝ) (left-zero-law-mul-ℝ x))
+  left-mul-real-ℕ 1 x = left-unit-law-mul-ℝ x
+  left-mul-real-ℕ (succ-ℕ n@(succ-ℕ _)) x =
+    equational-reasoning
+      real-ℕ (n +ℕ 1) *ℝ x
+      ＝ real-ℤ (int-ℕ n +ℤ one-ℤ) *ℝ x
+        by ap-mul-ℝ (ap real-ℤ (inv (add-int-ℕ n 1))) refl
+      ＝ real-ℚ (rational-ℕ n +ℚ one-ℚ) *ℝ x
+        by ap-mul-ℝ (ap real-ℚ (inv (add-rational-ℤ _ _))) refl
+      ＝ (real-ℕ n +ℝ one-ℝ) *ℝ x
+        by ap-mul-ℝ (inv (add-real-ℚ _ _)) refl
+      ＝ real-ℕ n *ℝ x +ℝ one-ℝ *ℝ x
+        by right-distributive-mul-add-ℝ _ _ _
+      ＝ multiple-Ab (ab-add-ℝ _) n x +ℝ x
+        by ap-add-ℝ (left-mul-real-ℕ n x) (left-unit-law-mul-ℝ x)
+```
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index 285729c4a4d..4e48764ed53 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -10,6 +10,7 @@ module real-numbers.rational-real-numbers where
 
 ```agda
 open import elementary-number-theory.integers
+open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
@@ -83,6 +84,13 @@ real-ℤ : ℤ → ℝ lzero
 real-ℤ x = real-ℚ (rational-ℤ x)
 ```
 
+### The canonical map from `ℕ` to `ℝ lzero`
+
+```agda
+real-ℕ : ℕ → ℝ lzero
+real-ℕ n = real-ℤ (int-ℕ n)
+```
+
 ### Zero as a real number
 
 ```agda
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 4f195703e9e..d64d8703277 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -32,7 +32,9 @@ open import foundation.propositional-truncations
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
+open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
@@ -304,3 +306,51 @@ abstract opaque
             ( upper-bound-square-even-interval-ℚ [-q,q] refl)
             ( refl-leq-ℚ (q *ℚ q))))
 ```
+
+### `(a + b)² = a² + 2ab + b²`
+
+```agda
+abstract
+  square-add-ℝ :
+    {l1 : Level} {l2 : Level} (x : ℝ l1) (y : ℝ l2) →
+    square-ℝ (x +ℝ y) ＝ square-ℝ x +ℝ real-ℕ 2 *ℝ (x *ℝ y) +ℝ square-ℝ y
+  square-add-ℝ x y =
+    equational-reasoning
+      square-ℝ (x +ℝ y)
+      ＝ (x +ℝ y) *ℝ x +ℝ (x +ℝ y) *ℝ y
+        by left-distributive-mul-add-ℝ (x +ℝ y) x y
+      ＝ (x *ℝ x +ℝ y *ℝ x) +ℝ (x *ℝ y +ℝ y *ℝ y)
+        by
+          ap-add-ℝ
+            ( right-distributive-mul-add-ℝ x y x)
+            ( right-distributive-mul-add-ℝ x y y)
+      ＝ (x *ℝ x +ℝ x *ℝ y) +ℝ (x *ℝ y +ℝ y *ℝ y)
+        by ap-add-ℝ (ap-add-ℝ refl (commutative-mul-ℝ y x)) refl
+      ＝ ((x *ℝ x +ℝ x *ℝ y) +ℝ x *ℝ y) +ℝ y *ℝ y
+        by inv (associative-add-ℝ _ _ _)
+      ＝ x *ℝ x +ℝ (x *ℝ y +ℝ x *ℝ y) +ℝ y *ℝ y
+        by ap-add-ℝ (associative-add-ℝ _ _ _) refl
+      ＝ square-ℝ x +ℝ real-ℕ 2 *ℝ (x *ℝ y) +ℝ square-ℝ y
+        by ap-add-ℝ (ap-add-ℝ refl (inv (left-mul-real-ℕ 2 (x *ℝ y)))) refl
+```
+
+### `(a - b)² = a² - 2ab + b²`
+
+```agda
+abstract
+  square-diff-ℝ :
+    {l1 : Level} {l2 : Level} (x : ℝ l1) (y : ℝ l2) →
+    square-ℝ (x -ℝ y) ＝ square-ℝ x -ℝ real-ℕ 2 *ℝ (x *ℝ y) +ℝ square-ℝ y
+  square-diff-ℝ x y =
+    equational-reasoning
+      square-ℝ (x -ℝ y)
+      ＝ square-ℝ x +ℝ real-ℕ 2 *ℝ (x *ℝ neg-ℝ y) +ℝ square-ℝ (neg-ℝ y)
+        by square-add-ℝ x (neg-ℝ y)
+      ＝ square-ℝ x +ℝ real-ℕ 2 *ℝ neg-ℝ (x *ℝ y) +ℝ square-ℝ y
+        by
+          ap-add-ℝ
+            ( ap-add-ℝ refl (ap-mul-ℝ refl (right-negative-law-mul-ℝ x y)))
+            ( square-neg-ℝ y)
+      ＝ square-ℝ x -ℝ real-ℕ 2 *ℝ (x *ℝ y) +ℝ square-ℝ y
+        by ap-add-ℝ (ap-add-ℝ refl (right-negative-law-mul-ℝ _ _)) refl
+```

From 71726c65ff082128ac8492f5994e56148ed7ff29 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 14:01:18 -0800
Subject: [PATCH 082/134] Progress

---
 ...ive-and-negative-rational-numbers.lagda.md | 30 +++++++++
 ...ositive-and-negative-real-numbers.lagda.md | 64 +++++++++++++++++++
 2 files changed, 94 insertions(+)

diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index 4cbe0a91112..fd865252b7f 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -18,6 +18,7 @@ open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 
 open import foundation.cartesian-product-types
+open import foundation.functoriality-coproduct-types
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
@@ -169,3 +170,32 @@ abstract
             ( y≠0)
             ( λ is-pos-y → inr (is-pos-x , is-pos-y)))
 ```
+
+### If `xy` is negative, one of `x` and `y` is positive and the other negative
+
+```agda
+abstract
+  different-signs-is-negative-mul-ℚ :
+    {x y : ℚ} → is-negative-ℚ (x *ℚ y) →
+    ( ( is-positive-ℚ x × is-negative-ℚ y) +
+      ( is-negative-ℚ x × is-positive-ℚ y))
+  different-signs-is-negative-mul-ℚ {x} {y} is-neg-xy =
+    map-coproduct
+      ( λ (is-neg-neg-x , is-neg-y) →
+        ( tr
+            ( is-positive-ℚ)
+            ( neg-neg-ℚ x)
+            ( is-positive-neg-is-negative-ℚ is-neg-neg-x) ,
+          is-neg-y))
+      ( λ (is-pos-neg-x , is-pos-y) →
+        ( tr
+            ( is-negative-ℚ)
+            ( neg-neg-ℚ x)
+            ( is-negative-neg-is-positive-ℚ is-pos-neg-x) ,
+          is-pos-y))
+      ( same-sign-is-positive-mul-ℚ
+        ( inv-tr
+          ( is-positive-ℚ)
+          ( left-negative-law-mul-ℚ x y)
+          ( is-positive-neg-is-negative-ℚ is-neg-xy)))
+```
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 353a564925a..12ada068a8c 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -9,7 +9,18 @@ module real-numbers.multiplication-positive-and-negative-real-numbers where
 ```agda
 open import foundation.dependent-pair-types
 open import foundation.transport-along-identifications
+open import foundation.disjunction
+open import foundation.coproduct-types
+open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.minimum-rational-numbers
+open import foundation.conjunction
+open import order-theory.posets
+open import elementary-number-theory.inequality-rational-numbers
+open import foundation.propositional-truncations
+open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.universe-levels
+open import elementary-number-theory.positive-rational-numbers
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.multiplication-positive-real-numbers
@@ -70,3 +81,56 @@ mul-negative-positive-ℝ :
 mul-negative-positive-ℝ (x , is-neg-x) (y , is-pos-y) =
   ( x *ℝ y , is-negative-mul-negative-positive-ℝ is-neg-x is-pos-y)
 ```
+
+### If the product of two real numbers is positive, both are negative or both are positive
+
+```agda
+abstract opaque
+  unfolding mul-ℝ
+
+  same-sign-is-positive-mul-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → is-positive-ℝ (x *ℝ y) →
+    type-disjunction-Prop
+      ( is-negative-prop-ℝ x ∧ is-negative-prop-ℝ y)
+      ( is-positive-prop-ℝ x ∧ is-positive-prop-ℝ y)
+  same-sign-is-positive-mul-ℝ x y 0<xy =
+    let
+      open
+        do-syntax-trunc-Prop
+          ( ( is-negative-prop-ℝ x ∧ is-negative-prop-ℝ y) ∨
+            ( is-positive-prop-ℝ x ∧ is-positive-prop-ℝ y))
+      open inequality-reasoning-Poset ℚ-Poset
+    in do
+      ( q⁺@(q , is-pos-q) , q<xy) ←
+        exists-ℚ⁺-in-lower-cut-is-positive-ℝ (x *ℝ y) 0<xy
+      ( ( ax<x<bx@([ax,bx]@((ax , bx) , _) , ax<x , x<bx) ,
+          ay<y<by@([ay,by]@((ay , by) , _) , ay<y , y<by)) ,
+          q<[ax,bx][ay,by]) ← q<xy
+      let
+        is-positive-axay =
+          is-positive-leq-ℚ⁺
+            ( q⁺)
+            ( chain-of-inequalities
+              q
+              ≤ lower-bound-mul-closed-interval-ℚ [ax,bx] [ay,by]
+                by leq-le-ℚ q<[ax,bx][ay,by]
+              ≤ min-ℚ (ax *ℚ ay) (ax *ℚ by)
+                by leq-left-min-ℚ _ _
+              ≤ ax *ℚ ay
+                by leq-left-min-ℚ _ _)
+        is-positive-bxby =
+          is-positive-leq-ℚ⁺
+            ( q⁺)
+            ( chain-of-inequalities
+              q
+              ≤ lower-bound-mul-closed-interval-ℚ [ax,bx] [ay,by]
+                by leq-le-ℚ q<[ax,bx][ay,by]
+              ≤ min-ℚ (bx *ℚ ay) (bx *ℚ by)
+                by leq-right-min-ℚ _ _
+              ≤ bx *ℚ by
+                by leq-right-min-ℚ _ _)
+      rec-coproduct
+        {!   !}
+        {!   !}
+        {!  same-sign !}
+```

From 0bb5730c8b6ac5a2d4a529fe15b0fdeaea75efe9 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 14:02:00 -0800
Subject: [PATCH 083/134] Update
 src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../geometric-sequences-commutative-rings.lagda.md          | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
index 70d3a210a8f..87f0c35680b 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -291,9 +291,9 @@ module _
 ```agda
 module _
   {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
-  (let _*_ = mul-Commutative-Ring R)
-  (let _+_ = add-Commutative-Ring R)
-  (let _-_ = right-subtraction-Commutative-Ring R)
+  (let _*R_ = mul-Commutative-Ring R)
+  (let _+R_ = add-Commutative-Ring R)
+  (let _-R_ = right-subtraction-Commutative-Ring R)
   (let zero-R = zero-Commutative-Ring R)
   (let one-R = one-Commutative-Ring R)
   where

From 62bf2b63642dd1a2d8f987a0abeba4fcb516370a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 14:04:00 -0800
Subject: [PATCH 084/134] Respond to review comment

---
 src/literature/100-theorems.lagda.md | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/literature/100-theorems.lagda.md b/src/literature/100-theorems.lagda.md
index d9ea4bf83ad..862d0a1cea8 100644
--- a/src/literature/100-theorems.lagda.md
+++ b/src/literature/100-theorems.lagda.md
@@ -122,7 +122,8 @@ open import foundation.cantors-theorem using
 
 ```agda
 open import real-numbers.geometric-sequences-real-numbers using
-  ( compute-sum-standard-geometric-series-ℝ)
+  ( compute-sum-standard-geometric-fin-sequence-ℝ ;
+    compute-sum-standard-geometric-series-ℝ)
 ```
 
 ### 68. Sum of an arithmetic series {#68}

From 855c5bb2ecaa61ff8ba7803f8046cd3d1210d042 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 14:08:04 -0800
Subject: [PATCH 085/134] Fix arithmetic op names

---
 ...etric-sequences-commutative-rings.lagda.md | 64 +++++++++----------
 1 file changed, 32 insertions(+), 32 deletions(-)

diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
index 87f0c35680b..b965ee88379 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -334,8 +334,8 @@ module _
       (1/⟨1-r⟩ , H) 0 =
       inv
         ( equational-reasoning
-          (a * 1/⟨1-r⟩) * (one-R - one-R)
-          ＝ (a * 1/⟨1-r⟩) * zero-R
+          (a *R 1/⟨1-r⟩) *R (one-R -R one-R)
+          ＝ (a *R 1/⟨1-r⟩) *R zero-R
             by
               ap-mul-Commutative-Ring R
                 ( refl)
@@ -347,7 +347,7 @@ module _
       equational-reasoning
         sum-standard-geometric-fin-sequence-Commutative-Ring R a r (succ-ℕ n)
         ＝
-          sum-standard-geometric-fin-sequence-Commutative-Ring R a r n +
+          sum-standard-geometric-fin-sequence-Commutative-Ring R a r n +R
           seq-standard-geometric-sequence-Commutative-Ring R a r n
           by
             cons-sum-fin-sequence-type-Commutative-Ring R
@@ -358,8 +358,8 @@ module _
                 ( succ-ℕ n))
               ( refl)
         ＝
-          ( (a * 1/⟨1-r⟩) * (one-R - power-Commutative-Ring R n r)) +
-          ( a * power-Commutative-Ring R n r)
+          ( (a *R 1/⟨1-r⟩) *R (one-R -R power-Commutative-Ring R n r)) +R
+          ( a *R power-Commutative-Ring R n r)
           by
             ap-add-Commutative-Ring R
               ( compute-sum-standard-geometric-fin-sequence-Commutative-Ring
@@ -371,8 +371,8 @@ module _
                   ( r)
                   ( n)))
         ＝
-          ( a * (1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r))) +
-          ( a * (one-R * power-Commutative-Ring R n r))
+          ( a *R (1/⟨1-r⟩ *R (one-R -R power-Commutative-Ring R n r))) +R
+          ( a *R (one-R *R power-Commutative-Ring R n r))
           by
             ap-add-Commutative-Ring R
               ( associative-mul-Commutative-Ring R _ _ _)
@@ -380,14 +380,14 @@ module _
                 ( refl)
                 ( inv (left-unit-law-mul-Commutative-Ring R _)))
         ＝
-          a *
-          ( (1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r)) +
-            (one-R * power-Commutative-Ring R n r))
+          a *R
+          ( (1/⟨1-r⟩ *R (one-R -R power-Commutative-Ring R n r)) +R
+            (one-R *R power-Commutative-Ring R n r))
           by inv (left-distributive-mul-add-Commutative-Ring R a _ _)
         ＝
-          a *
-          ( ( 1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r)) +
-            ( (1/⟨1-r⟩ * (one-R - r)) * power-Commutative-Ring R n r))
+          a *R
+          ( ( 1/⟨1-r⟩ *R (one-R -R power-Commutative-Ring R n r)) +R
+            ( (1/⟨1-r⟩ *R (one-R -R r)) *R power-Commutative-Ring R n r))
             by
               ap-mul-Commutative-Ring R
                 ( refl)
@@ -395,9 +395,9 @@ module _
                   ( refl)
                   ( ap-mul-Commutative-Ring R (inv (pr2 H)) refl))
         ＝
-          a *
-          ( ( 1/⟨1-r⟩ * (one-R - power-Commutative-Ring R n r)) +
-            ( 1/⟨1-r⟩ * ((one-R - r) * power-Commutative-Ring R n r)))
+          a *R
+          ( ( 1/⟨1-r⟩ *R (one-R -R power-Commutative-Ring R n r)) +R
+            ( 1/⟨1-r⟩ *R ((one-R -R r) *R power-Commutative-Ring R n r)))
           by
             ap-mul-Commutative-Ring R
               ( refl)
@@ -405,24 +405,24 @@ module _
                 ( refl)
                 ( associative-mul-Commutative-Ring R _ _ _))
         ＝
-          a *
-            ( 1/⟨1-r⟩ *
-              ( ( one-R - power-Commutative-Ring R n r) +
-                ( (one-R - r) * power-Commutative-Ring R n r)))
+          a *R
+            ( 1/⟨1-r⟩ *R
+              ( ( one-R -R power-Commutative-Ring R n r) +R
+                ( (one-R -R r) *R power-Commutative-Ring R n r)))
           by
             ap-mul-Commutative-Ring R
               ( refl)
               ( inv (left-distributive-mul-add-Commutative-Ring R _ _ _))
         ＝
-          ( a * 1/⟨1-r⟩) *
-          ( ( one-R - power-Commutative-Ring R n r) +
-            ( (one-R - r) * power-Commutative-Ring R n r))
+          ( a *R 1/⟨1-r⟩) *R
+          ( ( one-R -R power-Commutative-Ring R n r) +R
+            ( (one-R -R r) *R power-Commutative-Ring R n r))
           by inv (associative-mul-Commutative-Ring R _ _ _)
         ＝
-          ( a * 1/⟨1-r⟩) *
-          ( ( one-R - power-Commutative-Ring R n r) +
-            ( (one-R * power-Commutative-Ring R n r) -
-              (r * power-Commutative-Ring R n r)))
+          ( a *R 1/⟨1-r⟩) *R
+          ( ( one-R -R power-Commutative-Ring R n r) +R
+            ( (one-R *R power-Commutative-Ring R n r) -R
+              (r *R power-Commutative-Ring R n r)))
           by
             ap-mul-Commutative-Ring R
               ( refl)
@@ -433,9 +433,9 @@ module _
                   ( _)
                   ( _)))
         ＝
-          ( a * 1/⟨1-r⟩) *
-          ( ( one-R - power-Commutative-Ring R n r) +
-            ( power-Commutative-Ring R n r -
+          ( a *R 1/⟨1-r⟩) *R
+          ( ( one-R -R power-Commutative-Ring R n r) +R
+            ( power-Commutative-Ring R n r -R
               power-Commutative-Ring R (succ-ℕ n) r))
           by
             ap-mul-Commutative-Ring R
@@ -446,8 +446,8 @@ module _
                   ( left-unit-law-mul-Commutative-Ring R _)
                   ( inv (power-succ-Commutative-Ring' R n r))))
         ＝
-          ( a * 1/⟨1-r⟩) *
-          ( one-R - power-Commutative-Ring R (succ-ℕ n) r)
+          ( a *R 1/⟨1-r⟩) *R
+          ( one-R -R power-Commutative-Ring R (succ-ℕ n) r)
           by
             ap-mul-Commutative-Ring R
               ( refl)

From 9309668c0274c021f806a3ae94552e45d2acab3a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 15:18:39 -0800
Subject: [PATCH 086/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../geometric-sequences-commutative-rings.lagda.md          | 6 +++---
 src/real-numbers/nonzero-real-numbers.lagda.md              | 2 +-
 src/real-numbers/powers-real-numbers.lagda.md               | 4 ++--
 src/real-numbers/real-sequences-approximating-zero.lagda.md | 2 +-
 src/real-numbers/series-real-numbers.lagda.md               | 2 +-
 .../uniformly-continuous-functions-real-numbers.lagda.md    | 5 ++---
 6 files changed, 10 insertions(+), 11 deletions(-)

diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
index b965ee88379..5af1cfd3a26 100644
--- a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -406,9 +406,9 @@ module _
                 ( associative-mul-Commutative-Ring R _ _ _))
         ＝
           a *R
-            ( 1/⟨1-r⟩ *R
-              ( ( one-R -R power-Commutative-Ring R n r) +R
-                ( (one-R -R r) *R power-Commutative-Ring R n r)))
+          ( 1/⟨1-r⟩ *R
+            ( ( one-R -R power-Commutative-Ring R n r) +R
+              ( (one-R -R r) *R power-Commutative-Ring R n r)))
           by
             ap-mul-Commutative-Ring R
               ( refl)
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index 3a8df09cc9c..50a1218db66 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -122,7 +122,7 @@ module _
   nonzero-diff-apart-ℝ x#y = (x -ℝ y , is-nonzero-diff-is-apart-ℝ x#y)
 ```
 
-### If `x < y`, then `y - x` is nonzero
+### The nonzero difference of a pair of real numbers `x` and `y` such that `x < y`
 
 ```agda
 nonzero-diff-le-ℝ :
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 2f879cfc4ca..67caeae578b 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -355,7 +355,7 @@ abstract
         by leq-eq-ℝ (ap abs-ℝ (inv (power-succ-ℝ n y)))
 ```
 
-### If `x` and `y` are nonnegative, and `x ≤ y`, `xⁿ ≤ yⁿ`
+### If `x` and `y` are nonnegative such that `x ≤ y`, then `xⁿ ≤ yⁿ`
 
 ```agda
 abstract
@@ -388,7 +388,7 @@ abstract
         by leq-eq-ℝ (inv (power-succ-ℝ n y))
 ```
 
-### If `|r| < 1`, `rⁿ` approaches 0
+### If `|r| < 1`, then `rⁿ` approaches 0
 
 ```agda
 abstract
diff --git a/src/real-numbers/real-sequences-approximating-zero.lagda.md b/src/real-numbers/real-sequences-approximating-zero.lagda.md
index e36d1f6603c..4cf4a469c7e 100644
--- a/src/real-numbers/real-sequences-approximating-zero.lagda.md
+++ b/src/real-numbers/real-sequences-approximating-zero.lagda.md
@@ -110,5 +110,5 @@ abstract
                 by
                   preserves-leq-real-ℚ
                     ( leq-dist-neighborhood-ℚ ε _ _ (is-mod-μ ε n με≤n))))
-      ( lim-b=0)
+        ( lim-b=0)
 ```
diff --git a/src/real-numbers/series-real-numbers.lagda.md b/src/real-numbers/series-real-numbers.lagda.md
index c123f129447..f3d0ecf21e3 100644
--- a/src/real-numbers/series-real-numbers.lagda.md
+++ b/src/real-numbers/series-real-numbers.lagda.md
@@ -25,7 +25,7 @@ open import real-numbers.metric-abelian-group-of-real-numbers
 
 A {{#concept "series" Disambiguation="of real numbers" Agda=series-ℝ}} of
 [real numbers](real-numbers.dedekind-real-numbers.md) is an infinite sum
-$$\Sigma_{n=0}^∞ a_n$$, which is evaluated for convergence in the
+$$∑_{n=0}^∞ a_n$$, which is evaluated for convergence in the
 [metric abelian group of real numbers](real-numbers.metric-abelian-group-of-real-numbers.md).
 
 ## Definition
diff --git a/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md b/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
index c77135801b7..408e41e86f7 100644
--- a/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
+++ b/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
@@ -26,9 +26,8 @@ A
 from the [real numbers](real-numbers.dedekind-real-numbers.md) to the real
 numbers is a function `f : ℝ → ℝ` such that there
 [exists](foundation.existential-quantification.md) a modulus `μ : ℚ⁺ → ℚ⁺` such
-that for all `x y : ℝ` within a `μ ε`
-[neighborhood](real-numbers.metric-space-of-real-numbers.md) of each other,
-`f x` and `f y` are within an `ε` neighborhood.
+that for all `x y : ℝ` within a `μ ε`-[neighborhood](real-numbers.metric-space-of-real-numbers.md) of each other,
+`f x` and `f y` are within an `ε`-neighborhood.
 
 ## Definition
 

From d1447a82c3c24d6ca7c388ca643a17f886786621 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 15:22:27 -0800
Subject: [PATCH 087/134] Respond to comments

---
 .../geometric-sequences-rational-numbers.lagda.md |  2 +-
 .../triangular-numbers.lagda.md                   |  2 +-
 .../convergent-sequences-metric-spaces.lagda.md   |  6 +++---
 .../limits-of-sequences-metric-spaces.lagda.md    | 15 ++++++++-------
 .../geometric-sequences-real-numbers.lagda.md     |  2 +-
 .../limits-sequences-real-numbers.lagda.md        | 12 ++++++------
 src/real-numbers/nonzero-real-numbers.lagda.md    |  2 +-
 ...mly-continuous-functions-real-numbers.lagda.md |  5 +++--
 8 files changed, 24 insertions(+), 22 deletions(-)

diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 3773379808f..39b04e63dd4 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -162,7 +162,7 @@ module _
             ＝
               a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
               by right-unit-law-mul-ℚ _)
-          ( uniformly-continuous-map-limit-sequence-Metric-Space
+          ( preserves-limit-sequence-uniformly-continuous-function-Metric-Space
             ( metric-space-ℚ)
             ( metric-space-ℚ)
             ( comp-uniformly-continuous-function-Metric-Space
diff --git a/src/elementary-number-theory/triangular-numbers.lagda.md b/src/elementary-number-theory/triangular-numbers.lagda.md
index 23ae32125c6..2b5b9fdf3a9 100644
--- a/src/elementary-number-theory/triangular-numbers.lagda.md
+++ b/src/elementary-number-theory/triangular-numbers.lagda.md
@@ -284,7 +284,7 @@ abstract
           by ap-mul-ℚ refl (right-zero-law-diff-ℚ one-ℚ)
         ＝ rational-ℕ 2
           by right-unit-law-mul-ℚ _)
-      ( uniformly-continuous-map-limit-sequence-Metric-Space
+      ( preserves-limit-sequence-uniformly-continuous-function-Metric-Space
         ( metric-space-ℚ)
         ( metric-space-ℚ)
         ( comp-uniformly-continuous-function-Metric-Space
diff --git a/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md b/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md
index 073ef2f615b..7020270424b 100644
--- a/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md
@@ -111,7 +111,7 @@ module _
         ( seq-modulated-ucont-map-convergent-sequence-Metric-Space)
         ( limit-seq-modulated-ucont-map-convergent-sequence-Metric-Space)
     is-limit-seq-modulated-ucont-map-convergent-sequence-Metric-Space =
-      modulated-ucont-map-limit-sequence-Metric-Space
+      preserves-limit-sequence-modulated-ucont-map-Metric-Space
         ( A)
         ( B)
         ( f)
@@ -162,7 +162,7 @@ module _
         ( seq-uniformly-continuous-map-convergent-sequence-Metric-Space)
         ( limit-seq-uniformly-continuous-map-convergent-sequence-Metric-Space)
     is-limit-seq-uniformly-continuous-map-convergent-sequence-Metric-Space =
-      uniformly-continuous-map-limit-sequence-Metric-Space
+      preserves-limit-sequence-uniformly-continuous-function-Metric-Space
         ( A)
         ( B)
         ( f)
@@ -206,7 +206,7 @@ module _
   has-limit-seq-short-map-convergent-sequence-Metric-Space =
     ( map-short-function-Metric-Space A B f
       ( limit-convergent-sequence-Metric-Space A u)) ,
-    ( short-map-limit-sequence-Metric-Space
+    ( preserves-limit-sequence-short-function-Metric-Space
       ( A)
       ( B)
       ( f)
diff --git a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
index 219491a035a..61be4c7f1d9 100644
--- a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
@@ -292,7 +292,7 @@ module _
             ( n)
             ( N≤n)))
 
-    modulated-ucont-map-limit-sequence-Metric-Space :
+    preserves-limit-sequence-modulated-ucont-map-Metric-Space :
       is-limit-sequence-Metric-Space A u lim →
       is-limit-sequence-Metric-Space
         ( B)
@@ -300,7 +300,7 @@ module _
           ( map-modulated-ucont-map-Metric-Space A B f)
           ( u))
         ( map-modulated-ucont-map-Metric-Space A B f lim)
-    modulated-ucont-map-limit-sequence-Metric-Space =
+    preserves-limit-sequence-modulated-ucont-map-Metric-Space =
       map-is-inhabited modulated-ucont-map-limit-modulus-sequence-Metric-Space
 ```
 
@@ -317,7 +317,7 @@ module _
   where
 
   abstract
-    uniformly-continuous-map-limit-sequence-Metric-Space :
+    preserves-limit-sequence-uniformly-continuous-function-Metric-Space :
       is-limit-sequence-Metric-Space A u lim →
       is-limit-sequence-Metric-Space
         ( B)
@@ -325,11 +325,12 @@ module _
           ( map-uniformly-continuous-function-Metric-Space A B f)
           ( u))
         ( map-uniformly-continuous-function-Metric-Space A B f lim)
-    uniformly-continuous-map-limit-sequence-Metric-Space is-limit-lim =
+    preserves-limit-sequence-uniformly-continuous-function-Metric-Space
+      is-limit-lim =
       rec-trunc-Prop
         ( is-limit-prop-sequence-Metric-Space B _ _)
         ( λ m →
-          modulated-ucont-map-limit-sequence-Metric-Space
+          preserves-limit-sequence-modulated-ucont-map-Metric-Space
             ( A)
             ( B)
             ( map-uniformly-continuous-function-Metric-Space A B f , m)
@@ -370,7 +371,7 @@ module _
       ( u)
       ( lim)
 
-  short-map-limit-sequence-Metric-Space :
+  preserves-limit-sequence-short-function-Metric-Space :
     is-limit-sequence-Metric-Space A u lim →
     is-limit-sequence-Metric-Space
       ( B)
@@ -378,7 +379,7 @@ module _
         ( map-short-function-Metric-Space A B f)
         ( u))
       ( map-short-function-Metric-Space A B f lim)
-  short-map-limit-sequence-Metric-Space =
+  preserves-limit-sequence-short-function-Metric-Space =
     map-is-inhabited short-map-limit-modulus-sequence-Metric-Space
 ```
 
diff --git a/src/real-numbers/geometric-sequences-real-numbers.lagda.md b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
index 2db218fddaf..ed7444ffafe 100644
--- a/src/real-numbers/geometric-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
@@ -232,7 +232,7 @@ module _
               by ap-mul-ℝ refl (right-unit-law-diff-ℝ one-ℝ)
             ＝ a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)
               by right-unit-law-mul-ℝ _)
-          ( uniformly-continuous-map-limit-sequence-ℝ
+          ( preserves-limit-sequence-uniformly-continuous-function-ℝ
             ( comp-uniformly-continuous-function-ℝ
               ( uniformly-continuous-right-mul-ℝ
                 ( l)
diff --git a/src/real-numbers/limits-sequences-real-numbers.lagda.md b/src/real-numbers/limits-sequences-real-numbers.lagda.md
index b5d5af537b4..874278d681f 100644
--- a/src/real-numbers/limits-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/limits-sequences-real-numbers.lagda.md
@@ -61,13 +61,13 @@ module _
   where
 
   abstract
-    is-limit-add-sequence-ℝ :
+    preserves-limit-add-sequence-ℝ :
       is-limit-sequence-Metric-Space
         ( metric-space-ℝ (l1 ⊔ l2))
         ( binary-map-sequence add-ℝ u v)
         ( lim-u +ℝ lim-v)
-    is-limit-add-sequence-ℝ =
-      modulated-ucont-map-limit-sequence-Metric-Space
+    preserves-limit-add-sequence-ℝ =
+      preserves-limit-sequence-modulated-ucont-map-Metric-Space
         ( product-Metric-Space (metric-space-ℝ l1) (metric-space-ℝ l2))
         ( metric-space-ℝ (l1 ⊔ l2))
         ( modulated-ucont-add-pair-ℝ l1 l2)
@@ -91,15 +91,15 @@ module _
   where
 
   abstract
-    uniformly-continuous-map-limit-sequence-ℝ :
+    preserves-limit-sequence-uniformly-continuous-function-ℝ :
       is-limit-sequence-ℝ u lim →
       is-limit-sequence-ℝ
         ( map-sequence
           ( map-uniformly-continuous-function-ℝ f)
           ( u))
         ( map-uniformly-continuous-function-ℝ f lim)
-    uniformly-continuous-map-limit-sequence-ℝ =
-      uniformly-continuous-map-limit-sequence-Metric-Space
+    preserves-limit-sequence-uniformly-continuous-function-ℝ =
+      preserves-limit-sequence-uniformly-continuous-function-Metric-Space
         ( metric-space-ℝ l1)
         ( metric-space-ℝ l2)
         ( f)
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index 50a1218db66..c628c950a6b 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -130,7 +130,7 @@ nonzero-diff-le-ℝ :
 nonzero-diff-le-ℝ {x = x} {y = y} x<y = nonzero-ℝ⁺ (positive-diff-le-ℝ x<y)
 ```
 
