Add GroupHomomorphism

Author: nickdrozd

libs/contrib/Control/Algebra/GroupHomomorphism.idr

@@ -0,0 +1,28 @@
+import Prelude.Algebra as A
+import Control.Algebra
+import Control.Algebra.Laws
+
+||| A homomorphism is a mapping that preserves group structure.
+interface (VerifiedGroup a, VerifiedGroup b) => GroupHomomorphism a b where
+  to : a -> b
+  toGroup : (x, y : a) ->
+    to (x <+> y) = (to x) <+> (to y)
+
+||| Homomorphism preserves neutral.
+homoNeutral : GroupHomomorphism a b =>
+  to (the a A.neutral) = the b A.neutral
+homoNeutral {a} =
+  let ae = the a A.neutral in
+    selfSquareId (to ae) $
+      trans (sym $ toGroup ae ae) $
+        cong {f = to} $ monoidNeutralIsNeutralL ae
+
+||| Homomorphism preserves inverses.
+homoInverse : GroupHomomorphism a b => (x : a) ->
+  the b (to $ inverse x) = the b (inverse $ to x)
+homoInverse {a} {b} x =
+  cancelRight (to x) (to $ inverse x) (inverse $ to x) $
+    trans (sym $ toGroup (inverse x) x) $
+      trans (cong {f = to} $ groupInverseIsInverseR x) $
+        trans (homoNeutral {a=a} {b=b}) $
+          sym $ groupInverseIsInverseR (to x)

libs/contrib/contrib.ipkg

@@ -12,6 +12,7 @@ modules = CFFI
         , Control.Algebra.NumericImplementations
         , Control.Algebra.ComplexImplementations
         , Control.Algebra.VectorSpace
+	, Control.Algebra.GroupHomomorphism
 
         , Control.Delayed