From 84949606d80f30b1a7ada10f46ae13bdf17cacc2 Mon Sep 17 00:00:00 2001 From: Adam Megacz Date: Sat, 2 Apr 2011 13:17:18 -0700 Subject: [PATCH] add isos_forward_equal_then_backward_equal, iso_inv_inv --- src/Isomorphisms_ch1_5.v | 16 ++++++++++++++++ 1 file changed, 16 insertions(+) diff --git a/src/Isomorphisms_ch1_5.v b/src/Isomorphisms_ch1_5.v index fd9e19a..40a5f87 100644 --- a/src/Isomorphisms_ch1_5.v +++ b/src/Isomorphisms_ch1_5.v @@ -102,3 +102,19 @@ Lemma iso_shift_left' `{C:Category} : forall {a b c:C}(f:a~>b)(g:a~>c)(i:Isomorp symmetry. apply right_identity. Qed. + +Lemma isos_forward_equal_then_backward_equal `{C:Category}{a}{b}(i1 i2:a ≅ b) : #i1 ~~ #i2 -> #i1⁻¹ ~~ #i2⁻¹. + intro H. + setoid_rewrite <- left_identity at 1. + setoid_rewrite <- (iso_comp2 i2). + setoid_rewrite associativity. + setoid_rewrite <- H. + setoid_rewrite iso_comp1. + setoid_rewrite right_identity. + reflexivity. + Qed. + +Lemma iso_inv_inv `{C:Category}{a}{b}(i:a ≅ b) : #(i⁻¹)⁻¹ ~~ #i. + unfold iso_inv; simpl. + reflexivity. + Qed. -- 1.7.10.4