-### If `|x| < y`, then `y - x` is nonzero
+### The nonzero difference of a pair of real numbers `x` and `y` such that `|x| < y`
 
 ```agda
 nonzero-diff-le-abs-ℝ :
diff --git a/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md b/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
index 408e41e86f7..c929bb7da03 100644
--- a/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
+++ b/src/real-numbers/uniformly-continuous-functions-real-numbers.lagda.md
@@ -26,8 +26,9 @@ A
 from the [real numbers](real-numbers.dedekind-real-numbers.md) to the real
 numbers is a function `f : ℝ → ℝ` such that there
 [exists](foundation.existential-quantification.md) a modulus `μ : ℚ⁺ → ℚ⁺` such
-that for all `x y : ℝ` within a `μ ε`-[neighborhood](real-numbers.metric-space-of-real-numbers.md) of each other,
-`f x` and `f y` are within an `ε`-neighborhood.
+that for all `x y : ℝ` within a
+`μ ε`-[neighborhood](real-numbers.metric-space-of-real-numbers.md) of each
+other, `f x` and `f y` are within an `ε`-neighborhood.
 
 ## Definition
 

From fb3dec5a8dbcc32fc720fa528cfae4244cf6eec9 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 11 Nov 2025 00:33:30 +0100
Subject: [PATCH 088/134] plural `preserves-limits`

---
 .../geometric-sequences-rational-numbers.lagda.md  |  2 +-
 .../triangular-numbers.lagda.md                    |  2 +-
 ...etric-quotients-of-pseudometric-spaces.lagda.md |  4 ++--
 .../convergent-sequences-metric-spaces.lagda.md    |  6 +++---
 .../limits-of-sequences-metric-spaces.lagda.md     | 14 +++++++-------
 .../geometric-sequences-real-numbers.lagda.md      |  2 +-
 .../limits-sequences-real-numbers.lagda.md         | 12 ++++++------
 7 files changed, 21 insertions(+), 21 deletions(-)

diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 39b04e63dd4..5e81a68f072 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -162,7 +162,7 @@ module _
             ＝
               a *ℚ rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
               by right-unit-law-mul-ℚ _)
-          ( preserves-limit-sequence-uniformly-continuous-function-Metric-Space
+          ( preserves-limits-sequence-uniformly-continuous-function-Metric-Space
             ( metric-space-ℚ)
             ( metric-space-ℚ)
             ( comp-uniformly-continuous-function-Metric-Space
diff --git a/src/elementary-number-theory/triangular-numbers.lagda.md b/src/elementary-number-theory/triangular-numbers.lagda.md
index 2b5b9fdf3a9..0a3f74aae18 100644
--- a/src/elementary-number-theory/triangular-numbers.lagda.md
+++ b/src/elementary-number-theory/triangular-numbers.lagda.md
@@ -284,7 +284,7 @@ abstract
           by ap-mul-ℚ refl (right-zero-law-diff-ℚ one-ℚ)
         ＝ rational-ℕ 2
           by right-unit-law-mul-ℚ _)
-      ( preserves-limit-sequence-uniformly-continuous-function-Metric-Space
+      ( preserves-limits-sequence-uniformly-continuous-function-Metric-Space
         ( metric-space-ℚ)
         ( metric-space-ℚ)
         ( comp-uniformly-continuous-function-Metric-Space
diff --git a/src/metric-spaces/cauchy-approximations-metric-quotients-of-pseudometric-spaces.lagda.md b/src/metric-spaces/cauchy-approximations-metric-quotients-of-pseudometric-spaces.lagda.md
index 13705b93b6c..9184f29af6a 100644
--- a/src/metric-spaces/cauchy-approximations-metric-quotients-of-pseudometric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-approximations-metric-quotients-of-pseudometric-spaces.lagda.md
@@ -206,12 +206,12 @@ module _
     is-limit-cauchy-approximation-Pseudometric-Space M u lim)
   where
 
-  preserves-limit-map-metric-quotient-cauchy-approximation-Pseudometric-Space :
+  preserves-limits-map-metric-quotient-cauchy-approximation-Pseudometric-Space :
     is-limit-cauchy-approximation-Metric-Space
       ( metric-quotient-Pseudometric-Space M)
       ( map-metric-quotient-cauchy-approximation-Pseudometric-Space M u)
       ( map-metric-quotient-Pseudometric-Space M lim)
-  preserves-limit-map-metric-quotient-cauchy-approximation-Pseudometric-Space
+  preserves-limits-map-metric-quotient-cauchy-approximation-Pseudometric-Space
     ε δ (x , x∈uε) (y , y∈lim) =
     let
       lim~y : sim-Pseudometric-Space M lim y
diff --git a/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md b/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md
index 7020270424b..b2dc8712acd 100644
--- a/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/convergent-sequences-metric-spaces.lagda.md
@@ -111,7 +111,7 @@ module _
         ( seq-modulated-ucont-map-convergent-sequence-Metric-Space)
         ( limit-seq-modulated-ucont-map-convergent-sequence-Metric-Space)
     is-limit-seq-modulated-ucont-map-convergent-sequence-Metric-Space =
-      preserves-limit-sequence-modulated-ucont-map-Metric-Space
+      preserves-limits-sequence-modulated-ucont-map-Metric-Space
         ( A)
         ( B)
         ( f)
@@ -162,7 +162,7 @@ module _
         ( seq-uniformly-continuous-map-convergent-sequence-Metric-Space)
         ( limit-seq-uniformly-continuous-map-convergent-sequence-Metric-Space)
     is-limit-seq-uniformly-continuous-map-convergent-sequence-Metric-Space =
-      preserves-limit-sequence-uniformly-continuous-function-Metric-Space
+      preserves-limits-sequence-uniformly-continuous-function-Metric-Space
         ( A)
         ( B)
         ( f)
@@ -206,7 +206,7 @@ module _
   has-limit-seq-short-map-convergent-sequence-Metric-Space =
     ( map-short-function-Metric-Space A B f
       ( limit-convergent-sequence-Metric-Space A u)) ,
-    ( preserves-limit-sequence-short-function-Metric-Space
+    ( preserves-limits-sequence-short-function-Metric-Space
       ( A)
       ( B)
       ( f)
diff --git a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
index 61be4c7f1d9..2cfda0f25e0 100644
--- a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
@@ -292,7 +292,7 @@ module _
             ( n)
             ( N≤n)))
 
-    preserves-limit-sequence-modulated-ucont-map-Metric-Space :
+    preserves-limits-sequence-modulated-ucont-map-Metric-Space :
       is-limit-sequence-Metric-Space A u lim →
       is-limit-sequence-Metric-Space
         ( B)
@@ -300,7 +300,7 @@ module _
           ( map-modulated-ucont-map-Metric-Space A B f)
           ( u))
         ( map-modulated-ucont-map-Metric-Space A B f lim)
-    preserves-limit-sequence-modulated-ucont-map-Metric-Space =
+    preserves-limits-sequence-modulated-ucont-map-Metric-Space =
       map-is-inhabited modulated-ucont-map-limit-modulus-sequence-Metric-Space
 ```
 
@@ -317,7 +317,7 @@ module _
   where
 
   abstract
-    preserves-limit-sequence-uniformly-continuous-function-Metric-Space :
+    preserves-limits-sequence-uniformly-continuous-function-Metric-Space :
       is-limit-sequence-Metric-Space A u lim →
       is-limit-sequence-Metric-Space
         ( B)
@@ -325,12 +325,12 @@ module _
           ( map-uniformly-continuous-function-Metric-Space A B f)
           ( u))
         ( map-uniformly-continuous-function-Metric-Space A B f lim)
-    preserves-limit-sequence-uniformly-continuous-function-Metric-Space
+    preserves-limits-sequence-uniformly-continuous-function-Metric-Space
       is-limit-lim =
       rec-trunc-Prop
         ( is-limit-prop-sequence-Metric-Space B _ _)
         ( λ m →
-          preserves-limit-sequence-modulated-ucont-map-Metric-Space
+          preserves-limits-sequence-modulated-ucont-map-Metric-Space
             ( A)
             ( B)
             ( map-uniformly-continuous-function-Metric-Space A B f , m)
@@ -371,7 +371,7 @@ module _
       ( u)
       ( lim)
 
-  preserves-limit-sequence-short-function-Metric-Space :
+  preserves-limits-sequence-short-function-Metric-Space :
     is-limit-sequence-Metric-Space A u lim →
     is-limit-sequence-Metric-Space
       ( B)
@@ -379,7 +379,7 @@ module _
         ( map-short-function-Metric-Space A B f)
         ( u))
       ( map-short-function-Metric-Space A B f lim)
-  preserves-limit-sequence-short-function-Metric-Space =
+  preserves-limits-sequence-short-function-Metric-Space =
     map-is-inhabited short-map-limit-modulus-sequence-Metric-Space
 ```
 
diff --git a/src/real-numbers/geometric-sequences-real-numbers.lagda.md b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
index ed7444ffafe..a8688366577 100644
--- a/src/real-numbers/geometric-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/geometric-sequences-real-numbers.lagda.md
@@ -232,7 +232,7 @@ module _
               by ap-mul-ℝ refl (right-unit-law-diff-ℝ one-ℝ)
             ＝ a *ℝ real-inv-nonzero-ℝ (nonzero-diff-le-abs-ℝ |r|<1)
               by right-unit-law-mul-ℝ _)
-          ( preserves-limit-sequence-uniformly-continuous-function-ℝ
+          ( preserves-limits-sequence-uniformly-continuous-function-ℝ
             ( comp-uniformly-continuous-function-ℝ
               ( uniformly-continuous-right-mul-ℝ
                 ( l)
diff --git a/src/real-numbers/limits-sequences-real-numbers.lagda.md b/src/real-numbers/limits-sequences-real-numbers.lagda.md
index 874278d681f..4636b60dcda 100644
--- a/src/real-numbers/limits-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/limits-sequences-real-numbers.lagda.md
@@ -61,13 +61,13 @@ module _
   where
 
   abstract
-    preserves-limit-add-sequence-ℝ :
+    preserves-limits-add-sequence-ℝ :
       is-limit-sequence-Metric-Space
         ( metric-space-ℝ (l1 ⊔ l2))
         ( binary-map-sequence add-ℝ u v)
         ( lim-u +ℝ lim-v)
-    preserves-limit-add-sequence-ℝ =
-      preserves-limit-sequence-modulated-ucont-map-Metric-Space
+    preserves-limits-add-sequence-ℝ =
+      preserves-limits-sequence-modulated-ucont-map-Metric-Space
         ( product-Metric-Space (metric-space-ℝ l1) (metric-space-ℝ l2))
         ( metric-space-ℝ (l1 ⊔ l2))
         ( modulated-ucont-add-pair-ℝ l1 l2)
@@ -91,15 +91,15 @@ module _
   where
 
   abstract
-    preserves-limit-sequence-uniformly-continuous-function-ℝ :
+    preserves-limits-sequence-uniformly-continuous-function-ℝ :
       is-limit-sequence-ℝ u lim →
       is-limit-sequence-ℝ
         ( map-sequence
           ( map-uniformly-continuous-function-ℝ f)
           ( u))
         ( map-uniformly-continuous-function-ℝ f lim)
-    preserves-limit-sequence-uniformly-continuous-function-ℝ =
-      preserves-limit-sequence-uniformly-continuous-function-Metric-Space
+    preserves-limits-sequence-uniformly-continuous-function-ℝ =
+      preserves-limits-sequence-uniformly-continuous-function-Metric-Space
         ( metric-space-ℝ l1)
         ( metric-space-ℝ l2)
         ( f)

From b016840603269a603114e0919a64983132f06b6e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 16:35:09 -0800
Subject: [PATCH 089/134] Vector spaces

---
 src/linear-algebra/vector-spaces.lagda.md     | 55 ++++++++++++
 .../addition-negative-real-numbers.lagda.md   | 84 +++++++++++++++++++
 .../addition-nonzero-real-numbers.lagda.md    | 55 ++++++++++++
 .../local-ring-of-real-numbers.lagda.md       | 50 +++++++++++
 ...ive-inverses-nonzero-real-numbers.lagda.md | 14 ++++
 ...ositive-and-negative-real-numbers.lagda.md | 33 ++++++++
 6 files changed, 291 insertions(+)
 create mode 100644 src/linear-algebra/vector-spaces.lagda.md
 create mode 100644 src/real-numbers/addition-negative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/addition-nonzero-real-numbers.lagda.md
 create mode 100644 src/real-numbers/local-ring-of-real-numbers.lagda.md

diff --git a/src/linear-algebra/vector-spaces.lagda.md b/src/linear-algebra/vector-spaces.lagda.md
new file mode 100644
index 00000000000..d9e39a8ed7c
--- /dev/null
+++ b/src/linear-algebra/vector-spaces.lagda.md
@@ -0,0 +1,55 @@
+# Vector spaces
+
+```agda
+module linear-algebra.vector-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.sets
+open import foundation.universe-levels
+open import group-theory.abelian-groups
+open import commutative-algebra.local-commutative-rings
+open import linear-algebra.left-modules-commutative-rings
+```
+
+</details>
+
+## Idea
+
+A {{#concept "vector space" WD="vector space" WDID=Q125977}} is a
+[left module](linear-algebra.left-modules-rings.md) over a
+[local commutative ring](commutative-algebra.local-commutative-rings.md).
+
+## Definition
+
+```agda
+Vector-Space :
+  {l1 : Level} (l2 : Level) → Local-Commutative-Ring l1 → UU (l1 ⊔ lsuc l2)
+Vector-Space l2 R =
+  left-module-Commutative-Ring l2 (commutative-ring-Local-Commutative-Ring R)
+```
+
+## Properties
+
+```agda
+module _
+  {l1 l2 : Level} (R : Local-Commutative-Ring l1) (V : Vector-Space l2 R)
+  where
+
+  ab-Vector-Space : Ab l2
+  ab-Vector-Space =
+    ab-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  set-Vector-Space : Set l2
+  set-Vector-Space = set-Ab ab-Vector-Space
+
+  type-Vector-Space : UU l2
+  type-Vector-Space = type-Ab ab-Vector-Space
+
+  add-Vector-Space : type-Vector-Space → type-Vector-Space → type-Vector-Space
+  add-Vector-Space = add-Ab ab-Vector-Space
+```
diff --git a/src/real-numbers/addition-negative-real-numbers.lagda.md b/src/real-numbers/addition-negative-real-numbers.lagda.md
new file mode 100644
index 00000000000..66f7d5e1e54
--- /dev/null
+++ b/src/real-numbers/addition-negative-real-numbers.lagda.md
@@ -0,0 +1,84 @@
+# Addition of negative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.addition-negative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import real-numbers.negative-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import foundation.transport-along-identifications
+open import real-numbers.rational-real-numbers
+open import real-numbers.negation-real-numbers
+open import foundation.functoriality-disjunction
+open import real-numbers.addition-positive-real-numbers
+open import foundation.disjunction
+open import real-numbers.addition-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
+open import foundation.universe-levels
+open import foundation.dependent-pair-types
+```
+
+</details>
+
+## Idea
+
+The [negative](real-numbers.negative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are closed under
+[addition](real-numbers.addition-real-numbers.md).
+
+## Definition
+
+```agda
+abstract
+  is-negative-add-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → is-negative-ℝ x → is-negative-ℝ y →
+    is-negative-ℝ (x +ℝ y)
+  is-negative-add-ℝ {x = x} {y = y} x<0 y<0 =
+    transitive-le-ℝ
+      ( x +ℝ y)
+      ( x)
+      ( zero-ℝ)
+      ( x<0)
+      ( tr
+        ( le-ℝ (x +ℝ y))
+        ( right-unit-law-add-ℝ x)
+        ( preserves-le-left-add-ℝ x y zero-ℝ y<0))
+
+add-ℝ⁻ : {l1 l2 : Level} → ℝ⁻ l1 → ℝ⁻ l2 → ℝ⁻ (l1 ⊔ l2)
+add-ℝ⁻ (x , x<0) (y , y<0) = (add-ℝ x y , is-negative-add-ℝ x<0 y<0)
+
+infixl 35 _+ℝ⁻_
+
+_+ℝ⁻_ : {l1 l2 : Level} → ℝ⁻ l1 → ℝ⁻ l2 → ℝ⁻ (l1 ⊔ l2)
+_+ℝ⁻_ = add-ℝ⁻
+```
+
+## Properties
+
+### If `x + y` is negative, `x` or `y` is negative
+
+```agda
+abstract
+  is-negative-either-is-negative-add-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → is-negative-ℝ (x +ℝ y) →
+    disjunction-type (is-negative-ℝ x) (is-negative-ℝ y)
+  is-negative-either-is-negative-add-ℝ x y x+y<0 =
+    map-disjunction
+      ( is-negative-is-positive-neg-ℝ x)
+      ( is-negative-is-positive-neg-ℝ y)
+      ( is-positive-either-is-positive-add-ℝ
+        ( neg-ℝ x)
+        ( neg-ℝ y)
+        ( tr
+          ( is-positive-ℝ)
+          ( distributive-neg-add-ℝ x y)
+          ( neg-is-negative-ℝ (x +ℝ y) x+y<0)))
+```
diff --git a/src/real-numbers/addition-nonzero-real-numbers.lagda.md b/src/real-numbers/addition-nonzero-real-numbers.lagda.md
new file mode 100644
index 00000000000..3ab16e4f8b7
--- /dev/null
+++ b/src/real-numbers/addition-nonzero-real-numbers.lagda.md
@@ -0,0 +1,55 @@
+# Addition of nonzero real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.addition-nonzero-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import real-numbers.nonzero-real-numbers
+open import real-numbers.addition-positive-real-numbers
+open import real-numbers.addition-negative-real-numbers
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import foundation.functoriality-disjunction
+open import foundation.disjunction
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+```
+
+</details>
+
+## Idea
+
+On this page we describe properties of
+[addition](real-numbers.addition-real-numbers.md) of
+[nonzero](real-numbers.nonzero-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md).
+
+## Properties
+
+### If `x + y` is nonzero, `x` is nonzero or `y` is nonzero
+
+```agda
+abstract
+  is-nonzero-either-is-nonzero-add-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) →
+    is-nonzero-ℝ (x +ℝ y) →
+    disjunction-type (is-nonzero-ℝ x) (is-nonzero-ℝ y)
+  is-nonzero-either-is-nonzero-add-ℝ x y =
+    elim-disjunction
+      ( is-nonzero-prop-ℝ x ∨ is-nonzero-prop-ℝ y)
+      ( λ x+y<0 →
+        map-disjunction
+          ( inl-disjunction)
+          ( inl-disjunction)
+          ( is-negative-either-is-negative-add-ℝ x y x+y<0))
+      ( λ 0<x+y →
+        map-disjunction
+          ( inr-disjunction)
+          ( inr-disjunction)
+          ( is-positive-either-is-positive-add-ℝ x y 0<x+y))
+```
diff --git a/src/real-numbers/local-ring-of-real-numbers.lagda.md b/src/real-numbers/local-ring-of-real-numbers.lagda.md
new file mode 100644
index 00000000000..282f1dc2237
--- /dev/null
+++ b/src/real-numbers/local-ring-of-real-numbers.lagda.md
@@ -0,0 +1,50 @@
+# The local ring of Dedekind real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.local-ring-of-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.local-commutative-rings
+open import real-numbers.addition-real-numbers
+open import real-numbers.nonzero-real-numbers
+open import foundation.dependent-pair-types
+open import foundation.functoriality-disjunction
+open import real-numbers.addition-nonzero-real-numbers
+open import real-numbers.large-ring-of-real-numbers
+open import real-numbers.multiplicative-inverses-nonzero-real-numbers
+open import foundation.universe-levels
+```
+
+</details>
+
+## Idea
+
+The [ring of real numbers](real-numbers.large-ring-of-real-numbers.md) is a
+[local](commutative-algebra.local-commutative-rings.md)
+[commutative ring](commutative-algebra.commutative-rings.md). Notably, this is
+one of the strongest alternatives for the concept of field that applies to the
+[Dedekind real numbers](real-numbers.dedekind-real-numbers.md) constructively.
+
+## Definition
+
+```agda
+is-local-commutative-ring-ℝ :
+  (l : Level) → is-local-Commutative-Ring (commutative-ring-ℝ l)
+is-local-commutative-ring-ℝ l x y is-invertible-x+y =
+  map-disjunction
+    ( is-invertible-is-nonzero-ℝ x)
+    ( is-invertible-is-nonzero-ℝ y)
+    ( is-nonzero-either-is-nonzero-add-ℝ
+      ( x)
+      ( y)
+      ( is-nonzero-is-invertible-ℝ (x +ℝ y) is-invertible-x+y))
+
+local-commutative-ring-ℝ : (l : Level) → Local-Commutative-Ring (lsuc l)
+local-commutative-ring-ℝ l =
+  ( commutative-ring-ℝ l , is-local-commutative-ring-ℝ l)
+```
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index aedb75f632c..bd117101ec5 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -210,6 +210,20 @@ abstract opaque
   is-nonzero-has-left-inverse-mul-ℝ x y xy=1 =
     is-nonzero-has-right-inverse-mul-ℝ y x
       ( tr (λ z → sim-ℝ z one-ℝ) (commutative-mul-ℝ _ _) xy=1)
+
+  is-nonzero-is-invertible-ℝ :
+    {l : Level} (x : ℝ l) →
+    is-invertible-element-Commutative-Ring (commutative-ring-ℝ l) x →
+    is-nonzero-ℝ x
+  is-nonzero-is-invertible-ℝ {l} x (y , xy=1 , _) =
+    is-nonzero-has-right-inverse-mul-ℝ x y
+      ( inv-tr
+        ( λ z → sim-ℝ z one-ℝ)
+        ( xy=1)
+        ( symmetric-sim-ℝ
+          { x = one-ℝ}
+          { y = raise-ℝ l one-ℝ}
+          ( sim-raise-ℝ l one-ℝ)))
 ```
 
 ### The multiplicative inverse is unique
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index 18f7f33b894..b30852d3524 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -11,6 +11,7 @@ module real-numbers.positive-and-negative-real-numbers where
 ```agda
 open import foundation.dependent-pair-types
 open import foundation.transport-along-identifications
+open import foundation.logical-equivalences
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
@@ -78,6 +79,38 @@ neg-ℝ⁻ : {l : Level} → ℝ⁻ l → ℝ⁺ l
 neg-ℝ⁻ (x , is-neg-x) = (neg-ℝ x , neg-is-negative-ℝ x is-neg-x)
 ```
 
+### A real number is negative if and only if its negation is positive
+
+```agda
+abstract
+  is-negative-is-positive-neg-ℝ :
+    {l : Level} (x : ℝ l) → is-positive-ℝ (neg-ℝ x) → is-negative-ℝ x
+  is-negative-is-positive-neg-ℝ x 0<-x =
+    tr is-negative-ℝ (neg-neg-ℝ x) (neg-is-positive-ℝ (neg-ℝ x) 0<-x)
+
+is-negative-iff-neg-is-positive-ℝ :
+  {l : Level} (x : ℝ l) → is-negative-ℝ x ↔ is-positive-ℝ (neg-ℝ x)
+is-negative-iff-neg-is-positive-ℝ x =
+  ( neg-is-negative-ℝ x ,
+    is-negative-is-positive-neg-ℝ x)
+```
+
+### A real number is positive if and only if its negation is negative
+
+```agda
+abstract
+  is-positive-is-negative-neg-ℝ :
+    {l : Level} (x : ℝ l) → is-negative-ℝ (neg-ℝ x) → is-positive-ℝ x
+  is-positive-is-negative-neg-ℝ x -x<0 =
+    tr is-positive-ℝ (neg-neg-ℝ x) (neg-is-negative-ℝ (neg-ℝ x) -x<0)
+
+is-positive-iff-neg-is-negative-ℝ :
+  {l : Level} (x : ℝ l) → is-positive-ℝ x ↔ is-negative-ℝ (neg-ℝ x)
+is-positive-iff-neg-is-negative-ℝ x =
+  ( neg-is-positive-ℝ x ,
+    is-positive-is-negative-neg-ℝ x)
+```
+
 ### If a nonnegative real number `x` is less than a real number `y`, `y` is positive
 
 ```agda

From 3eaaa4620e0c17592c293ddec431901c66e82f74 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 16:42:52 -0800
Subject: [PATCH 090/134] The large ring of complex numbers

---
 .../addition-complex-numbers.lagda.md         | 13 +++
 ...additive-group-of-complex-numbers.lagda.md | 54 -----------
 ...additive-group-of-complex-numbers.lagda.md | 90 +++++++++++++++++++
 .../large-ring-of-complex-numbers.lagda.md    | 48 ++++++++++
 .../multiplication-complex-numbers.lagda.md   | 61 +++++++++++++
 ...g-universe-levels-complex-numbers.lagda.md | 44 +++++++++
 .../similarity-complex-numbers.lagda.md       | 90 +++++++++++++++++++
 7 files changed, 346 insertions(+), 54 deletions(-)
 delete mode 100644 src/complex-numbers/additive-group-of-complex-numbers.lagda.md
 create mode 100644 src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md
 create mode 100644 src/complex-numbers/large-ring-of-complex-numbers.lagda.md
 create mode 100644 src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md

diff --git a/src/complex-numbers/addition-complex-numbers.lagda.md b/src/complex-numbers/addition-complex-numbers.lagda.md
index 8b5dbbadd8f..63a33629054 100644
--- a/src/complex-numbers/addition-complex-numbers.lagda.md
+++ b/src/complex-numbers/addition-complex-numbers.lagda.md
@@ -91,3 +91,16 @@ abstract
   right-inverse-law-add-ℂ (a , b) =
     ( right-inverse-law-add-ℝ a , right-inverse-law-add-ℝ b)
 ```
+
+### Similarity is preserved by addition
+
+```agda
+abstract
+  preserves-sim-add-ℂ :
+    {l1 l2 l3 l4 : Level} {x : ℂ l1} {x' : ℂ l2} {y : ℂ l3} {y' : ℂ l4} →
+    sim-ℂ x x' → sim-ℂ y y' → sim-ℂ (x +ℂ y) (x' +ℂ y')
+  preserves-sim-add-ℂ
+    { x = a +iℂ b} {x' = a' +iℂ b'} {y = c +iℂ d} {y' = c' +iℂ d'}
+    ( a~a' , b~b') (c~c' , d~d') =
+    ( preserves-sim-add-ℝ a~a' c~c' , preserves-sim-add-ℝ b~b' d~d')
+```
diff --git a/src/complex-numbers/additive-group-of-complex-numbers.lagda.md b/src/complex-numbers/additive-group-of-complex-numbers.lagda.md
deleted file mode 100644
index 8c4dea6fd7e..00000000000
--- a/src/complex-numbers/additive-group-of-complex-numbers.lagda.md
+++ /dev/null
@@ -1,54 +0,0 @@
-# The additive group of complex numbers
-
-```agda
-module complex-numbers.additive-group-of-complex-numbers where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import complex-numbers.addition-complex-numbers
-open import complex-numbers.complex-numbers
-open import complex-numbers.similarity-complex-numbers
-
-open import foundation.dependent-pair-types
-open import foundation.function-types
-open import foundation.universe-levels
-
-open import group-theory.abelian-groups
-open import group-theory.groups
-open import group-theory.monoids
-open import group-theory.semigroups
-```
-
-</details>
-
-## Idea
-
-The type of [complex numbers](complex-numbers.complex-numbers.md) equipped with
-[addition](complex-numbers.addition-complex-numbers.md) is a
-[commutative group](group-theory.abelian-groups.md) with unit `zero-ℂ` and
-inverse given by `neg-ℂ`.
-
-## Definition
-
-```agda
-semigroup-add-ℂ-lzero : Semigroup (lsuc lzero)
-semigroup-add-ℂ-lzero =
-  ( ℂ-Set lzero ,
-    add-ℂ ,
-    associative-add-ℂ)
-
-group-add-ℂ-lzero : Group (lsuc lzero)
-group-add-ℂ-lzero =
-  ( semigroup-add-ℂ-lzero ,
-    ( zero-ℂ , left-unit-law-add-ℂ , right-unit-law-add-ℂ) ,
-    neg-ℂ ,
-    eq-sim-ℂ ∘ left-inverse-law-add-ℂ ,
-    eq-sim-ℂ ∘ right-inverse-law-add-ℂ)
-
-ab-add-ℂ-lzero : Ab (lsuc lzero)
-ab-add-ℂ-lzero =
-  ( group-add-ℂ-lzero ,
-    commutative-add-ℂ)
-```
diff --git a/src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md b/src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md
new file mode 100644
index 00000000000..c2385a5e253
--- /dev/null
+++ b/src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md
@@ -0,0 +1,90 @@
+# The large additive group of complex numbers
+
+```agda
+module complex-numbers.large-additive-group-of-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.addition-complex-numbers
+open import complex-numbers.complex-numbers
+open import complex-numbers.similarity-complex-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.function-types
+open import foundation.universe-levels
+open import complex-numbers.raising-universe-levels-complex-numbers
+
+open import group-theory.large-semigroups
+open import group-theory.large-monoids
+open import group-theory.large-commutative-monoids
+open import group-theory.large-groups
+open import group-theory.large-abelian-groups
+```
+
+</details>
+
+## Idea
+
+The type of [complex numbers](complex-numbers.complex-numbers.md) equipped with
+[addition](complex-numbers.addition-complex-numbers.md) is a
+[large abelian group](group-theory.large-abelian-groups.md).
+
+## Definition
+
+```agda
+large-semigroup-add-ℂ : Large-Semigroup lsuc
+large-semigroup-add-ℂ =
+  make-Large-Semigroup
+    ( ℂ-Set)
+    ( add-ℂ)
+    ( associative-add-ℂ)
+
+large-monoid-add-ℂ : Large-Monoid lsuc (_⊔_)
+large-monoid-add-ℂ =
+  make-Large-Monoid
+    ( large-semigroup-add-ℂ)
+    ( large-similarity-relation-ℂ)
+    ( raise-ℂ)
+    ( sim-raise-ℂ)
+    ( λ _ _ z~z' _ _ → preserves-sim-add-ℂ z~z')
+    ( zero-ℂ)
+    ( left-unit-law-add-ℂ)
+    ( right-unit-law-add-ℂ)
+
+large-commutative-monoid-add-ℂ : Large-Commutative-Monoid lsuc (_⊔_)
+large-commutative-monoid-add-ℂ =
+  make-Large-Commutative-Monoid
+    ( large-monoid-add-ℂ)
+    ( commutative-add-ℂ)
+
+large-group-add-ℂ : Large-Group lsuc (_⊔_)
+large-group-add-ℂ =
+  make-Large-Group
+    ( large-monoid-add-ℂ)
+    ( neg-ℂ)
+    ( λ _ _ → preserves-sim-neg-ℂ)
+    ( λ x →
+      eq-sim-ℂ
+        ( transitive-sim-ℂ
+          ( neg-ℂ x +ℂ x)
+          ( zero-ℂ)
+          ( raise-ℂ _ zero-ℂ)
+          ( sim-raise-ℂ _ zero-ℂ)
+          ( left-inverse-law-add-ℂ x)))
+    ( λ x →
+      eq-sim-ℂ
+        ( transitive-sim-ℂ
+          ( x +ℂ neg-ℂ x)
+          ( zero-ℂ)
+          ( raise-ℂ _ zero-ℂ)
+          ( sim-raise-ℂ _ zero-ℂ)
+          ( right-inverse-law-add-ℂ x)))
+
+large-ab-add-ℂ : Large-Ab lsuc (_⊔_)
+large-ab-add-ℂ =
+  make-Large-Ab
+    ( large-group-add-ℂ)
+    ( commutative-add-ℂ)
+```
diff --git a/src/complex-numbers/large-ring-of-complex-numbers.lagda.md b/src/complex-numbers/large-ring-of-complex-numbers.lagda.md
new file mode 100644
index 00000000000..4c38a6ec892
--- /dev/null
+++ b/src/complex-numbers/large-ring-of-complex-numbers.lagda.md
@@ -0,0 +1,48 @@
+# The large ring of complex numbers
+
+```agda
+module complex-numbers.large-ring-of-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import ring-theory.large-rings
+open import commutative-algebra.large-commutative-rings
+open import complex-numbers.large-additive-group-of-complex-numbers
+open import complex-numbers.complex-numbers
+open import foundation.universe-levels
+open import complex-numbers.multiplication-complex-numbers
+```
+
+</details>
+
+## Idea
+
+The [complex numbers](complex-numbers.complex-numbers.md) form a
+[large commutative ring](commutative-algebra.large-commutative-rings.md) under
+[addition](complex-numbers.addition-complex-numbers.md) and
+[multiplication](complex-numbers.multiplication-complex-numbers.md).
+
+## Definition
+
+```agda
+large-ring-ℂ : Large-Ring lsuc (_⊔_)
+large-ring-ℂ =
+  make-Large-Ring
+    ( large-ab-add-ℂ)
+    ( mul-ℂ)
+    ( λ _ _ z~z' _ _ → preserves-sim-mul-ℂ z~z')
+    ( one-ℂ)
+    ( associative-mul-ℂ)
+    ( left-unit-law-mul-ℂ)
+    ( right-unit-law-mul-ℂ)
+    ( left-distributive-mul-add-ℂ)
+    ( right-distributive-mul-add-ℂ)
+
+large-commutative-ring-ℂ : Large-Commutative-Ring lsuc (_⊔_)
+large-commutative-ring-ℂ =
+  make-Large-Commutative-Ring
+    ( large-ring-ℂ)
+    ( commutative-mul-ℂ)
+```
diff --git a/src/complex-numbers/multiplication-complex-numbers.lagda.md b/src/complex-numbers/multiplication-complex-numbers.lagda.md
index 12598cc434b..098ba6f3280 100644
--- a/src/complex-numbers/multiplication-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplication-complex-numbers.lagda.md
@@ -255,3 +255,64 @@ opaque
         ＝ zero-ℝ
           by left-unit-law-add-ℝ zero-ℝ)
 ```
+<<<<<<< Updated upstream
+=======
+
+### The canonical embedding of real numbers in the complex numbers preserves multiplication
+
+```agda
+abstract
+  mul-complex-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) →
+    complex-ℝ x *ℂ complex-ℝ y ＝ complex-ℝ (x *ℝ y)
+  mul-complex-ℝ {l1} {l2} x y =
+    eq-ℂ
+      ( equational-reasoning
+        x *ℝ y -ℝ raise-ℝ l1 zero-ℝ *ℝ raise-ℝ l2 zero-ℝ
+        ＝ x *ℝ y -ℝ zero-ℝ *ℝ zero-ℝ
+          by
+            eq-sim-ℝ
+              ( preserves-sim-diff-ℝ
+                ( refl-sim-ℝ (x *ℝ y))
+                ( symmetric-sim-ℝ
+                  ( preserves-sim-mul-ℝ (sim-raise-ℝ _ _) (sim-raise-ℝ _ _))))
+        ＝ x *ℝ y -ℝ zero-ℝ
+          by ap-diff-ℝ refl (eq-sim-ℝ (right-zero-law-mul-ℝ _))
+        ＝ x *ℝ y
+          by right-unit-law-diff-ℝ (x *ℝ y))
+      ( eq-sim-ℝ
+        ( similarity-reasoning-ℝ
+          x *ℝ raise-ℝ l2 zero-ℝ +ℝ raise-ℝ l1 zero-ℝ *ℝ y
+          ~ℝ x *ℝ zero-ℝ +ℝ zero-ℝ *ℝ y
+            by
+              symmetric-sim-ℝ
+                ( preserves-sim-add-ℝ
+                  ( preserves-sim-left-mul-ℝ _ _ _ (sim-raise-ℝ l2 zero-ℝ))
+                  ( preserves-sim-right-mul-ℝ _ _ _ (sim-raise-ℝ l1 zero-ℝ)))
+          ~ℝ zero-ℝ +ℝ zero-ℝ
+            by
+              preserves-sim-add-ℝ
+                ( right-zero-law-mul-ℝ x)
+                ( left-zero-law-mul-ℝ y)
+          ~ℝ zero-ℝ
+            by sim-eq-ℝ (left-unit-law-add-ℝ zero-ℝ)
+          ~ℝ raise-ℝ (l1 ⊔ l2) zero-ℝ
+            by sim-raise-ℝ (l1 ⊔ l2) zero-ℝ))
+```
+
+### Similarity is preserved by multiplication
+
+```agda
+abstract
+  preserves-sim-mul-ℂ :
+    {l1 l2 l3 l4 : Level} {x : ℂ l1} {x' : ℂ l2} {y : ℂ l3} {y' : ℂ l4} →
+    sim-ℂ x x' → sim-ℂ y y' → sim-ℂ (x *ℂ y) (x' *ℂ y')
+  preserves-sim-mul-ℂ (a~a' , b~b') (c~c' , d~d') =
+    ( preserves-sim-diff-ℝ
+        ( preserves-sim-mul-ℝ a~a' c~c')
+        ( preserves-sim-mul-ℝ b~b' d~d') ,
+      preserves-sim-add-ℝ
+        ( preserves-sim-mul-ℝ a~a' d~d')
+        ( preserves-sim-mul-ℝ b~b' c~c'))
+```
+>>>>>>> Stashed changes
diff --git a/src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md b/src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md
new file mode 100644
index 00000000000..2ad31cb0190
--- /dev/null
+++ b/src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md
@@ -0,0 +1,44 @@
+# Raising the universe level of complex numbers
+
+```agda
+module complex-numbers.raising-universe-levels-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.complex-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import foundation.dependent-pair-types
+open import complex-numbers.similarity-complex-numbers
+open import foundation.universe-levels
+```
+
+</details>
+
+## Idea
+
+For every [universe](foundation.universe-levels.md) `𝒰` there is a type of
+[complex numbers](complex-numbers.complex-numbers.md) `ℂ` relative to `𝒰`,
+`ℂ 𝒰`. Given a larger universe `𝒱`, then we may
+{{#concept "raise" Disambiguation="a complex number" Agda=raise-ℂ}} a complex
+number `z` from the universe `𝒰` to a
+[similar](complex-numbers.similarity-complex-numbers.md) complex number in the
+universe `𝒱`.
+
+## Definition
+
+```agda
+raise-ℂ : {l1 : Level} (l2 : Level) → ℂ l1 → ℂ (l1 ⊔ l2)
+raise-ℂ l (a +iℂ b) = raise-ℝ l a +iℂ raise-ℝ l b
+```
+
+## Properties
+
+### Complex numbers are similar to their raised-universe equivalents
+
+```agda
+abstract
+  sim-raise-ℂ : {l1 : Level} (l2 : Level) (z : ℂ l1) → sim-ℂ z (raise-ℂ l2 z)
+  sim-raise-ℂ l (a +iℂ b) = ( sim-raise-ℝ l a , sim-raise-ℝ l b)
+```
diff --git a/src/complex-numbers/similarity-complex-numbers.lagda.md b/src/complex-numbers/similarity-complex-numbers.lagda.md
index 2874755ca84..84294023d61 100644
--- a/src/complex-numbers/similarity-complex-numbers.lagda.md
+++ b/src/complex-numbers/similarity-complex-numbers.lagda.md
@@ -11,10 +11,19 @@ open import complex-numbers.complex-numbers
 
 open import foundation.conjunction
 open import foundation.dependent-pair-types
+open import foundation.large-equivalence-relations
+open import foundation.large-similarity-relations
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
 
+<<<<<<< Updated upstream
+=======
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.rational-real-numbers
+>>>>>>> Stashed changes
 open import real-numbers.similarity-real-numbers
 ```
 
