Add Algebra laws

Author: nickdrozd

libs/contrib/Control/Algebra/Laws.idr

@@ -0,0 +1,162 @@
+module Control.Algebra.Laws
+
+import Prelude
+import Control.Algebra
+
+%default total
+
+-- Monoids
+
+||| Inverses are unique.
+public export
+uniqueInverse : VMonoid ty => (x, y, z : ty) ->
+  y <+> x = neutral {ty} -> x <+> z = neutral {ty} -> y = z
+uniqueInverse x y z p q =
+  rewrite sym $ monoidNeutralIsNeutralL y in
+    rewrite sym q in
+      rewrite semigroupOpIsAssociative y x z in
+  rewrite p in
+    rewrite monoidNeutralIsNeutralR z in
+      Refl
+
+-- Groups
+
+||| Only identity is self-squaring.
+public export
+selfSquareId : VGroup ty => (x : ty) ->
+  x <+> x = x -> x = neutral {ty}
+
+||| Inverse elements commute.
+public export
+inverseCommute : VGroup ty => (x, y : ty) ->
+  y <+> x = neutral {ty} -> x <+> y = neutral {ty}
+
+||| Every element has a right inverse.
+public export
+groupInverseIsInverseL : VGroup ty => (x : ty) ->
+  x <+> inverse x = neutral {ty}
+
+||| -(-x) = x
+public export
+inverseSquaredIsIdentity : VGroup ty => (x : ty) ->
+  inverse (inverse x) = x
+
+||| If every square in a group is identity, the group is commutative.
+public export
+squareIdCommutative : VGroup ty => (x, y : ty) ->
+  ((a : ty) -> a <+> a = neutral {ty}) ->
+  x <+> y = y <+> x
+
+||| -0 = 0
+public export
+inverseNeutralIsNeutral : VGroup ty =>
+  inverse (neutral {ty}) = neutral {ty}
+
+||| -(x + y) = -y + -x
+public export
+inverseOfSum : VGroup ty => (l, r : ty) ->
+  inverse (l <+> r) = inverse r <+> inverse l
+
+||| y = z if x + y = x + z
+public export
+cancelLeft : VGroup ty => (x, y, z : ty) ->
+  x <+> y = x <+> z -> y = z
+cancelLeft x y z p =
+  rewrite sym $ monoidNeutralIsNeutralR y in
+    rewrite sym $ groupInverseIsInverseR x in
+      rewrite sym $ semigroupOpIsAssociative (inverse x) x y in
+        rewrite p in
+      rewrite semigroupOpIsAssociative (inverse x) x z in
+    rewrite groupInverseIsInverseR x in
+  monoidNeutralIsNeutralR z
+
+||| y = z if y + x = z + x.
+public export
+cancelRight : VGroup ty => (x, y, z : ty) ->
+  y <+> x = z <+> x -> y = z
+
+||| ab = 0 -> a = b^-1
+public export
+neutralProductInverseR : VGroup ty => (a, b : ty) ->
+  a <+> b = neutral {ty} -> a = inverse b
+
+||| ab = 0 -> a^-1 = b
+public export
+neutralProductInverseL : VGroup ty => (a, b : ty) ->
+  a <+> b = neutral {ty} -> inverse a = b
+neutralProductInverseL a b prf =
+  cancelLeft a (inverse a) b $
+    trans (groupInverseIsInverseL a) $ sym prf
+
+||| For any a and b, ax = b and ya = b have solutions.
+public export
+latinSquareProperty : VGroup ty => (a, b : ty) ->
+  ((x : ty ** a <+> x = b),
+    (y : ty ** y <+> a = b))
+
+||| For any a, b, x, the solution to ax = b is unique.
+public export
+uniqueSolutionR : VGroup ty => (a, b, x, y : ty) ->
+  a <+> x = b -> a <+> y = b -> x = y
+uniqueSolutionR a b x y p q = cancelLeft a x y $ trans p (sym q)
+
+||| For any a, b, y, the solution to ya = b is unique.
+public export
+uniqueSolutionL : VGroup t => (a, b, x, y : t) ->
+  x <+> a = b -> y <+> a = b -> x = y
+uniqueSolutionL a b x y p q = cancelRight a x y $ trans p (sym q)
+
+||| -(x + y) = -x + -y
+public export
+inverseDistributesOverGroupOp : AbelianGroup ty => (l, r : ty) ->
+  inverse (l <+> r) = inverse l <+> inverse r
+inverseDistributesOverGroupOp l r =
+  rewrite groupOpIsCommutative (inverse l) (inverse r) in
+    inverseOfSum l r
+
+||| Homomorphism preserves neutral.
+public export
+homoNeutral : GroupHomomorphism a b =>
+  to (neutral {ty=a}) = neutral {ty=b}
+
+||| Homomorphism preserves inverses.
+public export
+homoInverse : GroupHomomorphism a b => (x : a) ->
+  the b (to $ inverse x) = the b (inverse $ to x)
+
+-- Rings
+
+||| 0x = x
+public export
+multNeutralAbsorbingL : VRing ty => (r : ty) ->
+  neutral {ty} <.> r = neutral {ty}
+
+||| x0 = 0
+public export
+multNeutralAbsorbingR : VRing ty => (l : ty) ->
+  l <.> neutral {ty} = neutral {ty}
+
+||| (-x)y = -(xy)
+public export
+multInverseInversesL : VRing ty => (l, r : ty) ->
+  inverse l <.> r = inverse (l <.> r)
+
+||| x(-y) = -(xy)
+public export
+multInverseInversesR : VRing ty => (l, r : ty) ->
+  l <.> inverse r = inverse (l <.> r)
+
+||| (-x)(-y) = xy
+public export
+multNegativeByNegativeIsPositive : VRing ty => (l, r : ty) ->
+  inverse l <.> inverse r = l <.> r
+
+||| (-1)x = -x
+public export
+inverseOfUnityR : VRingWithUnity ty => (x : ty) ->
+  inverse (unity {ty}) <.> x = inverse x
+
+||| x(-1) = -x
+public export
+inverseOfUnityL : VRingWithUnity ty => (x : ty) ->
+  x <.> inverse (unity {ty}) = inverse x

libs/contrib/contrib.ipkg

@@ -1,7 +1,9 @@
 package contrib
 
 modules = Control.Delayed,
+
           Control.Algebra,
+          Control.Algebra.Laws,
 
           Data.List.TailRec,
           Data.List.Equalities,