Initial.

Isomorphism/Properties.lidr

+> module Isomorphism.Properties
+
+> import Control.Isomorphism
+
+> -- import Isomorphism.Operations
+> import Fun.Properties
+> import Basic.Operations
+
+> %default total
+> %access public export
+
+
+> |||
+> isoEq : {A, B : Type} -> A = B -> Iso A B
+> isoEq Refl = isoRefl
+> %freeze isoEq 
+
+
+> |||
+> isoCong : {A : Type} -> {x : A} -> {y : A} -> {P : A -> Type} -> x = y -> Iso (P x) (P y)
+> isoCong {x} {P} prf = replace {P = \ z => Iso (P x) (P z)} prf isoRefl
+> %freeze isoCong
+
+
+Injectivity of to and from
+
+> injectiveFrom : {A, B : Type} -> (iso : Iso A B) -> Injective1 (from iso)
+> injectiveFrom {A} {B} (MkIso to from toFrom fromTo) b1 b2 p = s3 where
+>   s1 : from b1 = from b2
+>   s1 = p
+>   s2 : to (from b1) = to (from b2)
+>   s2 = cong s1
+>   s3 : b1 = b2
+>   s3 = replace2 {a = B} {a1 = to (from b1)} {a2 = b1}
+>                 {b = B} {b1 = to (from b2)} {b2 = b2}
+>                 {P = \ a => \b => a = b}
+>                 (toFrom b1) (toFrom b2) s2
+> %freeze injectiveFrom
+
+
+> {-
+
+> ---}