@@ -74,3 +83,84 @@ abstract
   eq-sim-ℂ : {l : Level} {x y : ℂ l} → sim-ℂ x y → x ＝ y
   eq-sim-ℂ (a~c , b~d) = eq-ℂ (eq-sim-ℝ a~c) (eq-sim-ℝ b~d)
 ```
+<<<<<<< Updated upstream
+=======
+
+### Similarity is a large equivalence relation
+
+```agda
+large-equivalence-relation-sim-ℂ : Large-Equivalence-Relation (_⊔_) ℂ
+large-equivalence-relation-sim-ℂ =
+  make-Large-Equivalence-Relation
+    ( sim-prop-ℂ)
+    ( refl-sim-ℂ)
+    ( λ _ _ → symmetric-sim-ℂ)
+    ( transitive-sim-ℂ)
+```
+
+### Similarity is a large similarity relation
+
+```agda
+large-similarity-relation-ℂ : Large-Similarity-Relation (_⊔_) ℂ
+large-similarity-relation-ℂ =
+  make-Large-Similarity-Relation
+    ( large-equivalence-relation-sim-ℂ)
+    ( λ _ _ → eq-sim-ℂ)
+```
+
+### The canonical embedding of real numbers in the complex numbers preserves similarity
+
+```agda
+abstract
+  preserves-sim-complex-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → sim-ℝ x y →
+    sim-ℂ (complex-ℝ x) (complex-ℝ y)
+  preserves-sim-complex-ℝ {l1} {l2} x~y =
+    ( x~y ,
+      transitive-sim-ℝ
+        ( raise-ℝ l1 zero-ℝ)
+        ( zero-ℝ)
+        ( raise-ℝ l2 zero-ℝ)
+        ( sim-raise-ℝ l2 zero-ℝ)
+        ( symmetric-sim-ℝ (sim-raise-ℝ l1 zero-ℝ)))
+```
+
+### Similarity is preserved by negation
+
+```agda
+abstract
+  preserves-sim-neg-ℂ :
+    {l1 l2 : Level} {x : ℂ l1} {y : ℂ l2} →
+    sim-ℂ x y → sim-ℂ (neg-ℂ x) (neg-ℂ y)
+  preserves-sim-neg-ℂ (a~c , b~d) =
+    ( preserves-sim-neg-ℝ a~c , preserves-sim-neg-ℝ b~d)
+```
+
+### Similarity reasoning
+
+Similarities between complex numbers can be constructed by similarity reasoning
+in the following way:
+
+```text
+similarity-reasoning-ℂ
+  x ~ℂ y by sim-1
+    ~ℂ z by sim-2
+```
+
+```agda
+infixl 1 similarity-reasoning-ℂ_
+infixl 0 step-similarity-reasoning-ℂ
+
+abstract
+  similarity-reasoning-ℂ_ :
+    {l : Level} → (x : ℂ l) → sim-ℂ x x
+  similarity-reasoning-ℂ x = refl-sim-ℂ x
+
+  step-similarity-reasoning-ℂ :
+    {l1 l2 : Level} {x : ℂ l1} {y : ℂ l2} →
+    sim-ℂ x y → {l3 : Level} → (u : ℂ l3) → sim-ℂ y u → sim-ℂ x u
+  step-similarity-reasoning-ℂ {x = x} {y = y} p u q = transitive-sim-ℂ x y u q p
+
+  syntax step-similarity-reasoning-ℂ p u q = p ~ℂ u by q
+```
+>>>>>>> Stashed changes

From 53c47f3b41fafa3d0e40899fb0ef36e0e1f3a2c6 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 16:49:55 -0800
Subject: [PATCH 091/134] Large ring of complex numbers

---
 src/complex-numbers.lagda.md                  |  4 +-
 .../addition-complex-numbers.lagda.md         |  4 +-
 src/complex-numbers/complex-numbers.lagda.md  |  2 +
 ...additive-group-of-complex-numbers.lagda.md | 18 +++++--
 .../large-ring-of-complex-numbers.lagda.md    | 20 +++++--
 .../multiplication-complex-numbers.lagda.md   | 53 ++-----------------
 ...g-universe-levels-complex-numbers.lagda.md |  6 ++-
 .../similarity-complex-numbers.lagda.md       | 12 ++---
 8 files changed, 49 insertions(+), 70 deletions(-)

diff --git a/src/complex-numbers.lagda.md b/src/complex-numbers.lagda.md
index 70b0af5ab4d..56fb233cc1d 100644
--- a/src/complex-numbers.lagda.md
+++ b/src/complex-numbers.lagda.md
@@ -4,11 +4,13 @@
 module complex-numbers where
 
 open import complex-numbers.addition-complex-numbers public
-open import complex-numbers.additive-group-of-complex-numbers public
 open import complex-numbers.complex-numbers public
 open import complex-numbers.eisenstein-integers public
 open import complex-numbers.gaussian-integers public
+open import complex-numbers.large-additive-group-of-complex-numbers public
+open import complex-numbers.large-ring-of-complex-numbers public
 open import complex-numbers.multiplication-complex-numbers public
+open import complex-numbers.raising-universe-levels-complex-numbers public
 open import complex-numbers.ring-of-complex-numbers public
 open import complex-numbers.similarity-complex-numbers public
 ```
diff --git a/src/complex-numbers/addition-complex-numbers.lagda.md b/src/complex-numbers/addition-complex-numbers.lagda.md
index 63a33629054..904501938cf 100644
--- a/src/complex-numbers/addition-complex-numbers.lagda.md
+++ b/src/complex-numbers/addition-complex-numbers.lagda.md
@@ -99,8 +99,6 @@ abstract
   preserves-sim-add-ℂ :
     {l1 l2 l3 l4 : Level} {x : ℂ l1} {x' : ℂ l2} {y : ℂ l3} {y' : ℂ l4} →
     sim-ℂ x x' → sim-ℂ y y' → sim-ℂ (x +ℂ y) (x' +ℂ y')
-  preserves-sim-add-ℂ
-    { x = a +iℂ b} {x' = a' +iℂ b'} {y = c +iℂ d} {y' = c' +iℂ d'}
-    ( a~a' , b~b') (c~c' , d~d') =
+  preserves-sim-add-ℂ (a~a' , b~b') (c~c' , d~d') =
     ( preserves-sim-add-ℝ a~a' c~c' , preserves-sim-add-ℝ b~b' d~d')
 ```
diff --git a/src/complex-numbers/complex-numbers.lagda.md b/src/complex-numbers/complex-numbers.lagda.md
index ec8571f0de9..450fdc6bc9e 100644
--- a/src/complex-numbers/complex-numbers.lagda.md
+++ b/src/complex-numbers/complex-numbers.lagda.md
@@ -43,6 +43,8 @@ re-ℂ = pr1
 
 im-ℂ : {l : Level} → ℂ l → ℝ l
 im-ℂ = pr2
+
+pattern _+iℂ_ a b = (a , b)
 ```
 
 ## Properties
diff --git a/src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md b/src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md
index c2385a5e253..a356e7a35a9 100644
--- a/src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md
+++ b/src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md
@@ -9,18 +9,19 @@ module complex-numbers.large-additive-group-of-complex-numbers where
 ```agda
 open import complex-numbers.addition-complex-numbers
 open import complex-numbers.complex-numbers
+open import complex-numbers.raising-universe-levels-complex-numbers
 open import complex-numbers.similarity-complex-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.universe-levels
-open import complex-numbers.raising-universe-levels-complex-numbers
 
-open import group-theory.large-semigroups
-open import group-theory.large-monoids
+open import group-theory.abelian-groups
+open import group-theory.large-abelian-groups
 open import group-theory.large-commutative-monoids
 open import group-theory.large-groups
-open import group-theory.large-abelian-groups
+open import group-theory.large-monoids
+open import group-theory.large-semigroups
 ```
 
 </details>
@@ -88,3 +89,12 @@ large-ab-add-ℂ =
     ( large-group-add-ℂ)
     ( commutative-add-ℂ)
 ```
+
+## Properties
+
+### The small abelian group of complex numbers at a universe level
+
+```agda
+ab-add-ℂ : (l : Level) → Ab (lsuc l)
+ab-add-ℂ = ab-Large-Ab large-ab-add-ℂ
+```
diff --git a/src/complex-numbers/large-ring-of-complex-numbers.lagda.md b/src/complex-numbers/large-ring-of-complex-numbers.lagda.md
index 4c38a6ec892..f2d23c8f889 100644
--- a/src/complex-numbers/large-ring-of-complex-numbers.lagda.md
+++ b/src/complex-numbers/large-ring-of-complex-numbers.lagda.md
@@ -7,12 +7,16 @@ module complex-numbers.large-ring-of-complex-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import ring-theory.large-rings
+open import commutative-algebra.commutative-rings
 open import commutative-algebra.large-commutative-rings
-open import complex-numbers.large-additive-group-of-complex-numbers
+
 open import complex-numbers.complex-numbers
-open import foundation.universe-levels
+open import complex-numbers.large-additive-group-of-complex-numbers
 open import complex-numbers.multiplication-complex-numbers
+
+open import foundation.universe-levels
+
+open import ring-theory.large-rings
 ```
 
 </details>
@@ -46,3 +50,13 @@ large-commutative-ring-ℂ =
     ( large-ring-ℂ)
     ( commutative-mul-ℂ)
 ```
+
+## Properties
+
+### The small commutative ring of complex numbers at a universe level
+
+```agda
+commutative-ring-ℂ : (l : Level) → Commutative-Ring (lsuc l)
+commutative-ring-ℂ =
+  commutative-ring-Large-Commutative-Ring large-commutative-ring-ℂ
+```
diff --git a/src/complex-numbers/multiplication-complex-numbers.lagda.md b/src/complex-numbers/multiplication-complex-numbers.lagda.md
index 098ba6f3280..0617bb9d9bf 100644
--- a/src/complex-numbers/multiplication-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplication-complex-numbers.lagda.md
@@ -255,50 +255,6 @@ opaque
         ＝ zero-ℝ
           by left-unit-law-add-ℝ zero-ℝ)
 ```
-<<<<<<< Updated upstream
-=======
-
-### The canonical embedding of real numbers in the complex numbers preserves multiplication
-
-```agda
-abstract
-  mul-complex-ℝ :
-    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) →
-    complex-ℝ x *ℂ complex-ℝ y ＝ complex-ℝ (x *ℝ y)
-  mul-complex-ℝ {l1} {l2} x y =
-    eq-ℂ
-      ( equational-reasoning
-        x *ℝ y -ℝ raise-ℝ l1 zero-ℝ *ℝ raise-ℝ l2 zero-ℝ
-        ＝ x *ℝ y -ℝ zero-ℝ *ℝ zero-ℝ
-          by
-            eq-sim-ℝ
-              ( preserves-sim-diff-ℝ
-                ( refl-sim-ℝ (x *ℝ y))
-                ( symmetric-sim-ℝ
-                  ( preserves-sim-mul-ℝ (sim-raise-ℝ _ _) (sim-raise-ℝ _ _))))
-        ＝ x *ℝ y -ℝ zero-ℝ
-          by ap-diff-ℝ refl (eq-sim-ℝ (right-zero-law-mul-ℝ _))
-        ＝ x *ℝ y
-          by right-unit-law-diff-ℝ (x *ℝ y))
-      ( eq-sim-ℝ
-        ( similarity-reasoning-ℝ
-          x *ℝ raise-ℝ l2 zero-ℝ +ℝ raise-ℝ l1 zero-ℝ *ℝ y
-          ~ℝ x *ℝ zero-ℝ +ℝ zero-ℝ *ℝ y
-            by
-              symmetric-sim-ℝ
-                ( preserves-sim-add-ℝ
-                  ( preserves-sim-left-mul-ℝ _ _ _ (sim-raise-ℝ l2 zero-ℝ))
-                  ( preserves-sim-right-mul-ℝ _ _ _ (sim-raise-ℝ l1 zero-ℝ)))
-          ~ℝ zero-ℝ +ℝ zero-ℝ
-            by
-              preserves-sim-add-ℝ
-                ( right-zero-law-mul-ℝ x)
-                ( left-zero-law-mul-ℝ y)
-          ~ℝ zero-ℝ
-            by sim-eq-ℝ (left-unit-law-add-ℝ zero-ℝ)
-          ~ℝ raise-ℝ (l1 ⊔ l2) zero-ℝ
-            by sim-raise-ℝ (l1 ⊔ l2) zero-ℝ))
-```
 
 ### Similarity is preserved by multiplication
 
@@ -309,10 +265,9 @@ abstract
     sim-ℂ x x' → sim-ℂ y y' → sim-ℂ (x *ℂ y) (x' *ℂ y')
   preserves-sim-mul-ℂ (a~a' , b~b') (c~c' , d~d') =
     ( preserves-sim-diff-ℝ
-        ( preserves-sim-mul-ℝ a~a' c~c')
-        ( preserves-sim-mul-ℝ b~b' d~d') ,
+        ( preserves-sim-mul-ℝ _ _ a~a' _ _ c~c')
+        ( preserves-sim-mul-ℝ _ _ b~b' _ _ d~d') ,
       preserves-sim-add-ℝ
-        ( preserves-sim-mul-ℝ a~a' d~d')
-        ( preserves-sim-mul-ℝ b~b' c~c'))
+        ( preserves-sim-mul-ℝ _ _ a~a' _ _ d~d')
+        ( preserves-sim-mul-ℝ _ _ b~b' _ _ c~c'))
 ```
->>>>>>> Stashed changes
diff --git a/src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md b/src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md
index 2ad31cb0190..7f1a217b26a 100644
--- a/src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md
+++ b/src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md
@@ -8,10 +8,12 @@ module complex-numbers.raising-universe-levels-complex-numbers where
 
 ```agda
 open import complex-numbers.complex-numbers
-open import real-numbers.raising-universe-levels-real-numbers
-open import foundation.dependent-pair-types
 open import complex-numbers.similarity-complex-numbers
+
+open import foundation.dependent-pair-types
 open import foundation.universe-levels
+
+open import real-numbers.raising-universe-levels-real-numbers
 ```
 
 </details>
diff --git a/src/complex-numbers/similarity-complex-numbers.lagda.md b/src/complex-numbers/similarity-complex-numbers.lagda.md
index 84294023d61..868b61570ff 100644
--- a/src/complex-numbers/similarity-complex-numbers.lagda.md
+++ b/src/complex-numbers/similarity-complex-numbers.lagda.md
@@ -11,19 +11,16 @@ open import complex-numbers.complex-numbers
 
 open import foundation.conjunction
 open import foundation.dependent-pair-types
+open import foundation.identity-types
 open import foundation.large-equivalence-relations
 open import foundation.large-similarity-relations
-open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
 
-<<<<<<< Updated upstream
-=======
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.negation-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
->>>>>>> Stashed changes
 open import real-numbers.similarity-real-numbers
 ```
 
@@ -83,8 +80,6 @@ abstract
   eq-sim-ℂ : {l : Level} {x y : ℂ l} → sim-ℂ x y → x ＝ y
   eq-sim-ℂ (a~c , b~d) = eq-ℂ (eq-sim-ℝ a~c) (eq-sim-ℝ b~d)
 ```
-<<<<<<< Updated upstream
-=======
 
 ### Similarity is a large equivalence relation
 
@@ -163,4 +158,5 @@ abstract
 
   syntax step-similarity-reasoning-ℂ p u q = p ~ℂ u by q
 ```
->>>>>>> Stashed changes
+
+> > > > > > > Stashed changes

From 6b6689aa074ed678251d3afa6f76352d64e77326 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 16:55:07 -0800
Subject: [PATCH 092/134] Correct merge

---
 src/real-numbers/apartness-real-numbers.lagda.md | 7 +------
 1 file changed, 1 insertion(+), 6 deletions(-)

diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 5e22dd53385..47a55ed6a55 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -7,11 +7,10 @@ module real-numbers.apartness-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.apartness-relations
 open import foundation.binary-relations
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
-open import foundation.apartness-relations
-open import foundation.binary-transport
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
@@ -39,10 +38,6 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.absolute-value-real-numbers
-open import real-numbers.addition-real-numbers
-open import real-numbers.dedekind-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers

From 99a530bedf8292c702dbb2033d48c47e6e73ea48 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 16:57:50 -0800
Subject: [PATCH 093/134] Fix

---
 .../multiplication-complex-numbers.lagda.md               | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/complex-numbers/multiplication-complex-numbers.lagda.md b/src/complex-numbers/multiplication-complex-numbers.lagda.md
index 0617bb9d9bf..fcef6b3b346 100644
--- a/src/complex-numbers/multiplication-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplication-complex-numbers.lagda.md
@@ -265,9 +265,9 @@ abstract
     sim-ℂ x x' → sim-ℂ y y' → sim-ℂ (x *ℂ y) (x' *ℂ y')
   preserves-sim-mul-ℂ (a~a' , b~b') (c~c' , d~d') =
     ( preserves-sim-diff-ℝ
-        ( preserves-sim-mul-ℝ _ _ a~a' _ _ c~c')
-        ( preserves-sim-mul-ℝ _ _ b~b' _ _ d~d') ,
+        ( preserves-sim-mul-ℝ a~a' c~c')
+        ( preserves-sim-mul-ℝ b~b' d~d') ,
       preserves-sim-add-ℝ
-        ( preserves-sim-mul-ℝ _ _ a~a' _ _ d~d')
-        ( preserves-sim-mul-ℝ _ _ b~b' _ _ c~c'))
+        ( preserves-sim-mul-ℝ a~a' d~d')
+        ( preserves-sim-mul-ℝ b~b' c~c'))
 ```

From 1d02a41575c0da6681c41ab3c7934e6dfe7bd267 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:02:04 -0800
Subject: [PATCH 094/134] Progress

---
 .../real-vector-spaces.lagda.md               | 22 +++++++++++++++++++
 1 file changed, 22 insertions(+)
 create mode 100644 src/linear-algebra/real-vector-spaces.lagda.md

diff --git a/src/linear-algebra/real-vector-spaces.lagda.md b/src/linear-algebra/real-vector-spaces.lagda.md
new file mode 100644
index 00000000000..ab72ab74bd5
--- /dev/null
+++ b/src/linear-algebra/real-vector-spaces.lagda.md
@@ -0,0 +1,22 @@
+# Real vector spaces
+
+```agda
+module linear-algebra.real-vector-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import real-numbers.local-ring-of-real-numbers
+open import linear-algebra.vector-spaces
+open import foundation.universe-levels
+```
+
+</details>
+
+## Idea
+
+```agda
+ℝ-Vector-Space : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
+ℝ-Vector-Space l1 l2 = Vector-Space l2 (local-commutative-ring-ℝ l1)
+```

From b9dc6daacc24df48827683d387cc5d8674dccb5c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:03:37 -0800
Subject: [PATCH 095/134] Delete small ring of complex numbers from separate
 file

---
 src/complex-numbers.lagda.md                  |  1 -
 .../ring-of-complex-numbers.lagda.md          | 47 -------------------
 2 files changed, 48 deletions(-)
 delete mode 100644 src/complex-numbers/ring-of-complex-numbers.lagda.md

diff --git a/src/complex-numbers.lagda.md b/src/complex-numbers.lagda.md
index 56fb233cc1d..647fd88a7dd 100644
--- a/src/complex-numbers.lagda.md
+++ b/src/complex-numbers.lagda.md
@@ -11,6 +11,5 @@ open import complex-numbers.large-additive-group-of-complex-numbers public
 open import complex-numbers.large-ring-of-complex-numbers public
 open import complex-numbers.multiplication-complex-numbers public
 open import complex-numbers.raising-universe-levels-complex-numbers public
-open import complex-numbers.ring-of-complex-numbers public
 open import complex-numbers.similarity-complex-numbers public
 ```
diff --git a/src/complex-numbers/ring-of-complex-numbers.lagda.md b/src/complex-numbers/ring-of-complex-numbers.lagda.md
deleted file mode 100644
index 6b319d6c776..00000000000
--- a/src/complex-numbers/ring-of-complex-numbers.lagda.md
+++ /dev/null
@@ -1,47 +0,0 @@
-# The ring of complex numbers
-
-```agda
-module complex-numbers.ring-of-complex-numbers where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import commutative-algebra.commutative-rings
-
-open import complex-numbers.additive-group-of-complex-numbers
-open import complex-numbers.complex-numbers
-open import complex-numbers.multiplication-complex-numbers
-
-open import foundation.dependent-pair-types
-open import foundation.function-types
-open import foundation.universe-levels
-
-open import ring-theory.rings
-```
-
-</details>
-
-## Idea
-
-The type of [complex numbers](complex-numbers.complex-numbers.md) equipped with
-[multiplication](complex-numbers.multiplication-complex-numbers.md) and
-[addition](complex-numbers.addition-complex-numbers.md) is a
-[commutative ring](commutative-algebra.commutative-rings.md).
-
-## Definition
-
-```agda
-ring-ℂ-lzero : Ring (lsuc lzero)
-ring-ℂ-lzero =
-  ( ab-add-ℂ-lzero ,
-    ( mul-ℂ , associative-mul-ℂ) ,
-    ( one-ℂ , left-unit-law-mul-ℂ , right-unit-law-mul-ℂ) ,
-    left-distributive-mul-add-ℂ ,
-    right-distributive-mul-add-ℂ)
-
-commutative-ring-ℂ-lzero : Commutative-Ring (lsuc lzero)
-commutative-ring-ℂ-lzero =
-  ( ring-ℂ-lzero ,
-    commutative-mul-ℂ)
-```

From cac562e7e3e3bad1474e90e09faa57993cea33e1 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:06:38 -0800
Subject: [PATCH 096/134] Fix merge

---
 .../nonzero-real-numbers.lagda.md             | 30 +------------------
 1 file changed, 1 insertion(+), 29 deletions(-)

diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index 63f9fa0272a..6c785656b35 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -29,7 +29,7 @@ open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -97,34 +97,6 @@ eq-nonzero-ℝ :
 eq-nonzero-ℝ _ _ = eq-type-subtype is-nonzero-prop-ℝ
 ```
 
-### Two real numbers are apart if and only if their difference is nonzero
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
-  where
-
-  abstract
-    is-nonzero-diff-is-apart-ℝ : apart-ℝ x y → is-nonzero-ℝ (x -ℝ y)
-    is-nonzero-diff-is-apart-ℝ x#y =
-      apart-right-sim-ℝ
-        ( x -ℝ y)
-        ( y -ℝ y)
-        ( zero-ℝ)
-        ( right-inverse-law-add-ℝ y)
-        ( preserves-apart-right-add-ℝ (neg-ℝ y) x y x#y)
-
-    is-apart-is-nonzero-diff-ℝ : is-nonzero-ℝ (x -ℝ y) → apart-ℝ x y
-    is-apart-is-nonzero-diff-ℝ x-y#0 =
-      apart-sim-ℝ
-        ( cancel-right-diff-add-ℝ x y)
-        ( sim-eq-ℝ (left-unit-law-add-ℝ y))
-        ( preserves-apart-right-add-ℝ y _ _ x-y#0)
-
-  nonzero-diff-apart-ℝ : apart-ℝ x y → nonzero-ℝ (l1 ⊔ l2)
-  nonzero-diff-apart-ℝ x#y = (x -ℝ y , is-nonzero-diff-is-apart-ℝ x#y)
-```
-
 ### The nonzero difference of a pair of real numbers `x` and `y` such that `x < y`
 
 ```agda

From ed80b13e86774c930b14956c05f70ebb7bd21662 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:09:28 -0800
Subject: [PATCH 097/134] Fix table

---
 docs/tables/rings.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/docs/tables/rings.md b/docs/tables/rings.md
index f9721cae161..4de30be6964 100644
--- a/docs/tables/rings.md
+++ b/docs/tables/rings.md
@@ -1,6 +1,6 @@
 | Ring                                  | File                                                                                                        |
 | ------------------------------------- | ----------------------------------------------------------------------------------------------------------- |
-| Complex numbers                       | [`complex-numbers.ring-of-complex-numbers`](complex-numbers.ring-of-complex-numbers.md)                     |
+| Complex numbers                       | [`complex-numbers.large-ring-of-complex-numbers`](complex-numbers.large-ring-of-complex-numbers.md)         |
 | Eisenstein integers                   | [`complex-numbers.eisenstein-integers`](complex-numbers.eisenstein-integers.md)                             |
 | Endomorphism ring of an abelian group | [`group-theory.endomorphism-rings-abelian-groups`](group-theory.endomorphism-rings-abelian-groups.md)       |
 | Gaussian integers                     | [`complex-numbers.gaussian-integers`](complex-numbers.gaussian-integers.md)                                 |

From 15cab2640bc639e1160c20d25f16c4fe0c9456b5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:12:51 -0800
Subject: [PATCH 098/134] make pre-commit

