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[ add ] rule of signs for RingWithoutOne #2761

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Merged
merged 4 commits into from
Jul 31, 2025
Merged

[ add ] rule of signs for RingWithoutOne #2761

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43c7316
add: rule of signs for `RingWithoutOne`
jamesmckinna Jul 8, 2025
6b84352
reset: `CHANGELOG` for v2.4
jamesmckinna Jul 20, 2025
93e2215
Merge branch 'master' into rule-of-signs
jamesmckinna Jul 20, 2025
567a172
Merge branch 'master' into rule-of-signs
jamesmckinna Jul 31, 2025
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5 changes: 5 additions & 0 deletions CHANGELOG.md
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Expand Up @@ -26,3 +26,8 @@ New modules

Additions to existing modules
-----------------------------

* In `Algebra.Properties.RingWithoutOne`:
```agda
[-x][-y]≈xy : ∀ x y → - x * - y ≈ x * y
```
20 changes: 15 additions & 5 deletions src/Algebra/Properties/RingWithoutOne.agda
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Expand Up @@ -17,7 +17,7 @@ open import Function.Base using (_$_)
open import Relation.Binary.Reasoning.Setoid setoid

------------------------------------------------------------------------
-- Export properties of abelian groups
-- Re-export abelian group properties for addition

open AbelianGroupProperties +-abelianGroup public
renaming
Expand All @@ -36,6 +36,12 @@ open AbelianGroupProperties +-abelianGroup public
; ⁻¹-∙-comm to -‿+-comm
)

x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#
x+x≈x⇒x≈0 x eq = +-identityˡ-unique x x eq

------------------------------------------------------------------------
-- Consequences of distributivity

-‿distribˡ-* : ∀ x y → - (x * y) ≈ - x * y
-‿distribˡ-* x y = sym $ begin
- x * y ≈⟨ +-identityʳ (- x * y) ⟨
Expand All @@ -58,17 +64,21 @@ open AbelianGroupProperties +-abelianGroup public
- (x * y) + 0# ≈⟨ +-identityʳ (- (x * y)) ⟩
- (x * y) ∎

x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#
x+x≈x⇒x≈0 x eq = +-identityˡ-unique x x eq
[-x][-y]≈xy : ∀ x y → - x * - y ≈ x * y
[-x][-y]≈xy x y = begin
- x * - y ≈⟨ -‿distribˡ-* x (- y) ⟨
- (x * - y) ≈⟨ -‿cong (-‿distribʳ-* x y) ⟨
- (- (x * y)) ≈⟨ -‿involutive (x * y) ⟩
x * y ∎

x[y-z]≈xy-xz : ∀ x y z → x * (y - z) ≈ x * y - x * z
x[y-z]≈xy-xz x y z = begin
x * (y - z) ≈⟨ distribˡ x y (- z) ⟩
x * y + x * - z ≈⟨ +-congˡ (sym (-‿distribʳ-* x z)) ⟩
x * y + x * - z ≈⟨ +-congˡ (-‿distribʳ-* x z) ⟨
x * y - x * z ∎

[y-z]x≈yx-zx : ∀ x y z → (y - z) * x ≈ (y * x) - (z * x)
[y-z]x≈yx-zx x y z = begin
(y - z) * x ≈⟨ distribʳ x y (- z) ⟩
y * x + - z * x ≈⟨ +-congˡ (sym (-‿distribˡ-* z x)) ⟩
y * x + - z * x ≈⟨ +-congˡ (-‿distribˡ-* z x) ⟨
y * x - z * x ∎