---
 ...ive-and-negative-rational-numbers.lagda.md |  2 +-
 .../apartness-real-numbers.lagda.md           |  7 +-----
 ...ositive-and-negative-real-numbers.lagda.md | 22 ++++++++++---------
 3 files changed, 14 insertions(+), 17 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index fd865252b7f..1c3ee2661d6 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -18,10 +18,10 @@ open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 
 open import foundation.cartesian-product-types
-open import foundation.functoriality-coproduct-types
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
+open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
 ```
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index b7a7e0ec2db..5529df3497e 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -7,11 +7,10 @@ module real-numbers.apartness-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.apartness-relations
 open import foundation.binary-relations
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
-open import foundation.apartness-relations
-open import foundation.binary-transport
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
@@ -39,10 +38,6 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.absolute-value-real-numbers
-open import real-numbers.addition-real-numbers
-open import real-numbers.dedekind-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 12ada068a8c..0e27864d68e 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -7,20 +7,22 @@ module real-numbers.multiplication-positive-and-negative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.dependent-pair-types
-open import foundation.transport-along-identifications
-open import foundation.disjunction
-open import foundation.coproduct-types
-open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.minimum-rational-numbers
-open import foundation.conjunction
-open import order-theory.posets
 open import elementary-number-theory.inequality-rational-numbers
-open import foundation.propositional-truncations
+open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.conjunction
+open import foundation.coproduct-types
+open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.propositional-truncations
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
-open import elementary-number-theory.positive-rational-numbers
+
+open import order-theory.posets
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.multiplication-positive-real-numbers

From 46e57a4bb902a69133befa02d26621fdd3080aee Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:33:27 -0800
Subject: [PATCH 099/134] Progress

---
 ...closed-intervals-rational-numbers.lagda.md | 79 +++++++++++++++++++
 ...ositive-and-negative-real-numbers.lagda.md | 73 +++++++++++------
 2 files changed, 126 insertions(+), 26 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
index 698a8adb827..7b985554ddd 100644
--- a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
@@ -34,12 +34,14 @@ open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
 open import elementary-number-theory.negation-closed-intervals-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.poset-closed-intervals-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.proper-closed-intervals-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.squares-rational-numbers
+open import foundation.functoriality-coproduct-types
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
@@ -54,6 +56,7 @@ open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.images-subtypes
+open import foundation.coproduct-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
@@ -1287,3 +1290,79 @@ abstract
               ( neg-ℚ⁰⁺ (nonnegative-square-ℚ q))
               ( nonnegative-square-ℚ q))
 ```
+
+### If the lower bound of the product of `[a, b]` and `[c, d]` is positive, then `a`, `b`, `c`, and `d` are all positive or all negative
+
+```agda
+abstract
+  same-sign-bounds-is-positive-lower-bound-mul-closed-interval-ℚ :
+    ([a,b] [c,d] : closed-interval-ℚ) →
+    is-positive-ℚ (lower-bound-mul-closed-interval-ℚ [a,b] [c,d]) →
+    ( ( is-negative-ℚ (lower-bound-closed-interval-ℚ [a,b]) ×
+        is-negative-ℚ (upper-bound-closed-interval-ℚ [a,b]) ×
+        is-negative-ℚ (lower-bound-closed-interval-ℚ [c,d]) ×
+        is-negative-ℚ (upper-bound-closed-interval-ℚ [c,d])) +
+      ( is-positive-ℚ (lower-bound-closed-interval-ℚ [a,b]) ×
+        is-positive-ℚ (upper-bound-closed-interval-ℚ [a,b]) ×
+        is-positive-ℚ (lower-bound-closed-interval-ℚ [c,d]) ×
+        is-positive-ℚ (upper-bound-closed-interval-ℚ [c,d])))
+  same-sign-bounds-is-positive-lower-bound-mul-closed-interval-ℚ
+    [a,b]@((a , b) , _) [c,d]@((c , d) , _) is-pos-lb =
+    let
+      same-sign-a-c =
+        same-sign-is-positive-mul-ℚ
+          ( is-positive-leq-ℚ⁺
+            ( _ , is-pos-lb)
+            ( transitive-leq-ℚ _ _ _ (leq-left-min-ℚ _ _) (leq-left-min-ℚ _ _)))
+      same-sign-a-d =
+        same-sign-is-positive-mul-ℚ
+          ( is-positive-leq-ℚ⁺
+            ( _ , is-pos-lb)
+            ( transitive-leq-ℚ _ _ _
+              ( leq-right-min-ℚ _ _)
+              ( leq-left-min-ℚ _ _)))
+      same-sign-b-d =
+        same-sign-is-positive-mul-ℚ
+          ( is-positive-leq-ℚ⁺
+            ( _ , is-pos-lb)
+            ( transitive-leq-ℚ _ _ _
+              ( leq-right-min-ℚ _ _)
+              ( leq-right-min-ℚ _ _)))
+    in
+      map-coproduct
+        ( λ (is-neg-a , is-neg-c) →
+          let
+            is-neg-d =
+              rec-coproduct
+                ( pr2)
+                ( λ (is-pos-a , _) →
+                  ex-falso
+                    ( is-not-negative-and-positive-ℚ (is-neg-a , is-pos-a)))
+                ( same-sign-a-d)
+            is-neg-b =
+              rec-coproduct
+                ( pr1)
+                ( λ (_ , is-pos-d) →
+                  ex-falso
+                    ( is-not-negative-and-positive-ℚ (is-neg-d , is-pos-d)))
+                ( same-sign-b-d)
+          in (is-neg-a , is-neg-b , is-neg-c , is-neg-d))
+        ( λ (is-pos-a , is-pos-c) →
+          let
+            is-pos-d =
+              rec-coproduct
+                ( λ (is-neg-a , _) →
+                  ex-falso
+                    ( is-not-negative-and-positive-ℚ (is-neg-a , is-pos-a)))
+                ( pr2)
+                ( same-sign-a-d)
+            is-pos-b =
+              rec-coproduct
+                ( λ (_ , is-neg-d) →
+                  ex-falso
+                    ( is-not-negative-and-positive-ℚ (is-neg-d , is-pos-d)))
+                ( pr1)
+                ( same-sign-b-d)
+          in (is-pos-a , is-pos-b , is-pos-c , is-pos-d))
+        ( same-sign-a-c)
+```
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 0e27864d68e..f1b15142b40 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -1,6 +1,8 @@
 # Multiplication by positive, negative, and nonnegative real numbers
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module real-numbers.multiplication-positive-and-negative-real-numbers where
 ```
 
@@ -18,9 +20,11 @@ open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
+open import foundation.functoriality-disjunction
 open import foundation.propositional-truncations
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
+open import foundation.existential-quantification
 
 open import order-theory.posets
 
@@ -29,6 +33,7 @@ open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.positive-real-numbers
+open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -108,31 +113,47 @@ abstract opaque
       ( ( ax<x<bx@([ax,bx]@((ax , bx) , _) , ax<x , x<bx) ,
           ay<y<by@([ay,by]@((ay , by) , _) , ay<y , y<by)) ,
           q<[ax,bx][ay,by]) ← q<xy
-      let
-        is-positive-axay =
-          is-positive-leq-ℚ⁺
-            ( q⁺)
-            ( chain-of-inequalities
-              q
-              ≤ lower-bound-mul-closed-interval-ℚ [ax,bx] [ay,by]
-                by leq-le-ℚ q<[ax,bx][ay,by]
-              ≤ min-ℚ (ax *ℚ ay) (ax *ℚ by)
-                by leq-left-min-ℚ _ _
-              ≤ ax *ℚ ay
-                by leq-left-min-ℚ _ _)
-        is-positive-bxby =
-          is-positive-leq-ℚ⁺
-            ( q⁺)
-            ( chain-of-inequalities
-              q
-              ≤ lower-bound-mul-closed-interval-ℚ [ax,bx] [ay,by]
-                by leq-le-ℚ q<[ax,bx][ay,by]
-              ≤ min-ℚ (bx *ℚ ay) (bx *ℚ by)
-                by leq-right-min-ℚ _ _
-              ≤ bx *ℚ by
-                by leq-right-min-ℚ _ _)
       rec-coproduct
-        {!   !}
-        {!   !}
-        {!  same-sign !}
+        ( λ (_ , is-neg-bx , _ , is-neg-by) →
+          inl-disjunction
+            ( is-negative-exists-ℚ⁻-in-upper-cut-ℝ
+                ( x)
+                ( intro-exists (bx , is-neg-bx) x<bx) ,
+              is-negative-exists-ℚ⁻-in-upper-cut-ℝ
+                ( y)
+                ( intro-exists (by , is-neg-by) y<by)))
+        ( λ (is-pos-ax , _ , is-pos-ay , _) →
+          inr-disjunction
+            ( is-positive-exists-ℚ⁺-in-lower-cut-ℝ
+                ( x)
+                ( intro-exists (ax , is-pos-ax) ax<x) ,
+              is-positive-exists-ℚ⁺-in-lower-cut-ℝ
+                ( y)
+                ( intro-exists (ay , is-pos-ay) ay<y)))
+        ( same-sign-bounds-is-positive-lower-bound-mul-closed-interval-ℚ
+          ( [ax,bx])
+          ( [ay,by])
+          ( is-positive-le-ℚ⁺ q⁺ q<[ax,bx][ay,by]))
+```
+
+### If the product of two real numbers is negative, one is positive and the other negative
+
+```agda
+abstract
+  different-signs-is-negative-mul-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → is-negative-ℝ (x *ℝ y) →
+    type-disjunction-Prop
+      ( is-negative-prop-ℝ x ∧ is-positive-prop-ℝ y)
+      ( is-positive-prop-ℝ x ∧ is-negative-prop-ℝ y)
+  different-signs-is-negative-mul-ℝ x y xy<0 =
+    map-disjunction
+      {!   !}
+      {!   !}
+      ( same-sign-is-positive-mul-ℝ
+        ( neg-ℝ x)
+        ( y)
+        ( inv-tr
+          ( is-positive-ℝ)
+          ( left-negative-law-mul-ℝ x y)
+          {!   !}))
 ```

From d46c059634009e810bec2cc3823f2b44196910f2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:33:58 -0800
Subject: [PATCH 100/134] Progress

---
 ...ositive-and-negative-real-numbers.lagda.md | 33 +++++++++++++++++++
 1 file changed, 33 insertions(+)

diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index 18f7f33b894..b30852d3524 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -11,6 +11,7 @@ module real-numbers.positive-and-negative-real-numbers where
 ```agda
 open import foundation.dependent-pair-types
 open import foundation.transport-along-identifications
+open import foundation.logical-equivalences
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
@@ -78,6 +79,38 @@ neg-ℝ⁻ : {l : Level} → ℝ⁻ l → ℝ⁺ l
 neg-ℝ⁻ (x , is-neg-x) = (neg-ℝ x , neg-is-negative-ℝ x is-neg-x)
 ```
 
+### A real number is negative if and only if its negation is positive
+
+```agda
+abstract
+  is-negative-is-positive-neg-ℝ :
+    {l : Level} (x : ℝ l) → is-positive-ℝ (neg-ℝ x) → is-negative-ℝ x
+  is-negative-is-positive-neg-ℝ x 0<-x =
+    tr is-negative-ℝ (neg-neg-ℝ x) (neg-is-positive-ℝ (neg-ℝ x) 0<-x)
+
+is-negative-iff-neg-is-positive-ℝ :
+  {l : Level} (x : ℝ l) → is-negative-ℝ x ↔ is-positive-ℝ (neg-ℝ x)
+is-negative-iff-neg-is-positive-ℝ x =
+  ( neg-is-negative-ℝ x ,
+    is-negative-is-positive-neg-ℝ x)
+```
+
+### A real number is positive if and only if its negation is negative
+
+```agda
+abstract
+  is-positive-is-negative-neg-ℝ :
+    {l : Level} (x : ℝ l) → is-negative-ℝ (neg-ℝ x) → is-positive-ℝ x
+  is-positive-is-negative-neg-ℝ x -x<0 =
+    tr is-positive-ℝ (neg-neg-ℝ x) (neg-is-negative-ℝ (neg-ℝ x) -x<0)
+
+is-positive-iff-neg-is-negative-ℝ :
+  {l : Level} (x : ℝ l) → is-positive-ℝ x ↔ is-negative-ℝ (neg-ℝ x)
+is-positive-iff-neg-is-negative-ℝ x =
+  ( neg-is-positive-ℝ x ,
+    is-positive-is-negative-neg-ℝ x)
+```
+
 ### If a nonnegative real number `x` is less than a real number `y`, `y` is positive
 
 ```agda

From 453322e74496db065c214455fd2afb7308f1ba68 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 17:34:18 -0800
Subject: [PATCH 101/134] Revert accident

---
 ...ositive-and-negative-real-numbers.lagda.md | 33 -------------------
 1 file changed, 33 deletions(-)

diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index b30852d3524..18f7f33b894 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -11,7 +11,6 @@ module real-numbers.positive-and-negative-real-numbers where
 ```agda
 open import foundation.dependent-pair-types
 open import foundation.transport-along-identifications
-open import foundation.logical-equivalences
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
@@ -79,38 +78,6 @@ neg-ℝ⁻ : {l : Level} → ℝ⁻ l → ℝ⁺ l
 neg-ℝ⁻ (x , is-neg-x) = (neg-ℝ x , neg-is-negative-ℝ x is-neg-x)
 ```
 
-### A real number is negative if and only if its negation is positive
-
-```agda
-abstract
-  is-negative-is-positive-neg-ℝ :
-    {l : Level} (x : ℝ l) → is-positive-ℝ (neg-ℝ x) → is-negative-ℝ x
-  is-negative-is-positive-neg-ℝ x 0<-x =
-    tr is-negative-ℝ (neg-neg-ℝ x) (neg-is-positive-ℝ (neg-ℝ x) 0<-x)
-
-is-negative-iff-neg-is-positive-ℝ :
-  {l : Level} (x : ℝ l) → is-negative-ℝ x ↔ is-positive-ℝ (neg-ℝ x)
-is-negative-iff-neg-is-positive-ℝ x =
-  ( neg-is-negative-ℝ x ,
-    is-negative-is-positive-neg-ℝ x)
-```
-
-### A real number is positive if and only if its negation is negative
-
-```agda
-abstract
-  is-positive-is-negative-neg-ℝ :
-    {l : Level} (x : ℝ l) → is-negative-ℝ (neg-ℝ x) → is-positive-ℝ x
-  is-positive-is-negative-neg-ℝ x -x<0 =
-    tr is-positive-ℝ (neg-neg-ℝ x) (neg-is-negative-ℝ (neg-ℝ x) -x<0)
-
-is-positive-iff-neg-is-negative-ℝ :
-  {l : Level} (x : ℝ l) → is-positive-ℝ x ↔ is-negative-ℝ (neg-ℝ x)
-is-positive-iff-neg-is-negative-ℝ x =
-  ( neg-is-positive-ℝ x ,
-    is-positive-is-negative-neg-ℝ x)
-```
-
 ### If a nonnegative real number `x` is less than a real number `y`, `y` is positive
 
 ```agda

From b8e2f5f07fa0c382b37c7b41df67bfc60bcc1f43 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 18:36:54 -0800
Subject: [PATCH 102/134] Progress

---
 src/complex-numbers/complex-numbers.lagda.md  |  2 +
 .../magnitude-complex-numbers.lagda.md        | 68 +++++++++++++++++++
 ...-inverses-nonzero-complex-numbers.lagda.md | 68 +++++++++++++++++++
 .../nonzero-complex-numbers.lagda.md          | 22 ++++++
 ...ltiplication-nonzero-real-numbers.lagda.md | 38 ++++++++++-
 ...ositive-and-negative-real-numbers.lagda.md | 11 +--
 ...ive-inverses-nonzero-real-numbers.lagda.md | 62 ++++-------------
 .../negative-real-numbers.lagda.md            | 11 +++
 .../nonzero-real-numbers.lagda.md             | 42 ++++--------
 ...ositive-and-negative-real-numbers.lagda.md | 12 ++++
 .../positive-real-numbers.lagda.md            |  9 ++-
 ...re-roots-nonnegative-real-numbers.lagda.md | 30 ++++++--
 .../squares-real-numbers.lagda.md             | 19 ++++++
 13 files changed, 301 insertions(+), 93 deletions(-)

diff --git a/src/complex-numbers/complex-numbers.lagda.md b/src/complex-numbers/complex-numbers.lagda.md
index 7465731aca5..0b7cc5fcf8a 100644
--- a/src/complex-numbers/complex-numbers.lagda.md
+++ b/src/complex-numbers/complex-numbers.lagda.md
@@ -45,6 +45,8 @@ re-ℂ = pr1
 
 im-ℂ : {l : Level} → ℂ l → ℝ l
 im-ℂ = pr2
+
+pattern _+iℂ_ a b = (a , b)
 ```
 
 ## Properties
diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index 22a709c6efb..4a81647b463 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -12,14 +12,18 @@ module complex-numbers.magnitude-complex-numbers where
 open import complex-numbers.complex-numbers
 open import complex-numbers.conjugation-complex-numbers
 open import complex-numbers.multiplication-complex-numbers
+open import complex-numbers.similarity-complex-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.identity-types
+open import foundation.dependent-pair-types
 open import foundation.universe-levels
 
+open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.positive-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
@@ -231,3 +235,67 @@ abstract
       ＝ magnitude-ℂ z *ℝ magnitude-ℂ w
         by ap real-ℝ⁰⁺ (distributive-sqrt-mul-ℝ⁰⁺ _ _)
 ```
+
+### The magnitude of a real number as a complex number is its absolute value
+
+```agda
+abstract
+  magnitude-complex-ℝ :
+    {l : Level} (x : ℝ l) → magnitude-ℂ (complex-ℝ x) ＝ abs-ℝ x
+  magnitude-complex-ℝ {l} x =
+    equational-reasoning
+      real-sqrt-ℝ⁰⁺
+        ( nonnegative-square-ℝ x +ℝ⁰⁺ nonnegative-square-ℝ (raise-ℝ l zero-ℝ))
+      ＝
+        real-sqrt-ℝ⁰⁺
+          ( nonnegative-square-ℝ x +ℝ⁰⁺ nonnegative-square-ℝ zero-ℝ)
+        by
+          ap
+            ( real-sqrt-ℝ⁰⁺)
+            ( eq-ℝ⁰⁺ _ _
+              ( eq-sim-ℝ
+                ( preserves-sim-left-add-ℝ _ _ _
+                  ( preserves-sim-square-ℝ
+                    ( symmetric-sim-ℝ (sim-raise-ℝ l zero-ℝ))))))
+      ＝ real-sqrt-ℝ⁰⁺ (nonnegative-square-ℝ x +ℝ⁰⁺ zero-ℝ⁰⁺)
+        by ap real-sqrt-ℝ⁰⁺ (eq-ℝ⁰⁺ _ _ (ap-add-ℝ refl square-zero-ℝ))
+      ＝ real-sqrt-ℝ⁰⁺ (nonnegative-square-ℝ x)
+        by ap real-sqrt-ℝ⁰⁺ (eq-ℝ⁰⁺ _ _ (right-unit-law-add-ℝ _))
+      ＝ abs-ℝ x
+        by inv (eq-abs-sqrt-square-ℝ x)
+```
+
+### Similar complex numbers have similar magnitudes
+
+```agda
+abstract
+  preserves-sim-squared-magnitude-ℂ :
+    {l1 l2 : Level} {z : ℂ l1} {w : ℂ l2} →
+    sim-ℂ z w → sim-ℝ (squared-magnitude-ℂ z) (squared-magnitude-ℂ w)
+  preserves-sim-squared-magnitude-ℂ (a~c , b~d) =
+    preserves-sim-add-ℝ
+      ( preserves-sim-square-ℝ a~c)
+      ( preserves-sim-square-ℝ b~d)
+
+  preserves-sim-magnitude-ℂ :
+    {l1 l2 : Level} {z : ℂ l1} {w : ℂ l2} →
+    sim-ℂ z w → sim-ℝ (magnitude-ℂ z) (magnitude-ℂ w)
+  preserves-sim-magnitude-ℂ z~w =
+    preserves-sim-sqrt-ℝ⁰⁺ _ _ (preserves-sim-squared-magnitude-ℂ z~w)
+```
+
+### The magnitude of `one-ℂ` is `one-ℝ`
+
+```agda
+abstract
+  magnitude-one-ℂ : magnitude-ℂ one-ℂ ＝ one-ℝ
+  magnitude-one-ℂ =
+    equational-reasoning
+      magnitude-ℂ one-ℂ
+      ＝ magnitude-ℂ (complex-ℝ one-ℝ)
+        by ap magnitude-ℂ (inv eq-complex-one-ℝ)
+      ＝ abs-ℝ one-ℝ
+        by magnitude-complex-ℝ one-ℝ
+      ＝ one-ℝ
+        by abs-real-ℝ⁺ one-ℝ⁺
+```
diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 71dd292ad7f..7278fe13552 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -14,9 +14,18 @@ open import complex-numbers.multiplication-complex-numbers
 open import complex-numbers.nonzero-complex-numbers
 open import complex-numbers.similarity-complex-numbers
 
+open import real-numbers.nonzero-real-numbers
+open import real-numbers.nonnegative-real-numbers
 open import foundation.dependent-pair-types
+open import real-numbers.positive-and-negative-real-numbers
+open import foundation.function-types
+open import foundation.disjunction
+open import real-numbers.similarity-real-numbers
+open import real-numbers.positive-real-numbers
+open import foundation.empty-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
+open import real-numbers.multiplication-nonzero-real-numbers
 open import foundation.universe-levels
 
 open import real-numbers.multiplication-real-numbers
@@ -83,3 +92,62 @@ abstract
   right-inverse-law-mul-ℂ z =
     tr (λ w → sim-ℂ w one-ℂ) (commutative-mul-ℂ _ _) (left-inverse-law-mul-ℂ z)
 ```
+
+### If a complex number has a multiplicative inverse, it is nonzero
+
+```agda
+abstract
+  is-nonzero-has-right-inverse-mul-ℂ :
+    {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2) → sim-ℂ (z *ℂ w) one-ℂ →
+    is-nonzero-ℂ z
+  is-nonzero-has-right-inverse-mul-ℂ z w zw=1 =
+    is-nonzero-is-positive-magnitude-ℂ
+      ( z)
+      ( elim-disjunction
+        ( is-positive-prop-ℝ ∥ z ∥ℂ)
+        ( λ |z|<0 →
+          ex-falso
+            ( is-not-negative-and-nonnegative-ℝ
+              ( |z|<0 , is-nonnegative-real-ℝ⁰⁺ (nonnegative-magnitude-ℂ z))))
+        ( id)
+        ( pr1
+          ( is-nonzero-factors-is-nonzero-mul-ℝ
+            ( ∥ z ∥ℂ)
+            ( ∥ w ∥ℂ)
+            ( is-nonzero-sim-ℝ
+              ( symmetric-sim-ℝ
+                ( similarity-reasoning-ℝ
+                  ∥ z ∥ℂ *ℝ ∥ w ∥ℂ
+                  ~ℝ ∥ z *ℂ w ∥ℂ
+                    by sim-eq-ℝ (inv (distributive-magnitude-mul-ℂ z w))
+                  ~ℝ ∥ one-ℂ ∥ℂ
+                    by preserves-sim-magnitude-ℂ zw=1
+                  ~ℝ one-ℝ
+                    by sim-eq-ℝ magnitude-one-ℂ))
+              ( is-nonzero-is-positive-ℝ is-positive-one-ℝ)))))
+
+  is-nonzero-has-left-inverse-mul-ℂ :
+    {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2) → sim-ℂ (z *ℂ w) one-ℂ →
+    is-nonzero-ℂ w
+  is-nonzero-has-left-inverse-mul-ℂ z w zw=1 =
+    is-nonzero-has-right-inverse-mul-ℂ
+      ( w)
+      ( z)
+      ( tr (λ x → sim-ℂ x one-ℂ) (commutative-mul-ℂ z w) zw=1)
+```
+
+### The multiplicative inverse of a nonzero complex number is nonzero
+
+```agda
+abstract
+  is-nonzero-complex-inv-nonzero-ℂ :
+    {l : Level} (z : nonzero-ℂ l) → is-nonzero-ℂ (complex-inv-nonzero-ℂ z)
+  is-nonzero-complex-inv-nonzero-ℂ z =
+    is-nonzero-has-right-inverse-mul-ℂ
+      ( _)
+      ( _)
+      ( right-inverse-law-mul-ℂ z)
+
+inv-nonzero-ℂ : {l : Level} → nonzero-ℂ l → nonzero-ℂ l
+inv-nonzero-ℂ z = (complex-inv-nonzero-ℂ z , is-nonzero-complex-inv-nonzero-ℂ z)
+```
diff --git a/src/complex-numbers/nonzero-complex-numbers.lagda.md b/src/complex-numbers/nonzero-complex-numbers.lagda.md
index a10fec58972..a4cf4129b04 100644
--- a/src/complex-numbers/nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/nonzero-complex-numbers.lagda.md
@@ -25,6 +25,7 @@ open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.square-roots-nonnegative-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.squares-real-numbers
 open import real-numbers.strict-inequality-real-numbers
@@ -97,3 +98,24 @@ positive-squared-magnitude-nonzero-ℂ :
 positive-squared-magnitude-nonzero-ℂ (z , z≠0) =
   ( squared-magnitude-ℂ z , is-positive-squared-magnitude-is-nonzero-ℂ z z≠0)
 ```
+
+### A complex number is nonzero if and only if its magnitude is positive
+
+```agda
+abstract
+  is-nonzero-is-positive-magnitude-ℂ :
+    {l : Level} (z : ℂ l) → is-positive-ℝ ∥ z ∥ℂ → is-nonzero-ℂ z
+  is-nonzero-is-positive-magnitude-ℂ z 0<|z| =
+    is-nonzero-is-positive-squared-magnitude-ℂ
+      ( z)
+      ( is-positive-is-positive-sqrt-ℝ⁰⁺
+        ( nonnegative-squared-magnitude-ℂ z)
+        ( 0<|z|))
+
+  is-positive-magnitude-is-nonzero-ℂ :
+    {l : Level} (z : ℂ l) → is-nonzero-ℂ z → is-positive-ℝ ∥ z ∥ℂ
+  is-positive-magnitude-is-nonzero-ℂ z z≠0 =
+    is-positive-sqrt-is-positive-ℝ⁰⁺
+      ( nonnegative-squared-magnitude-ℂ z)
+      ( is-positive-squared-magnitude-is-nonzero-ℂ z z≠0)
+```
diff --git a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
index 5f6c8696300..e6a3c231ad5 100644
--- a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
@@ -10,6 +10,9 @@ module real-numbers.multiplication-nonzero-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
+open import foundation.cartesian-product-types
+open import foundation.conjunction
+open import foundation.functoriality-cartesian-product-types
 open import foundation.disjunction
 open import foundation.function-types
 open import foundation.universe-levels
@@ -19,7 +22,6 @@ open import real-numbers.multiplication-negative-real-numbers
 open import real-numbers.multiplication-positive-and-negative-real-numbers
 open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
-open import real-numbers.multiplicative-inverses-nonzero-real-numbers
 open import real-numbers.nonzero-real-numbers
 ```
 
@@ -66,3 +68,37 @@ mul-nonzero-ℝ :
 mul-nonzero-ℝ (x , x#0) (y , y#0) =
   ( x *ℝ y , is-nonzero-mul-ℝ x#0 y#0)
 ```
+
+## Properties
+
+### If the product of two real numbers is nonzero, they are both nonzero
+
+```agda
+module _
+  {l1 l2 : Level}
+  (x : ℝ l1)
+  (y : ℝ l2)
+  where
+
+  abstract
+    is-nonzero-factors-is-nonzero-mul-ℝ :
+      is-nonzero-ℝ (x *ℝ y) → is-nonzero-ℝ x × is-nonzero-ℝ y
+    is-nonzero-factors-is-nonzero-mul-ℝ =
+      let
+        motive = is-nonzero-prop-ℝ x ∧ is-nonzero-prop-ℝ y
+      in
+        elim-disjunction
+          ( motive)
+          ( λ xy<0 →
+            elim-disjunction
+              ( motive)
+              ( map-product inr-disjunction inl-disjunction)
+              ( map-product inl-disjunction inr-disjunction)
+              ( different-signs-is-negative-mul-ℝ x y xy<0))
+          ( λ 0<xy →
+            elim-disjunction
+              ( motive)
+              ( map-product inl-disjunction inl-disjunction)
+              ( map-product inr-disjunction inr-disjunction)
+              ( same-sign-is-positive-mul-ℝ x y 0<xy))
+```
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index f1b15142b40..2cf6bd648d9 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -19,6 +19,8 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
+open import foundation.function-types
+open import foundation.functoriality-cartesian-product-types
 open import foundation.disjunction
 open import foundation.functoriality-disjunction
 open import foundation.propositional-truncations
@@ -35,6 +37,7 @@ open import real-numbers.negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -143,17 +146,17 @@ abstract
   different-signs-is-negative-mul-ℝ :
     {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → is-negative-ℝ (x *ℝ y) →
     type-disjunction-Prop
-      ( is-negative-prop-ℝ x ∧ is-positive-prop-ℝ y)
       ( is-positive-prop-ℝ x ∧ is-negative-prop-ℝ y)
+      ( is-negative-prop-ℝ x ∧ is-positive-prop-ℝ y)
   different-signs-is-negative-mul-ℝ x y xy<0 =
     map-disjunction
-      {!   !}
-      {!   !}
+      ( map-product (is-positive-is-negative-neg-ℝ x) id)
+      ( map-product (is-negative-is-positive-neg-ℝ x) id)
       ( same-sign-is-positive-mul-ℝ
         ( neg-ℝ x)
         ( y)
         ( inv-tr
           ( is-positive-ℝ)
           ( left-negative-law-mul-ℝ x y)
-          {!   !}))
+          ( neg-is-negative-ℝ (x *ℝ y) xy<0)))
 ```
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index aedb75f632c..7ca1640e44d 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -40,6 +40,7 @@ open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-negative-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
 open import real-numbers.negative-real-numbers
+open import real-numbers.multiplication-nonzero-real-numbers
 open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
@@ -151,65 +152,26 @@ is-invertible-is-nonzero-ℝ x x≠0 =
 ### If a real number has a multiplicative inverse, it is nonzero
 
 ```agda
-abstract opaque
-  unfolding mul-ℝ real-ℚ sim-ℝ
-
+abstract
   is-nonzero-has-right-inverse-mul-ℝ :
     {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →
     is-nonzero-ℝ x
-  is-nonzero-has-right-inverse-mul-ℝ x y (_ , L1⊆Lxy) =
-    let open do-syntax-trunc-Prop (is-nonzero-prop-ℝ x)
-    in do
-      ( ( ([a,b]@((a , b) , a≤b) , a<x , x<b) ,
-          ([c,d]@((c , d) , c≤d) , c<y , y<d)) ,
-        0<[a,b][c,d]) ← L1⊆Lxy zero-ℚ le-zero-one-ℚ
-      let
-        is-positive-mul p q {r} lb1 lb2 =
-          is-positive-le-zero-ℚ
-            ( concatenate-le-leq-ℚ
-              ( zero-ℚ)
-              ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
-              ( p *ℚ q)
-              ( 0<[a,b][c,d])
-              ( transitive-leq-ℚ
-                ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
-                ( r)
-                ( p *ℚ q)
-                ( lb1)
-                ( lb2)))
-      rec-coproduct
-        ( λ (is-neg-a , is-neg-c) →
-          rec-coproduct
-            ( λ (is-neg-b , is-neg-d) →
-              inl-disjunction
-                ( is-negative-exists-ℚ⁻-in-upper-cut-ℝ
-                  ( x)
-                  ( intro-exists (b , is-neg-b) x<b)))
-            ( λ (is-pos-b , is-pos-d) →
-              ex-falso
-                ( is-not-negative-and-positive-ℚ
-                  ( is-negative-mul-negative-positive-ℚ is-neg-a is-pos-d ,
-                    is-positive-mul a d
-                      ( leq-right-min-ℚ _ _)
-                      ( leq-left-min-ℚ _ _))))
-            ( same-sign-is-positive-mul-ℚ
-              ( is-positive-mul b d
-                ( leq-right-min-ℚ _ _)
-                ( leq-right-min-ℚ _ _))))
-        ( λ (is-pos-a , is-pos-c) →
-          inr-disjunction
-            ( is-positive-exists-ℚ⁺-in-lower-cut-ℝ
-              ( x)
-              ( intro-exists (a , is-pos-a) a<x)))
-        ( same-sign-is-positive-mul-ℚ
-          ( is-positive-mul a c (leq-left-min-ℚ _ _) (leq-left-min-ℚ _ _)))
+  is-nonzero-has-right-inverse-mul-ℝ x y xy=1 =
+    pr1
+      ( is-nonzero-factors-is-nonzero-mul-ℝ
+        ( x)
+        ( y)
+        ( is-nonzero-is-positive-ℝ
+          ( is-positive-sim-ℝ
+            ( is-positive-one-ℝ)
+            ( symmetric-sim-ℝ xy=1))))
 
   is-nonzero-has-left-inverse-mul-ℝ :
     {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →
     is-nonzero-ℝ y
   is-nonzero-has-left-inverse-mul-ℝ x y xy=1 =
     is-nonzero-has-right-inverse-mul-ℝ y x
-      ( tr (λ z → sim-ℝ z one-ℝ) (commutative-mul-ℝ _ _) xy=1)
+      ( tr (λ z → sim-ℝ z one-ℝ) (commutative-mul-ℝ x y) xy=1)
 ```
 
 ### The multiplicative inverse is unique
diff --git a/src/real-numbers/negative-real-numbers.lagda.md b/src/real-numbers/negative-real-numbers.lagda.md
index 75b3944778a..b92bcee1aa7 100644
--- a/src/real-numbers/negative-real-numbers.lagda.md
+++ b/src/real-numbers/negative-real-numbers.lagda.md
@@ -22,6 +22,7 @@ open import foundation.universe-levels
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
+open import real-numbers.similarity-real-numbers
 ```
 
 </details>
@@ -92,3 +93,13 @@ module _
       ( exists-ℚ⁻-in-upper-cut-is-negative-ℝ ,
         is-negative-exists-ℚ⁻-in-upper-cut-ℝ)
 ```
+
+### Being nonnegative is preserved by similarity
+
+```agda
+abstract
+  is-negative-sim-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} →
+    sim-ℝ x y → is-negative-ℝ x → is-negative-ℝ y
+  is-negative-sim-ℝ = preserves-le-left-sim-ℝ _ _ _
+```
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index 6b0e6074cd7..fcb261fb251 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -19,6 +19,7 @@ open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
+open import foundation.functoriality-disjunction
 
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
@@ -101,34 +102,6 @@ eq-nonzero-ℝ :
 eq-nonzero-ℝ _ _ = eq-type-subtype is-nonzero-prop-ℝ
 ```
 
-### Two real numbers are apart if and only if their difference is nonzero
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
-  where
-
-  abstract
-    is-nonzero-diff-is-apart-ℝ : apart-ℝ x y → is-nonzero-ℝ (x -ℝ y)
-    is-nonzero-diff-is-apart-ℝ x#y =
-      apart-right-sim-ℝ
-        ( x -ℝ y)
-        ( y -ℝ y)
-        ( zero-ℝ)
-        ( right-inverse-law-add-ℝ y)
-        ( preserves-apart-right-add-ℝ (neg-ℝ y) x y x#y)
-
-    is-apart-is-nonzero-diff-ℝ : is-nonzero-ℝ (x -ℝ y) → apart-ℝ x y
-    is-apart-is-nonzero-diff-ℝ x-y#0 =
-      apart-sim-ℝ
-        ( cancel-right-diff-add-ℝ x y)
-        ( sim-eq-ℝ (left-unit-law-add-ℝ y))
-        ( preserves-apart-right-add-ℝ y _ _ x-y#0)
-
-  nonzero-diff-apart-ℝ : apart-ℝ x y → nonzero-ℝ (l1 ⊔ l2)
-  nonzero-diff-apart-ℝ x#y = (x -ℝ y , is-nonzero-diff-is-apart-ℝ x#y)
-```
-
 ### The nonzero difference of a pair of real numbers `x` and `y` such that `x < y`
 
 ```agda
@@ -229,5 +202,16 @@ module _
                     ( nonnegative-square-ℝ x)
                     ( refl))
             ＝ abs-ℝ x by inv (eq-abs-sqrt-square-ℝ x))
-          ( is-positive-sqrt-ℝ⁺ (square-ℝ x , 0<x²)))
+          ( is-positive-sqrt-is-positive-ℝ⁰⁺ _ 0<x²))
+```
+
+### Being nonzero is preserved by similarity
+
+```agda
+abstract
+  is-nonzero-sim-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} →
+    sim-ℝ x y → is-nonzero-ℝ x → is-nonzero-ℝ y
+  is-nonzero-sim-ℝ x~y =
+    map-disjunction (is-negative-sim-ℝ x~y) (is-positive-sim-ℝ x~y)
 ```
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index b30852d3524..17305bae625 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -12,6 +12,8 @@ module real-numbers.positive-and-negative-real-numbers where
 open import foundation.dependent-pair-types
 open import foundation.transport-along-identifications
 open import foundation.logical-equivalences
+open import foundation.cartesian-product-types
+open import foundation.negation
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
@@ -120,3 +122,13 @@ abstract
     is-positive-ℝ y
   is-positive-le-ℝ⁰⁺ (x , 0≤x) y = concatenate-leq-le-ℝ zero-ℝ x y 0≤x
 ```
+
+### Real numbers are not both negative and nonnegative
+
+```agda
+abstract
+  is-not-negative-and-nonnegative-ℝ :
+    {l : Level} {x : ℝ l} → ¬ (is-negative-ℝ x × is-nonnegative-ℝ x)
+  is-not-negative-and-nonnegative-ℝ {x = x} (x<0 , 0≤x) =
+    not-leq-le-ℝ x zero-ℝ x<0 0≤x
+```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index e0b81e741e2..d00d7fc4b6f 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -85,9 +85,9 @@ is-positive-real-ℝ⁺ = pr2
 abstract
   is-positive-sim-ℝ :
     {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} →
-    is-positive-ℝ x → sim-ℝ x y → is-positive-ℝ y
-  is-positive-sim-ℝ {x = x} {y = y} 0<x x~y =
-    preserves-le-right-sim-ℝ zero-ℝ x y x~y 0<x
+    sim-ℝ x y → is-positive-ℝ x → is-positive-ℝ y
+  is-positive-sim-ℝ {x = x} {y = y} =
+    preserves-le-right-sim-ℝ zero-ℝ x y
 ```
 
 ### Dedekind cuts of positive real numbers
@@ -251,6 +251,9 @@ positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , preserves-is-positive-real-ℚ
 
 one-ℝ⁺ : ℝ⁺ lzero
 one-ℝ⁺ = positive-real-ℚ⁺ one-ℚ⁺
+
+is-positive-one-ℝ : is-positive-ℝ one-ℝ
+is-positive-one-ℝ = is-positive-real-ℝ⁺ one-ℝ⁺
 ```
 
 ### `x` is positive if and only if there exists a positive rational number it is not less than or equal to
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index d7c62f65161..117bd2589af 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -51,6 +51,7 @@ open import foundation.universe-levels
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
+open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
@@ -651,20 +652,37 @@ abstract
                         ( is-section-square-ℝ⁰⁺ y)))))))
 ```
 
-### The square root of a positive real number is positive
+### The square root of a nonnegative real number is positive if and only if the real number is positive
 
 ```agda
 abstract opaque
   unfolding real-sqrt-ℝ⁰⁺
 
-  is-positive-sqrt-ℝ⁺ :
-    {l : Level} (x : ℝ⁺ l) → is-positive-ℝ (real-sqrt-ℝ⁰⁺ (nonnegative-ℝ⁺ x))
-  is-positive-sqrt-ℝ⁺ x⁺@(x , _) =
+  is-positive-sqrt-is-positive-ℝ⁰⁺ :
+    {l : Level} (x : ℝ⁰⁺ l) → is-positive-ℝ (real-ℝ⁰⁺ x) →
+    is-positive-ℝ (real-sqrt-ℝ⁰⁺ x)
+  is-positive-sqrt-is-positive-ℝ⁰⁺ x⁰⁺@(x , _) 0<x =
     is-positive-zero-in-lower-cut-ℝ
-      ( real-sqrt-ℝ⁰⁺ (nonnegative-ℝ⁺ x⁺))
+      ( real-sqrt-ℝ⁰⁺ x⁰⁺)
       ( λ _ →
         inv-tr
           ( is-in-lower-cut-ℝ x)
           ( left-zero-law-mul-ℚ zero-ℚ)
-          ( zero-in-lower-cut-ℝ⁺ x⁺))
+          ( zero-in-lower-cut-ℝ⁺ (x , 0<x)))
+
+  is-positive-is-positive-sqrt-ℝ⁰⁺ :
+    {l : Level} (x : ℝ⁰⁺ l) →
+    is-positive-ℝ (real-sqrt-ℝ⁰⁺ x) → is-positive-ℝ (real-ℝ⁰⁺ x)
+  is-positive-is-positive-sqrt-ℝ⁰⁺ x⁰⁺@(x , _) 0<√x =
+    tr
+      ( is-positive-ℝ)
+      ( eq-real-square-sqrt-ℝ⁰⁺ x⁰⁺)
+      ( is-positive-mul-ℝ 0<√x 0<√x)
+
+is-positive-sqrt-iff-is-positive-ℝ⁰⁺ :
+  {l : Level} (x : ℝ⁰⁺ l) →
+  is-positive-ℝ (real-sqrt-ℝ⁰⁺ x) ↔ is-positive-ℝ (real-ℝ⁰⁺ x)
+is-positive-sqrt-iff-is-positive-ℝ⁰⁺ x =
+  ( is-positive-is-positive-sqrt-ℝ⁰⁺ x ,
+    is-positive-sqrt-is-positive-ℝ⁰⁺ x)
 ```
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index d64d8703277..15ad6049564 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -46,6 +46,7 @@ open import real-numbers.nonnegative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -354,3 +355,21 @@ abstract
       ＝ square-ℝ x -ℝ real-ℕ 2 *ℝ (x *ℝ y) +ℝ square-ℝ y
         by ap-add-ℝ (ap-add-ℝ refl (right-negative-law-mul-ℝ _ _)) refl
 ```
+
+### Squaring preserves similarity
+
+```agda
+abstract
+  preserves-sim-square-ℝ :
+    {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} →
+    sim-ℝ x y → sim-ℝ (square-ℝ x) (square-ℝ y)
+  preserves-sim-square-ℝ x~y = preserves-sim-mul-ℝ x~y x~y
+```
+
+### `0² = 0`
+
+```agda
+abstract
+  square-zero-ℝ : square-ℝ zero-ℝ ＝ zero-ℝ
+  square-zero-ℝ = eq-sim-ℝ (left-zero-law-mul-ℝ zero-ℝ)
+```

From de80356f77650a959fc9e551544cb3deb44411d1 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 18:38:21 -0800
Subject: [PATCH 103/134] Merge

---
 .../magnitude-complex-numbers.lagda.md             |  4 ++--
 ...ative-inverses-nonzero-complex-numbers.lagda.md | 14 +++++++-------
 .../nonzero-complex-numbers.lagda.md               |  2 +-
 .../similarity-complex-numbers.lagda.md            |  4 ----
 ...tion-closed-intervals-rational-numbers.lagda.md |  6 +++---
 .../multiplication-nonzero-real-numbers.lagda.md   |  4 ++--
 ...ion-positive-and-negative-real-numbers.lagda.md |  8 ++++----
 ...licative-inverses-nonzero-real-numbers.lagda.md |  2 +-
 src/real-numbers/negative-real-numbers.lagda.md    |  2 +-
 src/real-numbers/nonzero-real-numbers.lagda.md     |  2 +-
 .../positive-and-negative-real-numbers.lagda.md    |  4 ++--
 11 files changed, 24 insertions(+), 28 deletions(-)

diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index 4a81647b463..87e2fcde0de 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -15,20 +15,20 @@ open import complex-numbers.multiplication-complex-numbers
 open import complex-numbers.similarity-complex-numbers
 
 open import foundation.action-on-identifications-functions
-open import foundation.identity-types
 open import foundation.dependent-pair-types
+open import foundation.identity-types
 open import foundation.universe-levels
 
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.positive-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.positive-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 7278fe13552..32a5a481247 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -14,23 +14,23 @@ open import complex-numbers.multiplication-complex-numbers
 open import complex-numbers.nonzero-complex-numbers
 open import complex-numbers.similarity-complex-numbers
 
-open import real-numbers.nonzero-real-numbers
-open import real-numbers.nonnegative-real-numbers
 open import foundation.dependent-pair-types
-open import real-numbers.positive-and-negative-real-numbers
-open import foundation.function-types
 open import foundation.disjunction
-open import real-numbers.similarity-real-numbers
-open import real-numbers.positive-real-numbers
 open import foundation.empty-types
+open import foundation.function-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
-open import real-numbers.multiplication-nonzero-real-numbers
 open import foundation.universe-levels
 
+open import real-numbers.multiplication-nonzero-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.nonzero-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
+open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
 ```
 
 </details>
diff --git a/src/complex-numbers/nonzero-complex-numbers.lagda.md b/src/complex-numbers/nonzero-complex-numbers.lagda.md
index a4cf4129b04..6a2af2cf3d1 100644
--- a/src/complex-numbers/nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/nonzero-complex-numbers.lagda.md
@@ -25,8 +25,8 @@ open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.square-roots-nonnegative-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.square-roots-nonnegative-real-numbers
 open import real-numbers.squares-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/complex-numbers/similarity-complex-numbers.lagda.md b/src/complex-numbers/similarity-complex-numbers.lagda.md
index 2da181a6090..15331bf67fd 100644
--- a/src/complex-numbers/similarity-complex-numbers.lagda.md
+++ b/src/complex-numbers/similarity-complex-numbers.lagda.md
@@ -161,7 +161,3 @@ abstract
 
   syntax step-similarity-reasoning-ℂ p u q = p ~ℂ u by q
 ```
-
-# <<<<<<< HEAD
-
-> > > > > > > Stashed changes complex-ring
diff --git a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
index 7b985554ddd..3c44fa5f0ce 100644
--- a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
@@ -26,6 +26,7 @@ open import elementary-number-theory.minima-and-maxima-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
@@ -34,29 +35,28 @@ open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
 open import elementary-number-theory.negation-closed-intervals-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
-open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.poset-closed-intervals-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.proper-closed-intervals-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.squares-rational-numbers
-open import foundation.functoriality-coproduct-types
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.cartesian-product-types
+open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.function-extensionality
 open import foundation.function-types
+open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.images-subtypes
-open import foundation.coproduct-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
diff --git a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
index e6a3c231ad5..fe9285555ba 100644
--- a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
@@ -9,12 +9,12 @@ module real-numbers.multiplication-nonzero-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.dependent-pair-types
 open import foundation.cartesian-product-types
 open import foundation.conjunction
-open import foundation.functoriality-cartesian-product-types
+open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.function-types
+open import foundation.functoriality-cartesian-product-types
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 2cf6bd648d9..72de9d1ce9c 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -19,25 +19,25 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
-open import foundation.disjunction
 open import foundation.functoriality-disjunction
 open import foundation.propositional-truncations
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
-open import foundation.existential-quantification
 
 open import order-theory.posets
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
+open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
+open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index 7ca1640e44d..798999e3303 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -36,11 +36,11 @@ open import foundation.universe-levels
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.large-ring-of-real-numbers
+open import real-numbers.multiplication-nonzero-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-negative-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
 open import real-numbers.negative-real-numbers
-open import real-numbers.multiplication-nonzero-real-numbers
 open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
diff --git a/src/real-numbers/negative-real-numbers.lagda.md b/src/real-numbers/negative-real-numbers.lagda.md
index b92bcee1aa7..ced2c41ca1b 100644
--- a/src/real-numbers/negative-real-numbers.lagda.md
+++ b/src/real-numbers/negative-real-numbers.lagda.md
@@ -21,8 +21,8 @@ open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index fcb261fb251..e28693637e9 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -13,13 +13,13 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.disjunction
+open import foundation.functoriality-disjunction
 open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
-open import foundation.functoriality-disjunction
 
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index 17305bae625..e5964a4b3d2 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -9,11 +9,11 @@ module real-numbers.positive-and-negative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.cartesian-product-types
 open import foundation.dependent-pair-types
-open import foundation.transport-along-identifications
 open import foundation.logical-equivalences
-open import foundation.cartesian-product-types
 open import foundation.negation
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers

From 83be7907c570625850e2380e929b14b84d52149d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 18:51:32 -0800
Subject: [PATCH 104/134] Progress

---
 .../addition-nonzero-complex-numbers.lagda.md | 51 +++++++++++++++++++
 .../local-ring-of-complex-numbers.lagda.md    | 18 +++++++
 ...-inverses-nonzero-complex-numbers.lagda.md | 33 ++++++++++++
 3 files changed, 102 insertions(+)
 create mode 100644 src/complex-numbers/addition-nonzero-complex-numbers.lagda.md
 create mode 100644 src/complex-numbers/local-ring-of-complex-numbers.lagda.md

diff --git a/src/complex-numbers/addition-nonzero-complex-numbers.lagda.md b/src/complex-numbers/addition-nonzero-complex-numbers.lagda.md
new file mode 100644
index 00000000000..272db134c60
--- /dev/null
+++ b/src/complex-numbers/addition-nonzero-complex-numbers.lagda.md
@@ -0,0 +1,51 @@
+# Addition of nonzero complex numbers
+
+```agda
+module complex-numbers.addition-nonzero-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.nonzero-complex-numbers
+open import complex-numbers.addition-complex-numbers
+open import foundation.functoriality-disjunction
+open import complex-numbers.complex-numbers
+open import real-numbers.addition-nonzero-real-numbers
+open import foundation.disjunction
+open import foundation.universe-levels
+open import real-numbers.nonzero-real-numbers
+```
+
+</details>
+
+## Idea
+
+This file describes properties of
+[addition](complex-numbers.addition-complex-numbers.md) of
+[nonzero](complex-numbers.nonzero-complex-numbers.md)
+[complex numbers](complex-numbers.complex-numbers.md).
+
+## Properties
+
+### If the sum of two complex numbers is nonzero, one of them is nonzero
+
+```agda
+abstract
+  is-nonzero-either-is-nonzero-add-ℂ :
+    {l1 l2 : Level} (z : ℂ l1) (w : ℂ l2) → is-nonzero-ℂ (z +ℂ w) →
+    disjunction-type (is-nonzero-ℂ z) (is-nonzero-ℂ w)
+  is-nonzero-either-is-nonzero-add-ℂ z@(a +iℂ b) w@(c +iℂ d) =
+    elim-disjunction
+      ( is-nonzero-prop-ℂ z ∨ is-nonzero-prop-ℂ w)
+      ( λ a+c≠0 →
+        map-disjunction
+          ( inl-disjunction)
+          ( inl-disjunction)
+          ( is-nonzero-either-is-nonzero-add-ℝ a c a+c≠0))
+      ( λ b+d≠0 →
+        map-disjunction
+          ( inr-disjunction)
+          ( inr-disjunction)
+          ( is-nonzero-either-is-nonzero-add-ℝ b d b+d≠0))
+```
diff --git a/src/complex-numbers/local-ring-of-complex-numbers.lagda.md b/src/complex-numbers/local-ring-of-complex-numbers.lagda.md
new file mode 100644
index 00000000000..836919bbd0d
--- /dev/null
+++ b/src/complex-numbers/local-ring-of-complex-numbers.lagda.md
@@ -0,0 +1,18 @@
+# The local ring of complex numbers
+
+```agda
+module complex-numbers.local-ring-of-complex-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.local-commutative-rings
+open import complex-numbers.large-ring-of-complex-numbers
+open import foundation.universe-levels
+open import complex-numbers.nonzero-complex-numbers
+```
+
+</details>
+
+## Idea
diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 32a5a481247..627ca231ef4 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -10,12 +10,16 @@ module complex-numbers.multiplicative-inverses-nonzero-complex-numbers where
 open import complex-numbers.complex-numbers
 open import complex-numbers.conjugation-complex-numbers
 open import complex-numbers.magnitude-complex-numbers
+open import commutative-algebra.invertible-elements-commutative-rings
 open import complex-numbers.multiplication-complex-numbers
+open import complex-numbers.raising-universe-levels-complex-numbers
 open import complex-numbers.nonzero-complex-numbers
 open import complex-numbers.similarity-complex-numbers
+open import complex-numbers.large-ring-of-complex-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.disjunction
+open import foundation.logical-equivalences
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.identity-types
@@ -151,3 +155,32 @@ abstract
 inv-nonzero-ℂ : {l : Level} → nonzero-ℂ l → nonzero-ℂ l
 inv-nonzero-ℂ z = (complex-inv-nonzero-ℂ z , is-nonzero-complex-inv-nonzero-ℂ z)
 ```
+
+### A complex number is invertible if and only if it is nonzero
+
+```agda
+abstract
+  is-invertible-iff-is-nonzero-ℂ :
+    {l : Level} (z : ℂ l) →
+    ( is-invertible-element-Commutative-Ring
+      ( commutative-ring-ℂ l)
+      ( z)) ↔
+    ( is-nonzero-ℂ z)
+  is-invertible-iff-is-nonzero-ℂ z =
+    ( ( λ (w , zw=1 , _) →
+        is-nonzero-has-right-inverse-mul-ℂ
+          ( z)
+          ( w)
+          ( transitive-sim-ℂ _ _ _
+            ( symmetric-sim-ℂ (sim-raise-ℂ _ one-ℂ))
+            ( sim-eq-ℂ zw=1))) ,
+      ( λ z≠0 →
+        ( complex-inv-nonzero-ℂ (z , z≠0) ,
+          eq-sim-ℂ
+            ( similarity-reasoning-ℂ
+              z *ℂ complex-inv-nonzero-ℂ (z , z≠0)
+              ~ℂ one-ℂ
+                by left-inverse-law-mul-ℂ (z , z≠0)
+              ~ℂ {!   !} by {!   !}) ,
+          {!   !})))
+```

From 23e4eab9ec094c41668e9b8356f7f28eda83d42e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 18:52:15 -0800
Subject: [PATCH 105/134] Fix lefts and rights

---
 ...cative-inverses-nonzero-complex-numbers.lagda.md | 13 ++++++++-----
 1 file changed, 8 insertions(+), 5 deletions(-)

diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 71dd292ad7f..183fbec3b2b 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -47,9 +47,9 @@ complex-inv-nonzero-ℂ (z , z≠0) =
 
 ```agda
 abstract
-  left-inverse-law-mul-ℂ : {l : Level} (z : nonzero-ℂ l) →
+  right-inverse-law-mul-nonzero-ℂ : {l : Level} (z : nonzero-ℂ l) →
     sim-ℂ (complex-nonzero-ℂ z *ℂ complex-inv-nonzero-ℂ z) one-ℂ
-  left-inverse-law-mul-ℂ (z@(a +iℂ b) , z≠0) =
+  right-inverse-law-mul-nonzero-ℂ (z@(a +iℂ b) , z≠0) =
     similarity-reasoning-ℂ
       z *ℂ
       ( conjugate-ℂ z *ℂ
@@ -78,8 +78,11 @@ abstract
       ~ℂ one-ℂ
         by sim-eq-ℂ eq-complex-one-ℝ
 
-  right-inverse-law-mul-ℂ : {l : Level} (z : nonzero-ℂ l) →
+  left-inverse-law-mul-nonzero-ℂ : {l : Level} (z : nonzero-ℂ l) →
     sim-ℂ (complex-inv-nonzero-ℂ z *ℂ complex-nonzero-ℂ z) one-ℂ
-  right-inverse-law-mul-ℂ z =
-    tr (λ w → sim-ℂ w one-ℂ) (commutative-mul-ℂ _ _) (left-inverse-law-mul-ℂ z)
+  left-inverse-law-mul-nonzero-ℂ z =
+    tr
+      ( λ w → sim-ℂ w one-ℂ)
+      ( commutative-mul-ℂ _ _)
+      ( right-inverse-law-mul-nonzero-ℂ z)
 ```

From 4114c71cbcadd367535d4c48a54792b3b8331197 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 19:00:19 -0800
Subject: [PATCH 106/134] The local ring of complex numbers

---
 src/complex-numbers.lagda.md                  |  9 ++-
 .../addition-nonzero-complex-numbers.lagda.md |  8 ++-
 .../local-ring-of-complex-numbers.lagda.md    | 34 +++++++++-
 ...-inverses-nonzero-complex-numbers.lagda.md | 64 +++++++++++--------
 src/linear-algebra.lagda.md                   |  2 +
 .../real-vector-spaces.lagda.md               |  6 +-
 src/linear-algebra/vector-spaces.lagda.md     |  5 +-
 src/real-numbers.lagda.md                     |  3 +
 .../addition-negative-real-numbers.lagda.md   | 23 +++----
 .../addition-nonzero-real-numbers.lagda.md    | 13 ++--
 .../local-ring-of-real-numbers.lagda.md       |  8 ++-
 11 files changed, 115 insertions(+), 60 deletions(-)

diff --git a/src/complex-numbers.lagda.md b/src/complex-numbers.lagda.md
index b1c16d18866..14b808adef3 100644
--- a/src/complex-numbers.lagda.md
+++ b/src/complex-numbers.lagda.md
@@ -4,20 +4,19 @@
 module complex-numbers where
 
 open import complex-numbers.addition-complex-numbers public
-open import complex-numbers.additive-group-of-complex-numbers public
+open import complex-numbers.addition-nonzero-complex-numbers public
 open import complex-numbers.apartness-complex-numbers public
 open import complex-numbers.complex-numbers public
 open import complex-numbers.conjugation-complex-numbers public
 open import complex-numbers.eisenstein-integers public
 open import complex-numbers.gaussian-integers public
+open import complex-numbers.large-additive-group-of-complex-numbers public
+open import complex-numbers.large-ring-of-complex-numbers public
+open import complex-numbers.local-ring-of-complex-numbers public
 open import complex-numbers.magnitude-complex-numbers public
 open import complex-numbers.multiplication-complex-numbers public
 open import complex-numbers.multiplicative-inverses-nonzero-complex-numbers public
 open import complex-numbers.nonzero-complex-numbers public
-open import complex-numbers.ring-of-complex-numbers public
-open import complex-numbers.large-additive-group-of-complex-numbers public
-open import complex-numbers.large-ring-of-complex-numbers public
-open import complex-numbers.multiplication-complex-numbers public
 open import complex-numbers.raising-universe-levels-complex-numbers public
 open import complex-numbers.similarity-complex-numbers public
 ```
diff --git a/src/complex-numbers/addition-nonzero-complex-numbers.lagda.md b/src/complex-numbers/addition-nonzero-complex-numbers.lagda.md
index 272db134c60..c74857b7bf9 100644
--- a/src/complex-numbers/addition-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/addition-nonzero-complex-numbers.lagda.md
@@ -7,13 +7,15 @@ module complex-numbers.addition-nonzero-complex-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import complex-numbers.nonzero-complex-numbers
 open import complex-numbers.addition-complex-numbers
-open import foundation.functoriality-disjunction
 open import complex-numbers.complex-numbers
-open import real-numbers.addition-nonzero-real-numbers
+open import complex-numbers.nonzero-complex-numbers
+
 open import foundation.disjunction
+open import foundation.functoriality-disjunction
 open import foundation.universe-levels
+
+open import real-numbers.addition-nonzero-real-numbers
 open import real-numbers.nonzero-real-numbers
 ```
 
diff --git a/src/complex-numbers/local-ring-of-complex-numbers.lagda.md b/src/complex-numbers/local-ring-of-complex-numbers.lagda.md
index 836919bbd0d..6f96731b109 100644
--- a/src/complex-numbers/local-ring-of-complex-numbers.lagda.md
+++ b/src/complex-numbers/local-ring-of-complex-numbers.lagda.md
@@ -8,11 +8,43 @@ module complex-numbers.local-ring-of-complex-numbers where
 
 ```agda
 open import commutative-algebra.local-commutative-rings
+
+open import complex-numbers.addition-complex-numbers
+open import complex-numbers.addition-nonzero-complex-numbers
 open import complex-numbers.large-ring-of-complex-numbers
-open import foundation.universe-levels
+open import complex-numbers.multiplicative-inverses-nonzero-complex-numbers
 open import complex-numbers.nonzero-complex-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.functoriality-disjunction
+open import foundation.universe-levels
 ```
 
 </details>
 
 ## Idea
+
+The
+[commutative ring of complex numbers](complex-numbers.large-ring-of-complex-numbers.md)
+is [local](commutative-algebra.local-commutative-rings.md).
+
+## Definition
+
+```agda
+is-local-commutative-ring-ℂ :
+  (l : Level) → is-local-Commutative-Ring (commutative-ring-ℂ l)
+is-local-commutative-ring-ℂ l z w is-invertible-z+w =
+  map-disjunction
+    ( is-invertible-is-nonzero-ℂ z)
+    ( is-invertible-is-nonzero-ℂ w)
+    ( is-nonzero-either-is-nonzero-add-ℂ
+      ( z)
+      ( w)
+      ( is-nonzero-is-invertible-ℂ
+        ( z +ℂ w)
+        ( is-invertible-z+w)))
+
+local-commutative-ring-ℂ : (l : Level) → Local-Commutative-Ring (lsuc l)
+local-commutative-ring-ℂ l =
+  ( commutative-ring-ℂ l , is-local-commutative-ring-ℂ l)
+```
diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 6f79ee601e7..10db59823f9 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -7,19 +7,19 @@ module complex-numbers.multiplicative-inverses-nonzero-complex-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import commutative-algebra.invertible-elements-commutative-rings
+
 open import complex-numbers.complex-numbers
 open import complex-numbers.conjugation-complex-numbers
+open import complex-numbers.large-ring-of-complex-numbers
 open import complex-numbers.magnitude-complex-numbers
-open import commutative-algebra.invertible-elements-commutative-rings
 open import complex-numbers.multiplication-complex-numbers
-open import complex-numbers.raising-universe-levels-complex-numbers
 open import complex-numbers.nonzero-complex-numbers
+open import complex-numbers.raising-universe-levels-complex-numbers
 open import complex-numbers.similarity-complex-numbers
-open import complex-numbers.large-ring-of-complex-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.disjunction
-open import foundation.logical-equivalences
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.identity-types
@@ -150,10 +150,10 @@ abstract
   is-nonzero-complex-inv-nonzero-ℂ :
     {l : Level} (z : nonzero-ℂ l) → is-nonzero-ℂ (complex-inv-nonzero-ℂ z)
   is-nonzero-complex-inv-nonzero-ℂ z =
-    is-nonzero-has-right-inverse-mul-ℂ
+    is-nonzero-has-left-inverse-mul-ℂ
       ( _)
       ( _)
-      ( right-inverse-law-mul-ℂ z)
+      ( right-inverse-law-mul-nonzero-ℂ z)
 
 inv-nonzero-ℂ : {l : Level} → nonzero-ℂ l → nonzero-ℂ l
 inv-nonzero-ℂ z = (complex-inv-nonzero-ℂ z , is-nonzero-complex-inv-nonzero-ℂ z)
@@ -163,27 +163,35 @@ inv-nonzero-ℂ z = (complex-inv-nonzero-ℂ z , is-nonzero-complex-inv-nonzero-
 
 ```agda
 abstract
-  is-invertible-iff-is-nonzero-ℂ :
+  is-nonzero-is-invertible-ℂ :
     {l : Level} (z : ℂ l) →
-    ( is-invertible-element-Commutative-Ring
-      ( commutative-ring-ℂ l)
-      ( z)) ↔
-    ( is-nonzero-ℂ z)
-  is-invertible-iff-is-nonzero-ℂ z =
-    ( ( λ (w , zw=1 , _) →
-        is-nonzero-has-right-inverse-mul-ℂ
-          ( z)
-          ( w)
-          ( transitive-sim-ℂ _ _ _
-            ( symmetric-sim-ℂ (sim-raise-ℂ _ one-ℂ))
-            ( sim-eq-ℂ zw=1))) ,
-      ( λ z≠0 →
-        ( complex-inv-nonzero-ℂ (z , z≠0) ,
-          eq-sim-ℂ
-            ( similarity-reasoning-ℂ
-              z *ℂ complex-inv-nonzero-ℂ (z , z≠0)
-              ~ℂ one-ℂ
-                by left-inverse-law-mul-ℂ (z , z≠0)
-              ~ℂ {!   !} by {!   !}) ,
-          {!   !})))
+    is-invertible-element-Commutative-Ring (commutative-ring-ℂ l) z →
+    is-nonzero-ℂ z
+  is-nonzero-is-invertible-ℂ z (w , zw=1 , _) =
+    is-nonzero-has-right-inverse-mul-ℂ
+      ( z)
+      ( w)
+      ( transitive-sim-ℂ _ _ _
+        ( symmetric-sim-ℂ (sim-raise-ℂ _ one-ℂ))
+        ( sim-eq-ℂ zw=1))
+
+  is-invertible-is-nonzero-ℂ :
+    {l : Level} (z : ℂ l) → is-nonzero-ℂ z →
+    is-invertible-element-Commutative-Ring (commutative-ring-ℂ l) z
+  is-invertible-is-nonzero-ℂ z z≠0 =
+    ( complex-inv-nonzero-ℂ (z , z≠0) ,
+      eq-sim-ℂ
+        ( similarity-reasoning-ℂ
+          z *ℂ complex-inv-nonzero-ℂ (z , z≠0)
+          ~ℂ one-ℂ
+            by right-inverse-law-mul-nonzero-ℂ (z , z≠0)
+          ~ℂ raise-ℂ _ one-ℂ
+            by sim-raise-ℂ _ one-ℂ) ,
+      eq-sim-ℂ
+        ( similarity-reasoning-ℂ
+          complex-inv-nonzero-ℂ (z , z≠0) *ℂ z
+          ~ℂ one-ℂ
+            by left-inverse-law-mul-nonzero-ℂ (z , z≠0)
+          ~ℂ raise-ℂ _ one-ℂ
+            by sim-raise-ℂ _ one-ℂ))
 ```
diff --git a/src/linear-algebra.lagda.md b/src/linear-algebra.lagda.md
index ba50e0c42c0..ba723c1a899 100644
--- a/src/linear-algebra.lagda.md
+++ b/src/linear-algebra.lagda.md
@@ -33,6 +33,7 @@ open import linear-algebra.matrices-on-rings public
 open import linear-algebra.multiplication-matrices public
 open import linear-algebra.preimages-of-left-module-structures-along-homomorphisms-of-rings public
 open import linear-algebra.rational-modules public
+open import linear-algebra.real-vector-spaces public
 open import linear-algebra.right-modules-rings public
 open import linear-algebra.scalar-multiplication-matrices public
 open import linear-algebra.scalar-multiplication-tuples public
@@ -46,4 +47,5 @@ open import linear-algebra.tuples-on-euclidean-domains public
 open import linear-algebra.tuples-on-monoids public
 open import linear-algebra.tuples-on-rings public
 open import linear-algebra.tuples-on-semirings public
+open import linear-algebra.vector-spaces public
 ```
diff --git a/src/linear-algebra/real-vector-spaces.lagda.md b/src/linear-algebra/real-vector-spaces.lagda.md
index ab72ab74bd5..869ad40d319 100644
--- a/src/linear-algebra/real-vector-spaces.lagda.md
+++ b/src/linear-algebra/real-vector-spaces.lagda.md
@@ -7,9 +7,11 @@ module linear-algebra.real-vector-spaces where
 <details><summary>Imports</summary>
 
 ```agda
-open import real-numbers.local-ring-of-real-numbers
-open import linear-algebra.vector-spaces
 open import foundation.universe-levels
+
+open import linear-algebra.vector-spaces
+
+open import real-numbers.local-ring-of-real-numbers
 ```
 
 </details>
diff --git a/src/linear-algebra/vector-spaces.lagda.md b/src/linear-algebra/vector-spaces.lagda.md
index d9e39a8ed7c..aa4caa42c75 100644
--- a/src/linear-algebra/vector-spaces.lagda.md
+++ b/src/linear-algebra/vector-spaces.lagda.md
@@ -7,10 +7,13 @@ module linear-algebra.vector-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import commutative-algebra.local-commutative-rings
+
 open import foundation.sets
 open import foundation.universe-levels
+
 open import group-theory.abelian-groups
-open import commutative-algebra.local-commutative-rings
+
 open import linear-algebra.left-modules-commutative-rings
 ```
 
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index a3ef18a8455..fd03949811f 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -9,7 +9,9 @@ open import real-numbers.absolute-value-closed-intervals-real-numbers public
 open import real-numbers.absolute-value-real-numbers public
 open import real-numbers.accumulation-points-subsets-real-numbers public
 open import real-numbers.addition-lower-dedekind-real-numbers public
+open import real-numbers.addition-negative-real-numbers public
 open import real-numbers.addition-nonnegative-real-numbers public
+open import real-numbers.addition-nonzero-real-numbers public
 open import real-numbers.addition-positive-real-numbers public
 open import real-numbers.addition-real-numbers public
 open import real-numbers.addition-upper-dedekind-real-numbers public
@@ -45,6 +47,7 @@ open import real-numbers.large-multiplicative-monoid-of-real-numbers public
 open import real-numbers.large-ring-of-real-numbers public
 open import real-numbers.limits-sequences-real-numbers public
 open import real-numbers.lipschitz-continuity-multiplication-real-numbers public
+open import real-numbers.local-ring-of-real-numbers public
 open import real-numbers.located-metric-space-of-real-numbers public
 open import real-numbers.lower-dedekind-real-numbers public
 open import real-numbers.maximum-finite-families-real-numbers public
diff --git a/src/real-numbers/addition-negative-real-numbers.lagda.md b/src/real-numbers/addition-negative-real-numbers.lagda.md
index 66f7d5e1e54..c3e3cef113d 100644
--- a/src/real-numbers/addition-negative-real-numbers.lagda.md
+++ b/src/real-numbers/addition-negative-real-numbers.lagda.md
@@ -9,21 +9,22 @@ module real-numbers.addition-negative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import real-numbers.negative-real-numbers
-open import real-numbers.dedekind-real-numbers
-open import foundation.transport-along-identifications
-open import real-numbers.rational-real-numbers
-open import real-numbers.negation-real-numbers
+open import foundation.dependent-pair-types
+open import foundation.disjunction
 open import foundation.functoriality-disjunction
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
 open import real-numbers.addition-positive-real-numbers
-open import foundation.disjunction
 open import real-numbers.addition-real-numbers
-open import real-numbers.strict-inequality-real-numbers
-open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
-open import real-numbers.positive-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.negative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
-open import foundation.universe-levels
-open import foundation.dependent-pair-types
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/addition-nonzero-real-numbers.lagda.md b/src/real-numbers/addition-nonzero-real-numbers.lagda.md
index 3ab16e4f8b7..e59f5b96f1c 100644
--- a/src/real-numbers/addition-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/addition-nonzero-real-numbers.lagda.md
@@ -9,15 +9,16 @@ module real-numbers.addition-nonzero-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import real-numbers.nonzero-real-numbers
-open import real-numbers.addition-positive-real-numbers
+open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.functoriality-disjunction
+open import foundation.universe-levels
+
 open import real-numbers.addition-negative-real-numbers
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import foundation.functoriality-disjunction
-open import foundation.disjunction
-open import foundation.dependent-pair-types
-open import foundation.universe-levels
+open import real-numbers.nonzero-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/local-ring-of-real-numbers.lagda.md b/src/real-numbers/local-ring-of-real-numbers.lagda.md
index 282f1dc2237..030a5b42de4 100644
--- a/src/real-numbers/local-ring-of-real-numbers.lagda.md
+++ b/src/real-numbers/local-ring-of-real-numbers.lagda.md
@@ -10,14 +10,16 @@ module real-numbers.local-ring-of-real-numbers where
 
 ```agda
 open import commutative-algebra.local-commutative-rings
-open import real-numbers.addition-real-numbers
-open import real-numbers.nonzero-real-numbers
+
 open import foundation.dependent-pair-types
 open import foundation.functoriality-disjunction
+open import foundation.universe-levels
+
 open import real-numbers.addition-nonzero-real-numbers
+open import real-numbers.addition-real-numbers
 open import real-numbers.large-ring-of-real-numbers
 open import real-numbers.multiplicative-inverses-nonzero-real-numbers
-open import foundation.universe-levels
+open import real-numbers.nonzero-real-numbers
 ```
 
 </details>

From ddf1a70559a0c355574eac761f5a6b57f4c4f68a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 19:23:47 -0800
Subject: [PATCH 107/134] Complex vector spaces

---
 src/linear-algebra.lagda.md                   |  1 +
 .../complex-vector-spaces.lagda.md            | 24 +++++++++++++++++++
 2 files changed, 25 insertions(+)
 create mode 100644 src/linear-algebra/complex-vector-spaces.lagda.md

diff --git a/src/linear-algebra.lagda.md b/src/linear-algebra.lagda.md
index ba723c1a899..474d03f52e1 100644
--- a/src/linear-algebra.lagda.md
+++ b/src/linear-algebra.lagda.md
@@ -5,6 +5,7 @@
 ```agda
 module linear-algebra where
 
+open import linear-algebra.complex-vector-spaces public
 open import linear-algebra.constant-matrices public
 open import linear-algebra.constant-tuples public
 open import linear-algebra.dependent-products-left-modules-commutative-rings public
diff --git a/src/linear-algebra/complex-vector-spaces.lagda.md b/src/linear-algebra/complex-vector-spaces.lagda.md
new file mode 100644
index 00000000000..fcff11674ad
--- /dev/null
+++ b/src/linear-algebra/complex-vector-spaces.lagda.md
@@ -0,0 +1,24 @@
+# Complex vector spaces
+
+```agda
+module linear-algebra.complex-vector-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import complex-numbers.local-ring-of-complex-numbers
+
+open import foundation.universe-levels
+
+open import linear-algebra.vector-spaces
+```
+
+</details>
+
+## Idea
+
+```agda
+ℂ-Vector-Space : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
+ℂ-Vector-Space l1 l2 = Vector-Space l2 (local-commutative-ring-ℂ l1)
+```

From 16e5f5cee9b9c39726e5cf7d092368689781c81f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 19:53:57 -0800
Subject: [PATCH 108/134] Progress

---
 .../local-commutative-rings.lagda.md          |  20 +++
 src/linear-algebra.lagda.md                   |   2 +
 .../real-vector-spaces.lagda.md               | 143 +++++++++++++++++-
 src/linear-algebra/vector-spaces.lagda.md     | 127 +++++++++++++++-
 src/real-numbers.lagda.md                     |   3 +
 .../addition-negative-real-numbers.lagda.md   |  23 +--
 .../addition-nonzero-real-numbers.lagda.md    |  13 +-
 .../local-ring-of-real-numbers.lagda.md       |   8 +-
 ...ositive-and-negative-real-numbers.lagda.md |  45 ++++++
 9 files changed, 361 insertions(+), 23 deletions(-)

diff --git a/src/commutative-algebra/local-commutative-rings.lagda.md b/src/commutative-algebra/local-commutative-rings.lagda.md
index 9de71b60fc1..717095eaa53 100644
--- a/src/commutative-algebra/local-commutative-rings.lagda.md
+++ b/src/commutative-algebra/local-commutative-rings.lagda.md
@@ -66,4 +66,24 @@ module _
   is-local-commutative-ring-Local-Commutative-Ring :
     is-local-Commutative-Ring commutative-ring-Local-Commutative-Ring
   is-local-commutative-ring-Local-Commutative-Ring = pr2 A
+
+  zero-Local-Commutative-Ring : type-Local-Commutative-Ring
+  zero-Local-Commutative-Ring = zero-Ring ring-Local-Commutative-Ring
+
+  one-Local-Commutative-Ring : type-Local-Commutative-Ring
+  one-Local-Commutative-Ring = one-Ring ring-Local-Commutative-Ring
+
+  add-Local-Commutative-Ring :
+    type-Local-Commutative-Ring → type-Local-Commutative-Ring →
+    type-Local-Commutative-Ring
+  add-Local-Commutative-Ring = add-Ring ring-Local-Commutative-Ring
+
+  mul-Local-Commutative-Ring :
+    type-Local-Commutative-Ring → type-Local-Commutative-Ring →
+    type-Local-Commutative-Ring
+  mul-Local-Commutative-Ring = mul-Ring ring-Local-Commutative-Ring
+
+  neg-Local-Commutative-Ring :
+    type-Local-Commutative-Ring → type-Local-Commutative-Ring
+  neg-Local-Commutative-Ring = neg-Ring ring-Local-Commutative-Ring
 ```
diff --git a/src/linear-algebra.lagda.md b/src/linear-algebra.lagda.md
index ba50e0c42c0..ba723c1a899 100644
--- a/src/linear-algebra.lagda.md
+++ b/src/linear-algebra.lagda.md
@@ -33,6 +33,7 @@ open import linear-algebra.matrices-on-rings public
 open import linear-algebra.multiplication-matrices public
 open import linear-algebra.preimages-of-left-module-structures-along-homomorphisms-of-rings public
 open import linear-algebra.rational-modules public
+open import linear-algebra.real-vector-spaces public
 open import linear-algebra.right-modules-rings public
 open import linear-algebra.scalar-multiplication-matrices public
 open import linear-algebra.scalar-multiplication-tuples public
@@ -46,4 +47,5 @@ open import linear-algebra.tuples-on-euclidean-domains public
 open import linear-algebra.tuples-on-monoids public
 open import linear-algebra.tuples-on-rings public
 open import linear-algebra.tuples-on-semirings public
+open import linear-algebra.vector-spaces public
 ```
diff --git a/src/linear-algebra/real-vector-spaces.lagda.md b/src/linear-algebra/real-vector-spaces.lagda.md
index ab72ab74bd5..1ee4b2f6451 100644
--- a/src/linear-algebra/real-vector-spaces.lagda.md
+++ b/src/linear-algebra/real-vector-spaces.lagda.md
@@ -7,16 +7,155 @@ module linear-algebra.real-vector-spaces where
 <details><summary>Imports</summary>
 
 ```agda
-open import real-numbers.local-ring-of-real-numbers
-open import linear-algebra.vector-spaces
+open import foundation.identity-types
+open import foundation.sets
 open import foundation.universe-levels
+
+open import group-theory.abelian-groups
+
+open import linear-algebra.vector-spaces
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.local-ring-of-real-numbers
+open import real-numbers.multiplication-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.rational-real-numbers
 ```
 
 </details>
 
 ## Idea
 
+A
+{{#concept "real vector space" WD="real vector space" WDID=Q46996054 Agda=ℝ-Vector-Space}}
+is a [vector space](linear-algebra.vector-spaces.md) over the
+[local commutative ring of real numbers](real-numbers.local-ring-of-real-numbers.md).
+
+## Definition
+
 ```agda
 ℝ-Vector-Space : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
 ℝ-Vector-Space l1 l2 = Vector-Space l2 (local-commutative-ring-ℝ l1)
 ```
+
+## Properties
+
+```agda
+module _
+  {l1 l2 : Level} (V : ℝ-Vector-Space l1 l2)
+  where
+
+  ab-ℝ-Vector-Space : Ab l2
+  ab-ℝ-Vector-Space = ab-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  set-ℝ-Vector-Space : Set l2
+  set-ℝ-Vector-Space = set-Ab ab-ℝ-Vector-Space
+
+  type-ℝ-Vector-Space : UU l2
+  type-ℝ-Vector-Space = type-Ab ab-ℝ-Vector-Space
+
+  add-ℝ-Vector-Space :
+    type-ℝ-Vector-Space → type-ℝ-Vector-Space → type-ℝ-Vector-Space
+  add-ℝ-Vector-Space = add-Ab ab-ℝ-Vector-Space
+
+  zero-ℝ-Vector-Space : type-ℝ-Vector-Space
+  zero-ℝ-Vector-Space = zero-Ab ab-ℝ-Vector-Space
+
+  neg-ℝ-Vector-Space : type-ℝ-Vector-Space → type-ℝ-Vector-Space
+  neg-ℝ-Vector-Space = neg-Ab ab-ℝ-Vector-Space
+
+  mul-ℝ-Vector-Space : ℝ l1 → type-ℝ-Vector-Space → type-ℝ-Vector-Space
+  mul-ℝ-Vector-Space = mul-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  associative-add-ℝ-Vector-Space :
+    (v w x : type-ℝ-Vector-Space) →
+    add-ℝ-Vector-Space (add-ℝ-Vector-Space v w) x ＝
+    add-ℝ-Vector-Space v (add-ℝ-Vector-Space w x)
+  associative-add-ℝ-Vector-Space = associative-add-Ab ab-ℝ-Vector-Space
+
+  left-unit-law-add-ℝ-Vector-Space :
+    (v : type-ℝ-Vector-Space) → add-ℝ-Vector-Space zero-ℝ-Vector-Space v ＝ v
+  left-unit-law-add-ℝ-Vector-Space = left-unit-law-add-Ab ab-ℝ-Vector-Space
+
+  right-unit-law-add-ℝ-Vector-Space :
+    (v : type-ℝ-Vector-Space) → add-ℝ-Vector-Space v zero-ℝ-Vector-Space ＝ v
+  right-unit-law-add-ℝ-Vector-Space = right-unit-law-add-Ab ab-ℝ-Vector-Space
+
+  left-inverse-law-add-ℝ-Vector-Space :
+    (v : type-ℝ-Vector-Space) →
+    add-ℝ-Vector-Space (neg-ℝ-Vector-Space v) v ＝ zero-ℝ-Vector-Space
+  left-inverse-law-add-ℝ-Vector-Space =
+    left-inverse-law-add-Ab ab-ℝ-Vector-Space
+
+  right-inverse-law-add-ℝ-Vector-Space :
+    (v : type-ℝ-Vector-Space) →
+    add-ℝ-Vector-Space v (neg-ℝ-Vector-Space v) ＝ zero-ℝ-Vector-Space
+  right-inverse-law-add-ℝ-Vector-Space =
+    right-inverse-law-add-Ab ab-ℝ-Vector-Space
+
+  commutative-add-ℝ-Vector-Space :
+    (v w : type-ℝ-Vector-Space) →
+    add-ℝ-Vector-Space v w ＝ add-ℝ-Vector-Space w v
+  commutative-add-ℝ-Vector-Space = commutative-add-Ab ab-ℝ-Vector-Space
+
+  left-unit-law-mul-ℝ-Vector-Space :
+    (v : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space (raise-ℝ l1 one-ℝ) v ＝ v
+  left-unit-law-mul-ℝ-Vector-Space =
+    left-unit-law-mul-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  left-distributive-mul-add-ℝ-Vector-Space :
+    (r : ℝ l1) (v w : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space r (add-ℝ-Vector-Space v w) ＝
+    add-ℝ-Vector-Space (mul-ℝ-Vector-Space r v) (mul-ℝ-Vector-Space r w)
+  left-distributive-mul-add-ℝ-Vector-Space =
+    left-distributive-mul-add-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  right-distributive-mul-add-ℝ-Vector-Space :
+    (r s : ℝ l1) (v : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space (r +ℝ s) v ＝
+    add-ℝ-Vector-Space (mul-ℝ-Vector-Space r v) (mul-ℝ-Vector-Space s v)
+  right-distributive-mul-add-ℝ-Vector-Space =
+    right-distributive-mul-add-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  associative-mul-ℝ-Vector-Space :
+    (r s : ℝ l1) (v : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space (r *ℝ s) v ＝
+    mul-ℝ-Vector-Space r (mul-ℝ-Vector-Space s v)
+  associative-mul-ℝ-Vector-Space =
+    associative-mul-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  left-zero-law-mul-ℝ-Vector-Space :
+    (v : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space (raise-ℝ l1 zero-ℝ) v ＝ zero-ℝ-Vector-Space
+  left-zero-law-mul-ℝ-Vector-Space =
+    left-zero-law-mul-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  right-zero-law-mul-ℝ-Vector-Space :
+    (r : ℝ l1) →
+    mul-ℝ-Vector-Space r zero-ℝ-Vector-Space ＝ zero-ℝ-Vector-Space
+  right-zero-law-mul-ℝ-Vector-Space =
+    right-zero-law-mul-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  left-negative-law-mul-ℝ-Vector-Space :
+    (r : ℝ l1) (v : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space (neg-ℝ r) v ＝
+    neg-ℝ-Vector-Space (mul-ℝ-Vector-Space r v)
+  left-negative-law-mul-ℝ-Vector-Space =
+    left-negative-law-mul-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  right-negative-law-mul-ℝ-Vector-Space :
+    (r : ℝ l1) (v : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space r (neg-ℝ-Vector-Space v) ＝
+    neg-ℝ-Vector-Space (mul-ℝ-Vector-Space r v)
+  right-negative-law-mul-ℝ-Vector-Space =
+    right-negative-law-mul-Vector-Space (local-commutative-ring-ℝ l1) V
+
+  mul-neg-one-ℝ-Vector-Space :
+    (v : type-ℝ-Vector-Space) →
+    mul-ℝ-Vector-Space (neg-ℝ (raise-ℝ l1 one-ℝ)) v ＝ neg-ℝ-Vector-Space v
+  mul-neg-one-ℝ-Vector-Space =
+    mul-neg-one-Vector-Space (local-commutative-ring-ℝ l1) V
+```
diff --git a/src/linear-algebra/vector-spaces.lagda.md b/src/linear-algebra/vector-spaces.lagda.md
index d9e39a8ed7c..9d253a8aba8 100644
--- a/src/linear-algebra/vector-spaces.lagda.md
+++ b/src/linear-algebra/vector-spaces.lagda.md
@@ -7,10 +7,14 @@ module linear-algebra.vector-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import commutative-algebra.local-commutative-rings
+
+open import foundation.identity-types
 open import foundation.sets
 open import foundation.universe-levels
+
 open import group-theory.abelian-groups
-open import commutative-algebra.local-commutative-rings
+
 open import linear-algebra.left-modules-commutative-rings
 ```
 
@@ -52,4 +56,125 @@ module _
 
   add-Vector-Space : type-Vector-Space → type-Vector-Space → type-Vector-Space
   add-Vector-Space = add-Ab ab-Vector-Space
+
+  zero-Vector-Space : type-Vector-Space
+  zero-Vector-Space = zero-Ab ab-Vector-Space
+
+  neg-Vector-Space : type-Vector-Space → type-Vector-Space
+  neg-Vector-Space = neg-Ab ab-Vector-Space
+
+  mul-Vector-Space :
+    type-Local-Commutative-Ring R → type-Vector-Space → type-Vector-Space
+  mul-Vector-Space =
+    mul-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  associative-add-Vector-Space :
+    (v w x : type-Vector-Space) →
+    add-Vector-Space (add-Vector-Space v w) x ＝
+    add-Vector-Space v (add-Vector-Space w x)
+  associative-add-Vector-Space = associative-add-Ab ab-Vector-Space
+
+  left-unit-law-add-Vector-Space :
+    (v : type-Vector-Space) → add-Vector-Space zero-Vector-Space v ＝ v
+  left-unit-law-add-Vector-Space = left-unit-law-add-Ab ab-Vector-Space
+
+  right-unit-law-add-Vector-Space :
+    (v : type-Vector-Space) → add-Vector-Space v zero-Vector-Space ＝ v
+  right-unit-law-add-Vector-Space = right-unit-law-add-Ab ab-Vector-Space
+
+  left-inverse-law-add-Vector-Space :
+    (v : type-Vector-Space) →
+    add-Vector-Space (neg-Vector-Space v) v ＝ zero-Vector-Space
+  left-inverse-law-add-Vector-Space = left-inverse-law-add-Ab ab-Vector-Space
+
+  right-inverse-law-add-Vector-Space :
+    (v : type-Vector-Space) →
+    add-Vector-Space v (neg-Vector-Space v) ＝ zero-Vector-Space
+  right-inverse-law-add-Vector-Space = right-inverse-law-add-Ab ab-Vector-Space
+
+  commutative-add-Vector-Space :
+    (v w : type-Vector-Space) → add-Vector-Space v w ＝ add-Vector-Space w v
+  commutative-add-Vector-Space = commutative-add-Ab ab-Vector-Space
+
+  left-unit-law-mul-Vector-Space :
+    (v : type-Vector-Space) →
+    mul-Vector-Space (one-Local-Commutative-Ring R) v ＝ v
+  left-unit-law-mul-Vector-Space =
+    left-unit-law-mul-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  left-distributive-mul-add-Vector-Space :
+    (r : type-Local-Commutative-Ring R) (v w : type-Vector-Space) →
+    mul-Vector-Space r (add-Vector-Space v w) ＝
+    add-Vector-Space (mul-Vector-Space r v) (mul-Vector-Space r w)
+  left-distributive-mul-add-Vector-Space =
+    left-distributive-mul-add-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  right-distributive-mul-add-Vector-Space :
+    (r s : type-Local-Commutative-Ring R) (v : type-Vector-Space) →
+    mul-Vector-Space (add-Local-Commutative-Ring R r s) v ＝
+    add-Vector-Space (mul-Vector-Space r v) (mul-Vector-Space s v)
+  right-distributive-mul-add-Vector-Space =
+    right-distributive-mul-add-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  associative-mul-Vector-Space :
+    (r s : type-Local-Commutative-Ring R) (v : type-Vector-Space) →
+    mul-Vector-Space (mul-Local-Commutative-Ring R r s) v ＝
+    mul-Vector-Space r (mul-Vector-Space s v)
+  associative-mul-Vector-Space =
+    associative-mul-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  left-zero-law-mul-Vector-Space :
+    (v : type-Vector-Space) →
+    mul-Vector-Space (zero-Local-Commutative-Ring R) v ＝ zero-Vector-Space
+  left-zero-law-mul-Vector-Space =
+    left-zero-law-mul-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  right-zero-law-mul-Vector-Space :
+    (r : type-Local-Commutative-Ring R) →
+    mul-Vector-Space r zero-Vector-Space ＝ zero-Vector-Space
+  right-zero-law-mul-Vector-Space =
+    right-zero-law-mul-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  left-negative-law-mul-Vector-Space :
+    (r : type-Local-Commutative-Ring R) (v : type-Vector-Space) →
+    mul-Vector-Space (neg-Local-Commutative-Ring R r) v ＝
+    neg-Vector-Space (mul-Vector-Space r v)
+  left-negative-law-mul-Vector-Space =
+    left-negative-law-mul-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  right-negative-law-mul-Vector-Space :
+    (r : type-Local-Commutative-Ring R) (v : type-Vector-Space) →
+    mul-Vector-Space r (neg-Vector-Space v) ＝
+    neg-Vector-Space (mul-Vector-Space r v)
+  right-negative-law-mul-Vector-Space =
+    right-negative-law-mul-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
+
+  mul-neg-one-Vector-Space :
+    (v : type-Vector-Space) →
+    mul-Vector-Space
+      ( neg-Local-Commutative-Ring R (one-Local-Commutative-Ring R))
+      ( v) ＝
+    neg-Vector-Space v
+  mul-neg-one-Vector-Space =
+    mul-neg-one-left-module-Commutative-Ring
+      ( commutative-ring-Local-Commutative-Ring R)
+      ( V)
 ```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index a3ef18a8455..fd03949811f 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -9,7 +9,9 @@ open import real-numbers.absolute-value-closed-intervals-real-numbers public
 open import real-numbers.absolute-value-real-numbers public
 open import real-numbers.accumulation-points-subsets-real-numbers public
 open import real-numbers.addition-lower-dedekind-real-numbers public
+open import real-numbers.addition-negative-real-numbers public
 open import real-numbers.addition-nonnegative-real-numbers public
+open import real-numbers.addition-nonzero-real-numbers public
 open import real-numbers.addition-positive-real-numbers public
 open import real-numbers.addition-real-numbers public
 open import real-numbers.addition-upper-dedekind-real-numbers public
@@ -45,6 +47,7 @@ open import real-numbers.large-multiplicative-monoid-of-real-numbers public
 open import real-numbers.large-ring-of-real-numbers public
 open import real-numbers.limits-sequences-real-numbers public
 open import real-numbers.lipschitz-continuity-multiplication-real-numbers public
+open import real-numbers.local-ring-of-real-numbers public
 open import real-numbers.located-metric-space-of-real-numbers public
 open import real-numbers.lower-dedekind-real-numbers public
 open import real-numbers.maximum-finite-families-real-numbers public
diff --git a/src/real-numbers/addition-negative-real-numbers.lagda.md b/src/real-numbers/addition-negative-real-numbers.lagda.md
index 66f7d5e1e54..c3e3cef113d 100644
--- a/src/real-numbers/addition-negative-real-numbers.lagda.md
+++ b/src/real-numbers/addition-negative-real-numbers.lagda.md
@@ -9,21 +9,22 @@ module real-numbers.addition-negative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import real-numbers.negative-real-numbers
-open import real-numbers.dedekind-real-numbers
-open import foundation.transport-along-identifications
-open import real-numbers.rational-real-numbers
-open import real-numbers.negation-real-numbers
+open import foundation.dependent-pair-types
+open import foundation.disjunction
 open import foundation.functoriality-disjunction
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
 open import real-numbers.addition-positive-real-numbers
-open import foundation.disjunction
 open import real-numbers.addition-real-numbers
-open import real-numbers.strict-inequality-real-numbers
-open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
-open import real-numbers.positive-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.negative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
-open import foundation.universe-levels
-open import foundation.dependent-pair-types
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
+open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/addition-nonzero-real-numbers.lagda.md b/src/real-numbers/addition-nonzero-real-numbers.lagda.md
index 3ab16e4f8b7..e59f5b96f1c 100644
--- a/src/real-numbers/addition-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/addition-nonzero-real-numbers.lagda.md
@@ -9,15 +9,16 @@ module real-numbers.addition-nonzero-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import real-numbers.nonzero-real-numbers
-open import real-numbers.addition-positive-real-numbers
+open import foundation.dependent-pair-types
+open import foundation.disjunction
+open import foundation.functoriality-disjunction
+open import foundation.universe-levels
+
 open import real-numbers.addition-negative-real-numbers
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import foundation.functoriality-disjunction
-open import foundation.disjunction
-open import foundation.dependent-pair-types
-open import foundation.universe-levels
+open import real-numbers.nonzero-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/local-ring-of-real-numbers.lagda.md b/src/real-numbers/local-ring-of-real-numbers.lagda.md
index 282f1dc2237..030a5b42de4 100644
--- a/src/real-numbers/local-ring-of-real-numbers.lagda.md
+++ b/src/real-numbers/local-ring-of-real-numbers.lagda.md
@@ -10,14 +10,16 @@ module real-numbers.local-ring-of-real-numbers where
 
 ```agda
 open import commutative-algebra.local-commutative-rings
-open import real-numbers.addition-real-numbers
-open import real-numbers.nonzero-real-numbers
+
 open import foundation.dependent-pair-types
 open import foundation.functoriality-disjunction
+open import foundation.universe-levels
+
 open import real-numbers.addition-nonzero-real-numbers
+open import real-numbers.addition-real-numbers
 open import real-numbers.large-ring-of-real-numbers
 open import real-numbers.multiplicative-inverses-nonzero-real-numbers
-open import foundation.universe-levels
+open import real-numbers.nonzero-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index 18f7f33b894..e5964a4b3d2 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -9,7 +9,10 @@ module real-numbers.positive-and-negative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.cartesian-product-types
 open import foundation.dependent-pair-types
+open import foundation.logical-equivalences
+open import foundation.negation
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
@@ -78,6 +81,38 @@ neg-ℝ⁻ : {l : Level} → ℝ⁻ l → ℝ⁺ l
 neg-ℝ⁻ (x , is-neg-x) = (neg-ℝ x , neg-is-negative-ℝ x is-neg-x)
 ```
 
+### A real number is negative if and only if its negation is positive
+
+```agda
+abstract
+  is-negative-is-positive-neg-ℝ :
+    {l : Level} (x : ℝ l) → is-positive-ℝ (neg-ℝ x) → is-negative-ℝ x
+  is-negative-is-positive-neg-ℝ x 0<-x =
+    tr is-negative-ℝ (neg-neg-ℝ x) (neg-is-positive-ℝ (neg-ℝ x) 0<-x)
+
+is-negative-iff-neg-is-positive-ℝ :
+  {l : Level} (x : ℝ l) → is-negative-ℝ x ↔ is-positive-ℝ (neg-ℝ x)
+is-negative-iff-neg-is-positive-ℝ x =
+  ( neg-is-negative-ℝ x ,
+    is-negative-is-positive-neg-ℝ x)
+```
+
+### A real number is positive if and only if its negation is negative
+
+```agda
+abstract
+  is-positive-is-negative-neg-ℝ :
+    {l : Level} (x : ℝ l) → is-negative-ℝ (neg-ℝ x) → is-positive-ℝ x
+  is-positive-is-negative-neg-ℝ x -x<0 =
+    tr is-positive-ℝ (neg-neg-ℝ x) (neg-is-negative-ℝ (neg-ℝ x) -x<0)
+
+is-positive-iff-neg-is-negative-ℝ :
+  {l : Level} (x : ℝ l) → is-positive-ℝ x ↔ is-negative-ℝ (neg-ℝ x)
+is-positive-iff-neg-is-negative-ℝ x =
+  ( neg-is-positive-ℝ x ,
+    is-positive-is-negative-neg-ℝ x)
+```
+
 ### If a nonnegative real number `x` is less than a real number `y`, `y` is positive
 
 ```agda
@@ -87,3 +122,13 @@ abstract
     is-positive-ℝ y
   is-positive-le-ℝ⁰⁺ (x , 0≤x) y = concatenate-leq-le-ℝ zero-ℝ x y 0≤x
 ```
+
+### Real numbers are not both negative and nonnegative
+
+```agda
+abstract
+  is-not-negative-and-nonnegative-ℝ :
+    {l : Level} {x : ℝ l} → ¬ (is-negative-ℝ x × is-nonnegative-ℝ x)
+  is-not-negative-and-nonnegative-ℝ {x = x} (x<0 , 0≤x) =
+    not-leq-le-ℝ x zero-ℝ x<0 0≤x
+```

From 71b83ece629c4dc99ca62884730939ebb125f81f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Nov 2025 20:10:28 -0800
Subject: [PATCH 109/134] Add properties for complex vector spaces

---
 .../complex-vector-spaces.lagda.md            | 120 ++++++++++++++++++
 .../real-vector-spaces.lagda.md               |   5 -
 2 files changed, 120 insertions(+), 5 deletions(-)

diff --git a/src/linear-algebra/complex-vector-spaces.lagda.md b/src/linear-algebra/complex-vector-spaces.lagda.md
index fcff11674ad..23c97e5303b 100644
--- a/src/linear-algebra/complex-vector-spaces.lagda.md
+++ b/src/linear-algebra/complex-vector-spaces.lagda.md
@@ -22,3 +22,123 @@ open import linear-algebra.vector-spaces
 ℂ-Vector-Space : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
 ℂ-Vector-Space l1 l2 = Vector-Space l2 (local-commutative-ring-ℂ l1)
 ```
+
+## Properties
+
+```agda
+module _
+  {l1 l2 : Level} (V : ℂ-Vector-Space l1 l2)
+  where
+
+  ab-ℂ-Vector-Space : Ab l2
+  ab-ℂ-Vector-Space = ab-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  set-ℂ-Vector-Space : Set l2
+  set-ℂ-Vector-Space = set-Ab ab-ℂ-Vector-Space
+
+  type-ℂ-Vector-Space : UU l2
+  type-ℂ-Vector-Space = type-Ab ab-ℂ-Vector-Space
+
+  add-ℂ-Vector-Space :
+    type-ℂ-Vector-Space → type-ℂ-Vector-Space → type-ℂ-Vector-Space
+  add-ℂ-Vector-Space = add-Ab ab-ℂ-Vector-Space
+
+  zero-ℂ-Vector-Space : type-ℂ-Vector-Space
+  zero-ℂ-Vector-Space = zero-Ab ab-ℂ-Vector-Space
+
+  neg-ℂ-Vector-Space : type-ℂ-Vector-Space → type-ℂ-Vector-Space
+  neg-ℂ-Vector-Space = neg-Ab ab-ℂ-Vector-Space
+
+  mul-ℂ-Vector-Space : ℂ l1 → type-ℂ-Vector-Space → type-ℂ-Vector-Space
+  mul-ℂ-Vector-Space = mul-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  associative-add-ℂ-Vector-Space :
+    (v w x : type-ℂ-Vector-Space) →
+    add-ℂ-Vector-Space (add-ℂ-Vector-Space v w) x ＝
+    add-ℂ-Vector-Space v (add-ℂ-Vector-Space w x)
+  associative-add-ℂ-Vector-Space = associative-add-Ab ab-ℂ-Vector-Space
+
+  left-unit-law-add-ℂ-Vector-Space :
+    (v : type-ℂ-Vector-Space) → add-ℂ-Vector-Space zero-ℂ-Vector-Space v ＝ v
+  left-unit-law-add-ℂ-Vector-Space = left-unit-law-add-Ab ab-ℂ-Vector-Space
+
+  right-unit-law-add-ℂ-Vector-Space :
+    (v : type-ℂ-Vector-Space) → add-ℂ-Vector-Space v zero-ℂ-Vector-Space ＝ v
+  right-unit-law-add-ℂ-Vector-Space = right-unit-law-add-Ab ab-ℂ-Vector-Space
+
+  left-inverse-law-add-ℂ-Vector-Space :
+    (v : type-ℂ-Vector-Space) →
+    add-ℂ-Vector-Space (neg-ℂ-Vector-Space v) v ＝ zero-ℂ-Vector-Space
+  left-inverse-law-add-ℂ-Vector-Space =
+    left-inverse-law-add-Ab ab-ℂ-Vector-Space
+
+  right-inverse-law-add-ℂ-Vector-Space :
+    (v : type-ℂ-Vector-Space) →
+    add-ℂ-Vector-Space v (neg-ℂ-Vector-Space v) ＝ zero-ℂ-Vector-Space
+  right-inverse-law-add-ℂ-Vector-Space =
+    right-inverse-law-add-Ab ab-ℂ-Vector-Space
+
+  commutative-add-ℂ-Vector-Space :
+    (v w : type-ℂ-Vector-Space) →
+    add-ℂ-Vector-Space v w ＝ add-ℂ-Vector-Space w v
+  commutative-add-ℂ-Vector-Space = commutative-add-Ab ab-ℂ-Vector-Space
+
+  left-unit-law-mul-ℂ-Vector-Space :
+    (v : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space (raise-ℂ l1 one-ℂ) v ＝ v
+  left-unit-law-mul-ℂ-Vector-Space =
+    left-unit-law-mul-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  left-distributive-mul-add-ℂ-Vector-Space :
+    (r : ℂ l1) (v w : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space r (add-ℂ-Vector-Space v w) ＝
+    add-ℂ-Vector-Space (mul-ℂ-Vector-Space r v) (mul-ℂ-Vector-Space r w)
+  left-distributive-mul-add-ℂ-Vector-Space =
+    left-distributive-mul-add-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  right-distributive-mul-add-ℂ-Vector-Space :
+    (r s : ℂ l1) (v : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space (r +ℂ s) v ＝
+    add-ℂ-Vector-Space (mul-ℂ-Vector-Space r v) (mul-ℂ-Vector-Space s v)
+  right-distributive-mul-add-ℂ-Vector-Space =
+    right-distributive-mul-add-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  associative-mul-ℂ-Vector-Space :
+    (r s : ℂ l1) (v : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space (r *ℂ s) v ＝
+    mul-ℂ-Vector-Space r (mul-ℂ-Vector-Space s v)
+  associative-mul-ℂ-Vector-Space =
+    associative-mul-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  left-zero-law-mul-ℂ-Vector-Space :
+    (v : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space (raise-ℂ l1 zero-ℂ) v ＝ zero-ℂ-Vector-Space
+  left-zero-law-mul-ℂ-Vector-Space =
+    left-zero-law-mul-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  right-zero-law-mul-ℂ-Vector-Space :
+    (r : ℂ l1) →
+    mul-ℂ-Vector-Space r zero-ℂ-Vector-Space ＝ zero-ℂ-Vector-Space
+  right-zero-law-mul-ℂ-Vector-Space =
+    right-zero-law-mul-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  left-negative-law-mul-ℂ-Vector-Space :
+    (r : ℂ l1) (v : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space (neg-ℂ r) v ＝
+    neg-ℂ-Vector-Space (mul-ℂ-Vector-Space r v)
+  left-negative-law-mul-ℂ-Vector-Space =
+    left-negative-law-mul-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  right-negative-law-mul-ℂ-Vector-Space :
+    (r : ℂ l1) (v : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space r (neg-ℂ-Vector-Space v) ＝
+    neg-ℂ-Vector-Space (mul-ℂ-Vector-Space r v)
+  right-negative-law-mul-ℂ-Vector-Space =
+    right-negative-law-mul-Vector-Space (local-commutative-ring-ℂ l1) V
+
+  mul-neg-one-ℂ-Vector-Space :
+    (v : type-ℂ-Vector-Space) →
+    mul-ℂ-Vector-Space (neg-ℂ (raise-ℂ l1 one-ℂ)) v ＝ neg-ℂ-Vector-Space v
+  mul-neg-one-ℂ-Vector-Space =
+    mul-neg-one-Vector-Space (local-commutative-ring-ℂ l1) V
+```
diff --git a/src/linear-algebra/real-vector-spaces.lagda.md b/src/linear-algebra/real-vector-spaces.lagda.md
index 98fb181caa8..1ee4b2f6451 100644
--- a/src/linear-algebra/real-vector-spaces.lagda.md
+++ b/src/linear-algebra/real-vector-spaces.lagda.md
@@ -7,11 +7,6 @@ module linear-algebra.real-vector-spaces where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.universe-levels
-
-open import linear-algebra.vector-spaces
-
-open import real-numbers.local-ring-of-real-numbers
 open import foundation.identity-types
 open import foundation.sets
 open import foundation.universe-levels

From 8ac326cf123b89fb2244d50f53fbfa7f43a6967e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 11 Nov 2025 07:55:28 -0800
Subject: [PATCH 110/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/metric-spaces/apartness-located-metric-spaces.lagda.md | 6 +++---
 src/real-numbers/absolute-value-real-numbers.lagda.md      | 2 +-
 src/real-numbers/nonzero-real-numbers.lagda.md             | 2 +-
 3 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/metric-spaces/apartness-located-metric-spaces.lagda.md b/src/metric-spaces/apartness-located-metric-spaces.lagda.md
index 4aeb3403b34..2dc671eb847 100644
--- a/src/metric-spaces/apartness-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/apartness-located-metric-spaces.lagda.md
@@ -47,7 +47,7 @@ module _
   apart-prop-Located-Metric-Space :
     Relation-Prop l2 (type-Located-Metric-Space M)
   apart-prop-Located-Metric-Space x y =
-    ∃ ℚ⁺ (λ ε → ¬' (neighborhood-prop-Located-Metric-Space M ε x y))
+    ∃ ℚ⁺ (λ ε → ¬' neighborhood-prop-Located-Metric-Space M ε x y)
 
   apart-Located-Metric-Space : Relation l2 (type-Located-Metric-Space M)
   apart-Located-Metric-Space =
@@ -101,8 +101,8 @@ module _
     is-cotransitive-apart-Located-Metric-Space x y z x#y =
       let
         motive =
-          apart-prop-Located-Metric-Space M x z ∨
-          apart-prop-Located-Metric-Space M y z
+          ( apart-prop-Located-Metric-Space M x z) ∨
+          ( apart-prop-Located-Metric-Space M y z)
         open do-syntax-trunc-Prop motive
       in do
         (dxy , ¬Ndxy) ← x#y
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index b5ba7ee7b01..6e56c70a08e 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -419,7 +419,7 @@ abstract
         by inv (ap-mul-ℝ (eq-abs-sqrt-square-ℝ x) (eq-abs-sqrt-square-ℝ y))
 ```
 
-### For any `ε : ℚ⁺`, `(abs-ℝ x - ε < x) ∨ (abs-ℝ x - ε < -x)`
+### For any `ε : ℚ⁺`, `|x| - ε < x` or `|x| - ε < -x`
 
 ```agda
 abstract opaque
diff --git a/src/real-numbers/nonzero-real-numbers.lagda.md b/src/real-numbers/nonzero-real-numbers.lagda.md
index 6c785656b35..5512d93592e 100644
--- a/src/real-numbers/nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/nonzero-real-numbers.lagda.md
@@ -48,7 +48,7 @@ equivalently if it is [negative](real-numbers.negative-real-numbers.md)
 
 ```agda
 is-nonzero-prop-ℝ : {l : Level} → ℝ l → Prop l
-is-nonzero-prop-ℝ x = is-negative-prop-ℝ x ∨ is-positive-prop-ℝ x
+is-nonzero-prop-ℝ x = (is-negative-prop-ℝ x) ∨ (is-positive-prop-ℝ x)
 
 is-nonzero-ℝ : {l : Level} → ℝ l → UU l
 is-nonzero-ℝ x = type-Prop (is-nonzero-prop-ℝ x)

From 98a567d06bea89ca0a051cf3966c5baef29d284f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 11 Nov 2025 08:05:11 -0800
Subject: [PATCH 111/134] Fix

---
 src/complex-numbers/apartness-complex-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/complex-numbers/apartness-complex-numbers.lagda.md b/src/complex-numbers/apartness-complex-numbers.lagda.md
index 9a0aac2539b..e58a5ee7283 100644
--- a/src/complex-numbers/apartness-complex-numbers.lagda.md
+++ b/src/complex-numbers/apartness-complex-numbers.lagda.md
@@ -30,7 +30,7 @@ Two [complex numbers](complex-numbers.complex-numbers.md) are
 {{#concept "apart" Agda=apart-ℂ}} if their
 [real](real-numbers.dedekind-real-numbers.md) parts are
 [apart](real-numbers.apartness-real-numbers.md) [or](foundation.disjunction.md)
-their complex parts are [apart].
+their imaginary parts are [apart].
 
 ## Definition
 

From 53e7aeb755e2d6c043d1726da48771f1d455bb23 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Nov 2025 09:45:57 -0800
Subject: [PATCH 112/134] make pre-commit

---
 src/complex-numbers.lagda.md | 7 ++-----
 1 file changed, 2 insertions(+), 5 deletions(-)

diff --git a/src/complex-numbers.lagda.md b/src/complex-numbers.lagda.md
index b1c16d18866..bb5ba406cb7 100644
--- a/src/complex-numbers.lagda.md
+++ b/src/complex-numbers.lagda.md
@@ -4,20 +4,17 @@
 module complex-numbers where
 
 open import complex-numbers.addition-complex-numbers public
-open import complex-numbers.additive-group-of-complex-numbers public
 open import complex-numbers.apartness-complex-numbers public
 open import complex-numbers.complex-numbers public
 open import complex-numbers.conjugation-complex-numbers public
 open import complex-numbers.eisenstein-integers public
 open import complex-numbers.gaussian-integers public
+open import complex-numbers.large-additive-group-of-complex-numbers public
+open import complex-numbers.large-ring-of-complex-numbers public
 open import complex-numbers.magnitude-complex-numbers public
 open import complex-numbers.multiplication-complex-numbers public
 open import complex-numbers.multiplicative-inverses-nonzero-complex-numbers public
 open import complex-numbers.nonzero-complex-numbers public
-open import complex-numbers.ring-of-complex-numbers public
-open import complex-numbers.large-additive-group-of-complex-numbers public
-open import complex-numbers.large-ring-of-complex-numbers public
-open import complex-numbers.multiplication-complex-numbers public
 open import complex-numbers.raising-universe-levels-complex-numbers public
 open import complex-numbers.similarity-complex-numbers public
 ```

From e32bd414a1abb0914672bb24c1bbf2a7d201ae62 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Nov 2025 09:47:38 -0800
Subject: [PATCH 113/134] make pre-commit

---
 src/complex-numbers.lagda.md | 7 ++-----
 1 file changed, 2 insertions(+), 5 deletions(-)

diff --git a/src/complex-numbers.lagda.md b/src/complex-numbers.lagda.md
index b1c16d18866..bb5ba406cb7 100644
--- a/src/complex-numbers.lagda.md
+++ b/src/complex-numbers.lagda.md
@@ -4,20 +4,17 @@
 module complex-numbers where
 
 open import complex-numbers.addition-complex-numbers public
-open import complex-numbers.additive-group-of-complex-numbers public
 open import complex-numbers.apartness-complex-numbers public
 open import complex-numbers.complex-numbers public
 open import complex-numbers.conjugation-complex-numbers public
 open import complex-numbers.eisenstein-integers public
 open import complex-numbers.gaussian-integers public
+open import complex-numbers.large-additive-group-of-complex-numbers public
+open import complex-numbers.large-ring-of-complex-numbers public
 open import complex-numbers.magnitude-complex-numbers public
 open import complex-numbers.multiplication-complex-numbers public
 open import complex-numbers.multiplicative-inverses-nonzero-complex-numbers public
 open import complex-numbers.nonzero-complex-numbers public
-open import complex-numbers.ring-of-complex-numbers public
-open import complex-numbers.large-additive-group-of-complex-numbers public
-open import complex-numbers.large-ring-of-complex-numbers public
-open import complex-numbers.multiplication-complex-numbers public
 open import complex-numbers.raising-universe-levels-complex-numbers public
 open import complex-numbers.similarity-complex-numbers public
 ```

From a1a6a8c03680c8d79fc93a51c01fbdbe03de2ce4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Nov 2025 08:30:00 -0800
Subject: [PATCH 114/134] Update
 src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../cauchy-sequences-metric-spaces.lagda.md   | 32 +++++++++----------
 1 file changed, 16 insertions(+), 16 deletions(-)

diff --git a/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md b/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
index 4fe9086ee64..71c916ccf96 100644
--- a/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
@@ -511,23 +511,23 @@ module _
           in
             ( nat-nonzero-ℕ n⁺ ,
               λ m n≤m →
-                monotonic-neighborhood-Metric-Space M
-                  ( map-cauchy-sequence-cauchy-approximation-Metric-Space M x m)
+              monotonic-neighborhood-Metric-Space M
+                ( map-cauchy-sequence-cauchy-approximation-Metric-Space M x m)
+                ( lim)
+                ( positive-reciprocal-rational-succ-ℕ m)
+                ( ε)
+                ( transitive-le-ℚ _ (reciprocal-rational-ℕ⁺ n⁺) _
+                  ( 1/n⁺<ε)
+                  ( inv-le-ℚ⁺
+                    ( positive-rational-ℕ⁺ n⁺)
+                    ( positive-rational-ℕ⁺ (succ-nonzero-ℕ' m))
+                    ( preserves-le-rational-ℕ
+                      ( le-succ-leq-ℕ _ _ n≤m))))
+                ( saturated-is-limit-cauchy-approximation-Metric-Space M
+                  ( x)
                   ( lim)
-                  ( positive-reciprocal-rational-succ-ℕ m)
-                  ( ε)
-                  ( transitive-le-ℚ _ (reciprocal-rational-ℕ⁺ n⁺) _
-                    ( 1/n⁺<ε)
-                    ( inv-le-ℚ⁺
-                      ( positive-rational-ℕ⁺ n⁺)
-                      ( positive-rational-ℕ⁺ (succ-nonzero-ℕ' m))
-                      ( preserves-le-rational-ℕ
-                        ( le-succ-leq-ℕ _ _ n≤m))))
-                  ( saturated-is-limit-cauchy-approximation-Metric-Space M
-                    ( x)
-                    ( lim)
-                    ( is-lim)
-                    ( positive-reciprocal-rational-succ-ℕ m))))
+                  ( is-lim)
+                  ( positive-reciprocal-rational-succ-ℕ m))))
 ```
 
 ### If the Cauchy sequence associated with a Cauchy approximation has a limit modulus at `l`, its associated approximation converges to `l`

From 1bcb06aebe1a3a39abbcf29c11c8655b52f02f00 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Nov 2025 08:30:09 -0800
Subject: [PATCH 115/134] Update
 src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../accumulation-points-subsets-located-metric-spaces.lagda.md  | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
index 475db6744c7..4a1a8ba60fd 100644
--- a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
@@ -43,7 +43,7 @@ open import metric-spaces.subspaces-metric-spaces
 ## Idea
 
 An
-{{#concept "accumulation point" WDID=Q858223 WD="limit point" Disambiguation="of a metric space" Agda=accumulation-point-subset-Located-Metric-Space}}
+{{#concept "accumulation point" WDID=Q858223 WD="limit point" Disambiguation="of a subset of a located metric space" Agda=accumulation-point-subset-Located-Metric-Space}}
 of a subset `S` of a
 [located metric space](metric-spaces.located-metric-spaces.md) `X` is a point
 `x : X` such that there exists a

From a9bacddd1db6792e51213ec46b51f20ee460e020 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Nov 2025 08:30:20 -0800
Subject: [PATCH 116/134] Update
 src/real-numbers/binary-maximum-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/real-numbers/binary-maximum-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 17dfd98f51b..b225ca78909 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -384,7 +384,7 @@ module _
           ( is-located-lower-upper-cut-ℝ x q<r)
 ```
 
-### If `x < z` and `y < z`, then `max-ℝ x y < x`
+### If `x < z` and `y < z`, then `max-ℝ x y < z`
 
 ```agda
 abstract

From f5e8fba967d9cbba040a613fb455bf26969b6584 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Nov 2025 08:53:31 -0800
Subject: [PATCH 117/134] Progress

---
 ...mulation-points-subsets-located-metric-spaces.lagda.md | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

diff --git a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
index 475db6744c7..4c1b343680c 100644
--- a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
@@ -50,7 +50,8 @@ of a subset `S` of a
 [Cauchy approximation](metric-spaces.cauchy-approximations-metric-spaces.md) `a`
 with [limit](metric-spaces.limits-of-cauchy-approximations-metric-spaces.md) `x`
 such that for every `ε : ℚ⁺`, `a ε` is in `S` and is
-[apart](metric-spaces.apartness-located-metric-spaces.md) from `x`.
+[apart](metric-spaces.apartness-located-metric-spaces.md) from `x`. In
+particular, this implies an accumulation point is not isolated.
 
 ## Definition
 
@@ -313,3 +314,8 @@ module _
         ( S)
         ( x))
 ```
+
+## External links
+
+- [Accumulation point](https://en.wikipedia.org/wiki/Accumulation_point) on
+  Wikipedia

From 08bb7e36150e407b449c46ca27195c99b471cc5f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Nov 2025 15:09:52 -0800
Subject: [PATCH 118/134] Fixes

---
 ...uchy-approximations-metric-spaces.lagda.md |   5 +-
 ...per-closed-intervals-real-numbers.lagda.md | 171 +++++++++++++++++-
 2 files changed, 172 insertions(+), 4 deletions(-)

diff --git a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
index 6757f63e605..d9a5fe6ea14 100644
--- a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
@@ -156,13 +156,14 @@ module _
   where
 
   abstract
-    map-short-function-is-limit-cauchy-approximation-Metric-Space :
+    preserves-limit-cauchy-approximation-map-short-function-Metric-Space :
       is-limit-cauchy-approximation-Metric-Space A a lim →
       is-limit-cauchy-approximation-Metric-Space
         ( B)
         ( map-short-function-cauchy-approximation-Metric-Space A B f a)
         ( map-short-function-Metric-Space A B f lim)
-    map-short-function-is-limit-cauchy-approximation-Metric-Space is-lim-a ε δ =
+    preserves-limit-cauchy-approximation-map-short-function-Metric-Space
+      is-lim-a ε δ =
       is-short-map-short-function-Metric-Space A B
         ( f)
         ( ε +ℚ⁺ δ)
diff --git a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
index 51360ec483c..d52fd8ef171 100644
--- a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
@@ -185,6 +185,173 @@ Note that this cannot be made more universe-polymorphic.
 
 ```agda
 abstract
+  is-accumulation-point-le-upper-bound-proper-closed-interval-ℝ :
+    {l : Level} ([a,b] : proper-closed-interval-ℝ l l) →
+    (x : ℝ l) → is-in-proper-closed-interval-ℝ [a,b] x →
+    le-ℝ x (upper-bound-proper-closed-interval-ℝ [a,b]) →
+    is-accumulation-point-subset-ℝ
+      ( subtype-proper-closed-interval-ℝ l [a,b])
+      ( x)
+  is-accumulation-point-le-upper-bound-proper-closed-interval-ℝ
+    {l} [a,b]@(a , b , a<b) x (a≤x , x≤b) x<b =
+    let
+      open inequality-reasoning-Large-Poset ℝ-Large-Poset
+      motive =
+        is-accumulation-point-prop-subset-ℝ
+          ( subtype-proper-closed-interval-ℝ _ [a,b])
+          ( x)
+      short-clamp-add =
+        comp-short-function-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-ℝ lzero)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( comp-short-function-Metric-Space
+            ( metric-space-ℝ lzero)
+            ( metric-space-ℝ l)
+            ( metric-space-proper-closed-interval-ℝ l [a,b])
+            ( short-clamp-proper-closed-interval-ℝ [a,b])
+            ( short-left-add-ℝ x))
+          ( short-isometry-Metric-Space
+            ( metric-space-ℚ)
+            ( metric-space-ℝ lzero)
+            ( isometry-metric-space-real-ℚ))
+      short-inclusion =
+        short-inclusion-subspace-Metric-Space
+          ( metric-space-ℝ l)
+          ( subtype-proper-closed-interval-ℝ l [a,b])
+      approx-clamp-add =
+        map-short-function-cauchy-approximation-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( short-clamp-add)
+          ( cauchy-approximation-rational-ℚ⁺)
+    in
+      intro-exists
+        ( approx-clamp-add)
+        ( ( λ ε →
+            apart-located-metric-space-apart-ℝ _ _
+              ( apart-le-ℝ'
+                ( concatenate-le-leq-ℝ
+                  ( x)
+                  ( min-ℝ b (x +ℝ real-ℚ⁺ ε))
+                  ( max-ℝ a (min-ℝ b (x +ℝ real-ℚ⁺ ε)))
+                  ( le-min-le-le-ℝ
+                    ( x<b)
+                    ( le-left-add-real-ℝ⁺ x (positive-real-ℚ⁺ ε)))
+                  ( leq-right-max-ℝ _ _)))) ,
+          tr
+            ( is-limit-cauchy-approximation-Metric-Space
+              ( metric-space-ℝ l)
+              ( map-short-function-cauchy-approximation-Metric-Space
+                ( metric-space-proper-closed-interval-ℝ l [a,b])
+                ( metric-space-ℝ l)
+                ( short-inclusion)
+                ( approx-clamp-add)))
+            ( equational-reasoning
+              max-ℝ a (min-ℝ b (x +ℝ zero-ℝ))
+              ＝ max-ℝ a (min-ℝ b x)
+                by ap-max-ℝ refl (ap-min-ℝ refl (right-unit-law-add-ℝ x))
+              ＝ max-ℝ a x
+                by ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
+              ＝ x
+                by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
+            ( preserves-limit-cauchy-approximation-map-short-function-Metric-Space
+              ( metric-space-ℚ)
+              ( metric-space-ℝ l)
+              ( comp-short-function-Metric-Space
+                ( metric-space-ℚ)
+                ( metric-space-proper-closed-interval-ℝ l [a,b])
+                ( metric-space-ℝ l)
+                ( short-inclusion)
+                ( short-clamp-add))
+              ( cauchy-approximation-rational-ℚ⁺)
+              ( zero-ℚ)
+              ( is-zero-limit-rational-ℚ⁺)))
+
+  is-accumulation-point-le-lower-bound-proper-closed-interval-ℝ :
+    {l : Level} ([a,b] : proper-closed-interval-ℝ l l) →
+    (x : ℝ l) → is-in-proper-closed-interval-ℝ [a,b] x →
+    le-ℝ (lower-bound-proper-closed-interval-ℝ [a,b]) x →
+    is-accumulation-point-subset-ℝ
+      ( subtype-proper-closed-interval-ℝ l [a,b])
+      ( x)
+  is-accumulation-point-le-lower-bound-proper-closed-interval-ℝ
+    {l} [a,b]@(a , b , a<b) x (a≤x , x≤b) a<x =
+    let
+      open inequality-reasoning-Large-Poset ℝ-Large-Poset
+      motive =
+        is-accumulation-point-prop-subset-ℝ
+          ( subtype-proper-closed-interval-ℝ _ [a,b])
+          ( x)
+      short-clamp-diff =
+        comp-short-function-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-ℝ lzero)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( comp-short-function-Metric-Space
+            ( metric-space-ℝ lzero)
+            ( metric-space-ℝ l)
+            ( metric-space-proper-closed-interval-ℝ l [a,b])
+            ( short-clamp-proper-closed-interval-ℝ [a,b])
+            ( short-diff-ℝ x))
+          ( short-isometry-Metric-Space
+            ( metric-space-ℚ)
+            ( metric-space-ℝ lzero)
+            ( isometry-metric-space-real-ℚ))
+      short-inclusion =
+        short-inclusion-subspace-Metric-Space
+          ( metric-space-ℝ l)
+          ( subtype-proper-closed-interval-ℝ l [a,b])
+      approx-clamp-diff =
+        map-short-function-cauchy-approximation-Metric-Space
+          ( metric-space-ℚ)
+          ( metric-space-proper-closed-interval-ℝ l [a,b])
+          ( short-clamp-diff)
+          ( cauchy-approximation-rational-ℚ⁺)
+    in
+      intro-exists
+        ( approx-clamp-diff)
+        ( ( λ ε →
+            apart-located-metric-space-apart-ℝ _ _
+              ( apart-le-ℝ
+                ( le-max-le-le-ℝ
+                  ( a<x)
+                  ( concatenate-leq-le-ℝ
+                    ( min-ℝ b (x -ℝ real-ℚ⁺ ε))
+                    ( x -ℝ real-ℚ⁺ ε)
+                    ( x)
+                    ( leq-right-min-ℝ _ _)
+                    ( le-diff-real-ℝ⁺ x (positive-real-ℚ⁺ ε)))))) ,
+          tr
+            ( is-limit-cauchy-approximation-Metric-Space
+              ( metric-space-ℝ l)
+              ( map-short-function-cauchy-approximation-Metric-Space
+                ( metric-space-proper-closed-interval-ℝ l [a,b])
+                ( metric-space-ℝ l)
+                ( short-inclusion)
+                ( approx-clamp-diff)))
+            ( equational-reasoning
+              max-ℝ a (min-ℝ b (x -ℝ zero-ℝ))
+              ＝ max-ℝ a (min-ℝ b x)
+                by ap-max-ℝ refl (ap-min-ℝ refl (right-unit-law-diff-ℝ x))
+              ＝ max-ℝ a x
+                by
+                  ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
+              ＝ x
+                by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
+            ( preserves-limit-cauchy-approximation-map-short-function-Metric-Space
+              ( metric-space-ℚ)
+              ( metric-space-ℝ l)
+              ( comp-short-function-Metric-Space
+                ( metric-space-ℚ)
+                ( metric-space-proper-closed-interval-ℝ l [a,b])
+                ( metric-space-ℝ l)
+                ( short-inclusion)
+                ( short-clamp-diff))
+              ( cauchy-approximation-rational-ℚ⁺)
+              ( zero-ℚ)
+              ( is-zero-limit-rational-ℚ⁺)))
+
   is-accumulation-point-is-in-proper-closed-interval-ℝ :
     {l : Level} ([a,b] : proper-closed-interval-ℝ l l) →
     (x : ℝ l) → is-in-proper-closed-interval-ℝ [a,b] x →
@@ -279,7 +446,7 @@ abstract
                       ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
                   ＝ x
                     by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
-                ( map-short-function-is-limit-cauchy-approximation-Metric-Space
+                ( preserves-limit-cauchy-approximation-map-short-function-Metric-Space
                   ( metric-space-ℚ)
                   ( metric-space-ℝ l)
                   ( comp-short-function-Metric-Space
@@ -321,7 +488,7 @@ abstract
                     by ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
                   ＝ x
                     by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
-                ( map-short-function-is-limit-cauchy-approximation-Metric-Space
+                ( preserves-limit-cauchy-approximation-map-short-function-Metric-Space
                   ( metric-space-ℚ)
                   ( metric-space-ℝ l)
                   ( comp-short-function-Metric-Space

From db68b9d81c5b8e2d1ee0277d68b7dbe5c46138e5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Nov 2025 15:12:23 -0800
Subject: [PATCH 119/134] Simplify

---
 ...per-closed-intervals-real-numbers.lagda.md | 156 ++----------------
 1 file changed, 14 insertions(+), 142 deletions(-)

diff --git a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
index d52fd8ef171..fa15226a2a3 100644
--- a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
@@ -359,146 +359,18 @@ abstract
       ( subtype-proper-closed-interval-ℝ l [a,b])
       ( x)
   is-accumulation-point-is-in-proper-closed-interval-ℝ
-    {l} [a,b]@(a , b , a<b) x (a≤x , x≤b) =
-    let
-      open inequality-reasoning-Large-Poset ℝ-Large-Poset
-      motive =
-        is-accumulation-point-prop-subset-ℝ
-          ( subtype-proper-closed-interval-ℝ _ [a,b])
-          ( x)
-      short-clamp-diff =
-        comp-short-function-Metric-Space
-          ( metric-space-ℚ)
-          ( metric-space-ℝ lzero)
-          ( metric-space-proper-closed-interval-ℝ l [a,b])
-          ( comp-short-function-Metric-Space
-            ( metric-space-ℝ lzero)
-            ( metric-space-ℝ l)
-            ( metric-space-proper-closed-interval-ℝ l [a,b])
-            ( short-clamp-proper-closed-interval-ℝ [a,b])
-            ( short-diff-ℝ x))
-          ( short-isometry-Metric-Space
-            ( metric-space-ℚ)
-            ( metric-space-ℝ lzero)
-            ( isometry-metric-space-real-ℚ))
-      short-clamp-add =
-        comp-short-function-Metric-Space
-          ( metric-space-ℚ)
-          ( metric-space-ℝ lzero)
-          ( metric-space-proper-closed-interval-ℝ l [a,b])
-          ( comp-short-function-Metric-Space
-            ( metric-space-ℝ lzero)
-            ( metric-space-ℝ l)
-            ( metric-space-proper-closed-interval-ℝ l [a,b])
-            ( short-clamp-proper-closed-interval-ℝ [a,b])
-            ( short-left-add-ℝ x))
-          ( short-isometry-Metric-Space
-            ( metric-space-ℚ)
-            ( metric-space-ℝ lzero)
-            ( isometry-metric-space-real-ℚ))
-      short-inclusion =
-        short-inclusion-subspace-Metric-Space
-          ( metric-space-ℝ l)
-          ( subtype-proper-closed-interval-ℝ l [a,b])
-      approx-clamp-diff =
-        map-short-function-cauchy-approximation-Metric-Space
-          ( metric-space-ℚ)
-          ( metric-space-proper-closed-interval-ℝ l [a,b])
-          ( short-clamp-diff)
-          ( cauchy-approximation-rational-ℚ⁺)
-      approx-clamp-add =
-        map-short-function-cauchy-approximation-Metric-Space
-          ( metric-space-ℚ)
-          ( metric-space-proper-closed-interval-ℝ l [a,b])
-          ( short-clamp-add)
-          ( cauchy-approximation-rational-ℚ⁺)
-    in
-      elim-disjunction
-        ( motive)
-        ( λ a<x →
-          intro-exists
-            ( approx-clamp-diff)
-            ( ( λ ε →
-                apart-located-metric-space-apart-ℝ _ _
-                  ( apart-le-ℝ
-                    ( le-max-le-le-ℝ
-                      ( a<x)
-                      ( concatenate-leq-le-ℝ
-                        ( min-ℝ b (x -ℝ real-ℚ⁺ ε))
-                        ( x -ℝ real-ℚ⁺ ε)
-                        ( x)
-                        ( leq-right-min-ℝ _ _)
-                        ( le-diff-real-ℝ⁺ x (positive-real-ℚ⁺ ε)))))) ,
-              tr
-                ( is-limit-cauchy-approximation-Metric-Space
-                  ( metric-space-ℝ l)
-                  ( map-short-function-cauchy-approximation-Metric-Space
-                    ( metric-space-proper-closed-interval-ℝ l [a,b])
-                    ( metric-space-ℝ l)
-                    ( short-inclusion)
-                    ( approx-clamp-diff)))
-                ( equational-reasoning
-                  max-ℝ a (min-ℝ b (x -ℝ zero-ℝ))
-                  ＝ max-ℝ a (min-ℝ b x)
-                    by ap-max-ℝ refl (ap-min-ℝ refl (right-unit-law-diff-ℝ x))
-                  ＝ max-ℝ a x
-                    by
-                      ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
-                  ＝ x
-                    by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
-                ( preserves-limit-cauchy-approximation-map-short-function-Metric-Space
-                  ( metric-space-ℚ)
-                  ( metric-space-ℝ l)
-                  ( comp-short-function-Metric-Space
-                    ( metric-space-ℚ)
-                    ( metric-space-proper-closed-interval-ℝ l [a,b])
-                    ( metric-space-ℝ l)
-                    ( short-inclusion)
-                    ( short-clamp-diff))
-                  ( cauchy-approximation-rational-ℚ⁺)
-                  ( zero-ℚ)
-                  ( is-zero-limit-rational-ℚ⁺))))
-        ( λ x<b →
-          intro-exists
-            ( approx-clamp-add)
-            ( ( λ ε →
-                apart-located-metric-space-apart-ℝ _ _
-                  ( apart-le-ℝ'
-                    ( concatenate-le-leq-ℝ
-                      ( x)
-                      ( min-ℝ b (x +ℝ real-ℚ⁺ ε))
-                      ( max-ℝ a (min-ℝ b (x +ℝ real-ℚ⁺ ε)))
-                      ( le-min-le-le-ℝ
-                        ( x<b)
-                        ( le-left-add-real-ℝ⁺ x (positive-real-ℚ⁺ ε)))
-                      ( leq-right-max-ℝ _ _)))) ,
-              tr
-                ( is-limit-cauchy-approximation-Metric-Space
-                  ( metric-space-ℝ l)
-                  ( map-short-function-cauchy-approximation-Metric-Space
-                    ( metric-space-proper-closed-interval-ℝ l [a,b])
-                    ( metric-space-ℝ l)
-                    ( short-inclusion)
-                    ( approx-clamp-add)))
-                ( equational-reasoning
-                  max-ℝ a (min-ℝ b (x +ℝ zero-ℝ))
-                  ＝ max-ℝ a (min-ℝ b x)
-                    by ap-max-ℝ refl (ap-min-ℝ refl (right-unit-law-add-ℝ x))
-                  ＝ max-ℝ a x
-                    by ap-max-ℝ refl (eq-sim-ℝ (right-leq-left-min-ℝ x≤b))
-                  ＝ x
-                    by eq-sim-ℝ (left-leq-right-max-ℝ a≤x))
-                ( preserves-limit-cauchy-approximation-map-short-function-Metric-Space
-                  ( metric-space-ℚ)
-                  ( metric-space-ℝ l)
-                  ( comp-short-function-Metric-Space
-                    ( metric-space-ℚ)
-                    ( metric-space-proper-closed-interval-ℝ l [a,b])
-                    ( metric-space-ℝ l)
-                    ( short-inclusion)
-                    ( short-clamp-add))
-                  ( cauchy-approximation-rational-ℚ⁺)
-                  ( zero-ℚ)
-                  ( is-zero-limit-rational-ℚ⁺))))
-        ( cotransitive-le-ℝ a b x a<b)
+    {l} [a,b]@(a , b , a<b) x a≤x≤b =
+    elim-disjunction
+      ( is-accumulation-point-prop-subset-ℝ
+        ( subtype-proper-closed-interval-ℝ _ [a,b])
+        ( x))
+      ( is-accumulation-point-le-lower-bound-proper-closed-interval-ℝ
+        ( [a,b])
+        ( x)
+        ( a≤x≤b))
+      ( is-accumulation-point-le-upper-bound-proper-closed-interval-ℝ
+        ( [a,b])
+        ( x)
+        ( a≤x≤b))
+      ( cotransitive-le-ℝ a b x a<b)
 ```

From 4a1d1c9350613509af81c8d574fa642b839186ae Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 14 Nov 2025 16:53:27 -0800
Subject: [PATCH 120/134] Fix braces

---
 src/complex-numbers/magnitude-complex-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index 22a709c6efb..ee2b26b10c4 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -39,7 +39,7 @@ open import real-numbers.squares-real-numbers
 The
 {{#concept "magnitude" WD="magnitude of a complex number" WDID=Q3317982 Agda=magnitude-ℂ}}
 of a [complex number](complex-numbers.complex-numbers.md) `a + bi` is defined as
-$$\sqrt{a^2 + b^2}}$$.
+$$\sqrt{a^2 + b^2}$$.
 
 ## Definition
 

From ecfd72dae26027ec681f5dccb5b4de183f7c8f39 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 15 Nov 2025 09:16:15 -0800
Subject: [PATCH 121/134] Update
 src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../accumulation-points-subsets-located-metric-spaces.lagda.md  | 2 --
 1 file changed, 2 deletions(-)

diff --git a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
index ebf9d7fa5eb..8f84ddb5099 100644
--- a/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
+++ b/src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
@@ -209,8 +209,6 @@ module _
 
 ### If `x` is an accumulation point of `S`, it is a sequential accumulation point of `S`
 
-The converse has yet to be proved.
-
 ```agda
 module _
   {l1 l2 l3 : Level}

From 1c592429b69c08410c7ad71f37414c6df2a7ce27 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 15 Nov 2025 09:29:18 -0800
Subject: [PATCH 122/134] Fix imports

---
 src/real-numbers/maximum-finite-families-real-numbers.lagda.md   | 1 +
 src/real-numbers/proper-closed-intervals-real-numbers.lagda.md   | 1 +
 .../short-function-binary-maximum-real-numbers.lagda.md          | 1 +
 src/real-numbers/suprema-families-real-numbers.lagda.md          | 1 +
 4 files changed, 4 insertions(+)

diff --git a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
index 543950300ed..62fe4c3cebf 100644
--- a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
@@ -33,6 +33,7 @@ open import order-theory.joins-finite-families-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
 open import order-theory.upper-bounds-large-posets
 
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
index fa15226a2a3..21f5ab7e363 100644
--- a/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
@@ -32,6 +32,7 @@ open import metric-spaces.subspaces-metric-spaces
 open import order-theory.large-posets
 
 open import real-numbers.accumulation-points-subsets-real-numbers
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.apartness-real-numbers
 open import real-numbers.binary-maximum-real-numbers
diff --git a/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
index d0716d0b192..70621782e2b 100644
--- a/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
@@ -21,6 +21,7 @@ open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import order-theory.least-upper-bounds-large-posets
 
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/suprema-families-real-numbers.lagda.md b/src/real-numbers/suprema-families-real-numbers.lagda.md
index d51a2e2a299..ca69a732052 100644
--- a/src/real-numbers/suprema-families-real-numbers.lagda.md
+++ b/src/real-numbers/suprema-families-real-numbers.lagda.md
@@ -33,6 +33,7 @@ open import foundation.universe-levels
 open import order-theory.least-upper-bounds-large-posets
 open import order-theory.upper-bounds-large-posets
 
+open import real-numbers.addition-positive-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers

From f09fd5f32347306e2ef6cd86a495f733f0403421 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 15 Nov 2025 09:35:18 -0800
Subject: [PATCH 123/134] The reals are a vector space over themselves

---
 src/linear-algebra/real-vector-spaces.lagda.md |  9 +++++++++
 src/linear-algebra/vector-spaces.lagda.md      | 12 +++++++++++-
 2 files changed, 20 insertions(+), 1 deletion(-)

diff --git a/src/linear-algebra/real-vector-spaces.lagda.md b/src/linear-algebra/real-vector-spaces.lagda.md
index 1ee4b2f6451..8a536fde9e7 100644
--- a/src/linear-algebra/real-vector-spaces.lagda.md
+++ b/src/linear-algebra/real-vector-spaces.lagda.md
@@ -159,3 +159,12 @@ module _
   mul-neg-one-ℝ-Vector-Space =
     mul-neg-one-Vector-Space (local-commutative-ring-ℝ l1) V
 ```
+
+### The real numbers are a real vector space
+
+```agda
+real-vector-space-ℝ : (l : Level) → ℝ-Vector-Space l (lsuc l)
+real-vector-space-ℝ l =
+  vector-space-local-commutative-ring-Local-Commutative-Ring
+    ( local-commutative-ring-ℝ l)
+```
diff --git a/src/linear-algebra/vector-spaces.lagda.md b/src/linear-algebra/vector-spaces.lagda.md
index 9d253a8aba8..dba90b457b5 100644
--- a/src/linear-algebra/vector-spaces.lagda.md
+++ b/src/linear-algebra/vector-spaces.lagda.md
@@ -1,4 +1,4 @@
-# Vector spaces
+m# Vector spaces
 
 ```agda
 module linear-algebra.vector-spaces where
@@ -178,3 +178,13 @@ module _
       ( commutative-ring-Local-Commutative-Ring R)
       ( V)
 ```
+
+### Any local commutative ring is a vector space over itself
+
+```agda
+vector-space-local-commutative-ring-Local-Commutative-Ring :
+  {l : Level} (R : Local-Commutative-Ring l) → Vector-Space l R
+vector-space-local-commutative-ring-Local-Commutative-Ring R =
+  left-module-commutative-ring-Commutative-Ring
+    ( commutative-ring-Local-Commutative-Ring R)
+```

From 5ef21f217a7811ecfcadf312709ad0412bdf5c0d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 15 Nov 2025 10:08:42 -0800
Subject: [PATCH 124/134] Fix title in vector-spaces

---
 src/linear-algebra/vector-spaces.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/linear-algebra/vector-spaces.lagda.md b/src/linear-algebra/vector-spaces.lagda.md
index dba90b457b5..55c407c4c5b 100644
--- a/src/linear-algebra/vector-spaces.lagda.md
+++ b/src/linear-algebra/vector-spaces.lagda.md
@@ -1,4 +1,4 @@
-m# Vector spaces
+# Vector spaces
 
 ```agda
 module linear-algebra.vector-spaces where

From de71c712bcb74768dfb0237dd30565ba98c5c06a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 15 Nov 2025 19:29:03 -0800
Subject: [PATCH 125/134] Add more cross links

---
 src/linear-algebra/real-vector-spaces.lagda.md | 4 ++++
 src/linear-algebra/vector-spaces.lagda.md      | 8 ++++++++
 2 files changed, 12 insertions(+)

diff --git a/src/linear-algebra/real-vector-spaces.lagda.md b/src/linear-algebra/real-vector-spaces.lagda.md
index 8a536fde9e7..21ac77abb4d 100644
--- a/src/linear-algebra/real-vector-spaces.lagda.md
+++ b/src/linear-algebra/real-vector-spaces.lagda.md
@@ -168,3 +168,7 @@ real-vector-space-ℝ l =
   vector-space-local-commutative-ring-Local-Commutative-Ring
     ( local-commutative-ring-ℝ l)
 ```
+
+## See also
+
+- [Vector spaces](linear-algebra.vector-spaces.md)
diff --git a/src/linear-algebra/vector-spaces.lagda.md b/src/linear-algebra/vector-spaces.lagda.md
index 55c407c4c5b..c7c658f41f8 100644
--- a/src/linear-algebra/vector-spaces.lagda.md
+++ b/src/linear-algebra/vector-spaces.lagda.md
@@ -188,3 +188,11 @@ vector-space-local-commutative-ring-Local-Commutative-Ring R =
   left-module-commutative-ring-Commutative-Ring
     ( commutative-ring-Local-Commutative-Ring R)
 ```
+
+## See also
+
+- [Real vector spaces](linear-algebra.real-vector-spaces.md)
+
+## External links
+
+- [Vector space](https://en.wikipedia.org/wiki/Vector_space) on Wikipedia

From ef5fb9d35c9b1b57cd1252e4b1d0a5710e106d99 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 09:23:38 -0800
Subject: [PATCH 126/134] Update
 src/complex-numbers/apartness-complex-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/complex-numbers/apartness-complex-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/complex-numbers/apartness-complex-numbers.lagda.md b/src/complex-numbers/apartness-complex-numbers.lagda.md
index e58a5ee7283..4b1cddfa5d9 100644
--- a/src/complex-numbers/apartness-complex-numbers.lagda.md
+++ b/src/complex-numbers/apartness-complex-numbers.lagda.md
@@ -41,7 +41,7 @@ module _
 
   apart-prop-ℂ : Prop (l1 ⊔ l2)
   apart-prop-ℂ =
-    apart-prop-ℝ (re-ℂ z) (re-ℂ w) ∨ apart-prop-ℝ (im-ℂ z) (im-ℂ w)
+    (apart-prop-ℝ (re-ℂ z) (re-ℂ w)) ∨ (apart-prop-ℝ (im-ℂ z) (im-ℂ w))
 
   apart-ℂ : UU (l1 ⊔ l2)
   apart-ℂ = type-Prop apart-prop-ℂ

From f734c77ce6295fd995ac7eec523d337d0646dfd1 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 09:23:45 -0800
Subject: [PATCH 127/134] Update
 src/complex-numbers/magnitude-complex-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/complex-numbers/magnitude-complex-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index ee2b26b10c4..e35f0098103 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -39,7 +39,7 @@ open import real-numbers.squares-real-numbers
 The
 {{#concept "magnitude" WD="magnitude of a complex number" WDID=Q3317982 Agda=magnitude-ℂ}}
 of a [complex number](complex-numbers.complex-numbers.md) `a + bi` is defined as
-$$\sqrt{a^2 + b^2}$$.
+$\sqrt{a^2 + b^2}$.
 
 ## Definition
 

From b069ea8b432e1b8c50a27cff7f5ceef073c61db2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 09:23:52 -0800
Subject: [PATCH 128/134] Update
 src/complex-numbers/magnitude-complex-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/complex-numbers/magnitude-complex-numbers.lagda.md | 3 ---
 1 file changed, 3 deletions(-)

diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index e35f0098103..ea3a19a6a40 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -51,9 +51,6 @@ nonnegative-squared-magnitude-ℂ (a +iℂ b) =
 squared-magnitude-ℂ : {l : Level} → ℂ l → ℝ l
 squared-magnitude-ℂ z = real-ℝ⁰⁺ (nonnegative-squared-magnitude-ℂ z)
 
-∥_∥²ℂ : {l : Level} → ℂ l → ℝ l
-∥ z ∥²ℂ = squared-magnitude-ℂ z
-
 nonnegative-magnitude-ℂ : {l : Level} → ℂ l → ℝ⁰⁺ l
 nonnegative-magnitude-ℂ z = sqrt-ℝ⁰⁺ (nonnegative-squared-magnitude-ℂ z)
 

From e090719216820858666635b223f53449d240df5d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 09:24:09 -0800
Subject: [PATCH 129/134] Update
 src/complex-numbers/magnitude-complex-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/complex-numbers/magnitude-complex-numbers.lagda.md | 3 ---
 1 file changed, 3 deletions(-)

diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index ea3a19a6a40..3f1410f24d2 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -56,9 +56,6 @@ nonnegative-magnitude-ℂ z = sqrt-ℝ⁰⁺ (nonnegative-squared-magnitude-ℂ
 
 magnitude-ℂ : {l : Level} → ℂ l → ℝ l
 magnitude-ℂ z = real-ℝ⁰⁺ (nonnegative-magnitude-ℂ z)
-
-∥_∥ℂ : {l : Level} → ℂ l → ℝ l
-∥ z ∥ℂ = magnitude-ℂ z
 ```
 
 ## Properties

From 6783b3d1295cabd15fdea1f8fc810d8f2c03027e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 09:24:32 -0800
Subject: [PATCH 130/134] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/complex-numbers/apartness-complex-numbers.lagda.md | 2 +-
 src/complex-numbers/magnitude-complex-numbers.lagda.md | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/complex-numbers/apartness-complex-numbers.lagda.md b/src/complex-numbers/apartness-complex-numbers.lagda.md
index 4b1cddfa5d9..03c9a2a2512 100644
--- a/src/complex-numbers/apartness-complex-numbers.lagda.md
+++ b/src/complex-numbers/apartness-complex-numbers.lagda.md
@@ -27,7 +27,7 @@ open import real-numbers.apartness-real-numbers
 ## Idea
 
 Two [complex numbers](complex-numbers.complex-numbers.md) are
-{{#concept "apart" Agda=apart-ℂ}} if their
+{{#concept "apart" Disambiguation="complex numbers" Agda=apart-ℂ}} if their
 [real](real-numbers.dedekind-real-numbers.md) parts are
 [apart](real-numbers.apartness-real-numbers.md) [or](foundation.disjunction.md)
 their imaginary parts are [apart].
diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index 3f1410f24d2..f151528dc49 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -38,7 +38,7 @@ open import real-numbers.squares-real-numbers
 
 The
 {{#concept "magnitude" WD="magnitude of a complex number" WDID=Q3317982 Agda=magnitude-ℂ}}
-of a [complex number](complex-numbers.complex-numbers.md) `a + bi` is defined as
+of a [complex number](complex-numbers.complex-numbers.md) $a + bi$ is defined as
 $\sqrt{a^2 + b^2}$.
 
 ## Definition

From c02c5933d06269342195309f5f37beaf43baefda Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 09:35:03 -0800
Subject: [PATCH 131/134] Respond to review comments

---
 src/complex-numbers/nonzero-complex-numbers.lagda.md | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/src/complex-numbers/nonzero-complex-numbers.lagda.md b/src/complex-numbers/nonzero-complex-numbers.lagda.md
index a10fec58972..9f978a56f18 100644
--- a/src/complex-numbers/nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/nonzero-complex-numbers.lagda.md
@@ -64,10 +64,11 @@ complex-nonzero-ℂ = pr1
 ```agda
 abstract
   is-positive-squared-magnitude-is-nonzero-ℂ :
-    {l : Level} (z : ℂ l) → is-nonzero-ℂ z → is-positive-ℝ ∥ z ∥²ℂ
+    {l : Level} (z : ℂ l) → is-nonzero-ℂ z →
+    is-positive-ℝ (squared-magnitude-ℂ z)
   is-positive-squared-magnitude-is-nonzero-ℂ (a +iℂ b) =
     elim-disjunction
-      ( is-positive-prop-ℝ ∥ a +iℂ b ∥²ℂ)
+      ( is-positive-prop-ℝ (squared-magnitude-ℂ (a +iℂ b)))
       ( λ a≠0 →
         concatenate-le-leq-ℝ
           ( zero-ℝ)
@@ -84,7 +85,8 @@ abstract
           ( leq-right-add-real-ℝ⁰⁺ _ (nonnegative-square-ℝ a)))
 
   is-nonzero-is-positive-squared-magnitude-ℂ :
-    {l : Level} (z : ℂ l) → is-positive-ℝ ∥ z ∥²ℂ → is-nonzero-ℂ z
+    {l : Level} (z : ℂ l) → is-positive-ℝ (squared-magnitude-ℂ z) →
+    is-nonzero-ℂ z
   is-nonzero-is-positive-squared-magnitude-ℂ (a +iℂ b) 0<a²+b² =
     elim-disjunction
       ( is-nonzero-prop-ℂ (a +iℂ b))

From 6cec6d04d182657d4393fc4c909a69ee64ac2960 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 11:43:49 -0800
Subject: [PATCH 132/134] make pre-commit

---
 .../magnitude-complex-numbers.lagda.md                |  5 -----
 ...licative-inverses-nonzero-complex-numbers.lagda.md | 11 -----------
 .../multiplication-nonzero-real-numbers.lagda.md      |  3 ---
 3 files changed, 19 deletions(-)

diff --git a/src/complex-numbers/magnitude-complex-numbers.lagda.md b/src/complex-numbers/magnitude-complex-numbers.lagda.md
index aa3deb8b65d..3c033afa8dc 100644
--- a/src/complex-numbers/magnitude-complex-numbers.lagda.md
+++ b/src/complex-numbers/magnitude-complex-numbers.lagda.md
@@ -20,11 +20,6 @@ open import foundation.identity-types
 open import foundation.universe-levels
 
 open import real-numbers.absolute-value-real-numbers
-
-open import foundation.action-on-identifications-functions
-open import foundation.identity-types
-open import foundation.universe-levels
-
 open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 826104685b5..10db59823f9 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -22,14 +22,6 @@ open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
-open import complex-numbers.complex-numbers
-open import complex-numbers.conjugation-complex-numbers
-open import complex-numbers.magnitude-complex-numbers
-open import complex-numbers.multiplication-complex-numbers
-open import complex-numbers.nonzero-complex-numbers
-open import complex-numbers.similarity-complex-numbers
-
-open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -43,9 +35,6 @@ open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.multiplication-real-numbers
-open import real-numbers.multiplicative-inverses-positive-real-numbers
-open import real-numbers.rational-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
index 6025556798e..47bc27ae7f2 100644
--- a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
@@ -15,9 +15,6 @@ open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
-open import foundation.dependent-pair-types
-open import foundation.disjunction
-open import foundation.function-types
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers

From a4cec6ecf7fc983bde0e09fceff023c869ea94d3 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 11:48:57 -0800
Subject: [PATCH 133/134] Merge properly

---
 ...-inverses-nonzero-complex-numbers.lagda.md | 12 ++---
 ...ltiplication-nonzero-real-numbers.lagda.md | 46 +++++++++++++++++++
 ...ive-inverses-nonzero-real-numbers.lagda.md | 40 ----------------
 3 files changed, 52 insertions(+), 46 deletions(-)

diff --git a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
index 10db59823f9..4d7c38d8961 100644
--- a/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
+++ b/src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md
@@ -111,7 +111,7 @@ abstract
     is-nonzero-is-positive-magnitude-ℂ
       ( z)
       ( elim-disjunction
-        ( is-positive-prop-ℝ ∥ z ∥ℂ)
+        ( is-positive-prop-ℝ (magnitude-ℂ z))
         ( λ |z|<0 →
           ex-falso
             ( is-not-negative-and-nonnegative-ℝ
@@ -119,15 +119,15 @@ abstract
         ( id)
         ( pr1
           ( is-nonzero-factors-is-nonzero-mul-ℝ
-            ( ∥ z ∥ℂ)
-            ( ∥ w ∥ℂ)
+            ( (magnitude-ℂ z))
+            ( (magnitude-ℂ w))
             ( is-nonzero-sim-ℝ
               ( symmetric-sim-ℝ
                 ( similarity-reasoning-ℝ
-                  ∥ z ∥ℂ *ℝ ∥ w ∥ℂ
-                  ~ℝ ∥ z *ℂ w ∥ℂ
+                  (magnitude-ℂ z) *ℝ (magnitude-ℂ w)
+                  ~ℝ magnitude-ℂ (z *ℂ w)
                     by sim-eq-ℝ (inv (distributive-magnitude-mul-ℂ z w))
-                  ~ℝ ∥ one-ℂ ∥ℂ
+                  ~ℝ magnitude-ℂ one-ℂ
                     by preserves-sim-magnitude-ℂ zw=1
                   ~ℝ one-ℝ
                     by sim-eq-ℝ magnitude-one-ℂ))
diff --git a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
index 47bc27ae7f2..d8642acca13 100644
--- a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
@@ -14,16 +14,23 @@ open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.function-types
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.large-ring-of-real-numbers
 open import foundation.functoriality-cartesian-product-types
 open import foundation.universe-levels
+open import commutative-algebra.invertible-elements-commutative-rings
+open import foundation.transport-along-identifications
 
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.rational-real-numbers
 open import real-numbers.multiplication-negative-real-numbers
 open import real-numbers.multiplication-positive-and-negative-real-numbers
 open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-nonzero-real-numbers
+open import real-numbers.similarity-real-numbers
 open import real-numbers.nonzero-real-numbers
+open import real-numbers.positive-real-numbers
 ```
 
 </details>
@@ -103,3 +110,42 @@ module _
               ( map-product inr-disjunction inr-disjunction)
               ( same-sign-is-positive-mul-ℝ x y 0<xy))
 ```
+
+### If a real number has a multiplicative inverse, it is nonzero
+
+```agda
+abstract
+  is-nonzero-has-right-inverse-mul-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →
+    is-nonzero-ℝ x
+  is-nonzero-has-right-inverse-mul-ℝ x y xy=1 =
+    pr1
+      ( is-nonzero-factors-is-nonzero-mul-ℝ
+        ( x)
+        ( y)
+        ( is-nonzero-is-positive-ℝ
+          ( is-positive-sim-ℝ
+            ( symmetric-sim-ℝ xy=1)
+            ( is-positive-one-ℝ))))
+
+  is-nonzero-has-left-inverse-mul-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →
+    is-nonzero-ℝ y
+  is-nonzero-has-left-inverse-mul-ℝ x y xy=1 =
+    is-nonzero-has-right-inverse-mul-ℝ y x
+      ( tr (λ z → sim-ℝ z one-ℝ) (commutative-mul-ℝ x y) xy=1)
+
+  is-nonzero-is-invertible-ℝ :
+    {l : Level} (x : ℝ l) →
+    is-invertible-element-Commutative-Ring (commutative-ring-ℝ l) x →
+    is-nonzero-ℝ x
+  is-nonzero-is-invertible-ℝ {l} x (y , xy=1 , _) =
+    is-nonzero-has-right-inverse-mul-ℝ x y
+      ( inv-tr
+        ( λ z → sim-ℝ z one-ℝ)
+        ( xy=1)
+        ( symmetric-sim-ℝ
+          { x = one-ℝ}
+          { y = raise-ℝ l one-ℝ}
+          ( sim-raise-ℝ l one-ℝ)))
+```
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index 841dc0bf2a7..a07ffadc383 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -36,7 +36,6 @@ open import foundation.universe-levels
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.large-ring-of-real-numbers
-open import real-numbers.multiplication-nonzero-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-negative-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
@@ -149,45 +148,6 @@ is-invertible-is-nonzero-ℝ x x≠0 =
         ( left-inverse-law-mul-nonzero-ℝ (x , x≠0))))
 ```
 
-### If a real number has a multiplicative inverse, it is nonzero
-
-```agda
-abstract
-  is-nonzero-has-right-inverse-mul-ℝ :
-    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →
-    is-nonzero-ℝ x
-  is-nonzero-has-right-inverse-mul-ℝ x y xy=1 =
-    pr1
-      ( is-nonzero-factors-is-nonzero-mul-ℝ
-        ( x)
-        ( y)
-        ( is-nonzero-is-positive-ℝ
-          ( is-positive-sim-ℝ
-            ( is-positive-one-ℝ)
-            ( symmetric-sim-ℝ xy=1))))
-
-  is-nonzero-has-left-inverse-mul-ℝ :
-    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →
-    is-nonzero-ℝ y
-  is-nonzero-has-left-inverse-mul-ℝ x y xy=1 =
-    is-nonzero-has-right-inverse-mul-ℝ y x
-      ( tr (λ z → sim-ℝ z one-ℝ) (commutative-mul-ℝ x y) xy=1)
-
-  is-nonzero-is-invertible-ℝ :
-    {l : Level} (x : ℝ l) →
-    is-invertible-element-Commutative-Ring (commutative-ring-ℝ l) x →
-    is-nonzero-ℝ x
-  is-nonzero-is-invertible-ℝ {l} x (y , xy=1 , _) =
-    is-nonzero-has-right-inverse-mul-ℝ x y
-      ( inv-tr
-        ( λ z → sim-ℝ z one-ℝ)
-        ( xy=1)
-        ( symmetric-sim-ℝ
-          { x = one-ℝ}
-          { y = raise-ℝ l one-ℝ}
-          ( sim-raise-ℝ l one-ℝ)))
-```
-
 ### The multiplicative inverse is unique
 
 ```agda

From 5edfafdec903c94984182a8815da2668c1f3079e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 16 Nov 2025 11:51:45 -0800
Subject: [PATCH 134/134] make pre-commit

---
 .../multiplication-nonzero-real-numbers.lagda.md    | 13 +++++++------
 1 file changed, 7 insertions(+), 6 deletions(-)

diff --git a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
index d8642acca13..42ea33baba2 100644
--- a/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonzero-real-numbers.lagda.md
@@ -9,28 +9,29 @@ module real-numbers.multiplication-nonzero-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import commutative-algebra.invertible-elements-commutative-rings
+
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.function-types
-open import real-numbers.raising-universe-levels-real-numbers
-open import real-numbers.large-ring-of-real-numbers
 open import foundation.functoriality-cartesian-product-types
-open import foundation.universe-levels
-open import commutative-algebra.invertible-elements-commutative-rings
 open import foundation.transport-along-identifications
+open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.rational-real-numbers
+open import real-numbers.large-ring-of-real-numbers
 open import real-numbers.multiplication-negative-real-numbers
 open import real-numbers.multiplication-positive-and-negative-real-numbers
 open import real-numbers.multiplication-positive-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-nonzero-real-numbers
-open import real-numbers.similarity-real-numbers
 open import real-numbers.nonzero-real-numbers
 open import real-numbers.positive-real-numbers
+open import real-numbers.raising-universe-levels-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
 ```
 
 </details>