X-Git-Url: http://git.megacz.com/?p=coq-categories.git;a=blobdiff_plain;f=src%2FIsomorphisms_ch1_5.v;h=422d4bbf75c4dfb0c8a138ad22f238a6f2f7d538;hp=65eaf4624265e65d2d567e8da5b31949cbce6bc7;hb=57dfee583c84915cb6b3aadfa1dba692681499b3;hpb=e928451c4c45cdbdd975bbfb229e8cc2616b8194

diff --git a/src/Isomorphisms_ch1_5.v b/src/Isomorphisms_ch1_5.v
index 65eaf46..422d4bb 100644
--- a/src/Isomorphisms_ch1_5.v
+++ b/src/Isomorphisms_ch1_5.v
@@ -1,6 +1,5 @@
 Generalizable All Variables.
-Require Import Preamble.
-Require Import General.
+Require Import Notations.
 Require Import Categories_ch1_3.
 Require Import Functors_ch1_4.
 
@@ -53,6 +52,7 @@ Definition iso_comp `{C:Category}{a b c:C}(i1:Isomorphic a b)(i2:Isomorphic b c)
     [ setoid_rewrite (iso_comp1 i1) | setoid_rewrite (iso_comp2 i2) ];
     reflexivity.
   Defined.
+Notation "a >>≅>> b" := (iso_comp a b).
 
 Definition functors_preserve_isos `{C1:Category}`{C2:Category}{Fo}(F:Functor C1 C2 Fo){a b:C1}(i:Isomorphic a b)
   : Isomorphic (F a) (F b).
@@ -118,3 +118,32 @@ Lemma iso_inv_inv `{C:Category}{a}{b}(i:a ≅ b) : #(i⁻¹)⁻¹ ~~ #i.
   unfold iso_inv; simpl.
   reflexivity.
   Qed.
+
+(* the next four lemmas are handy for setoid_rewrite; they let you avoid having to get the associativities right *)
+Lemma iso_comp2_right : forall `{C:Category}{a b}(i:a≅b) c (g:b~>c), iso_backward i >>> (iso_forward i >>> g) ~~ g.
+  intros.
+  setoid_rewrite <- associativity.
+  setoid_rewrite iso_comp2.
+  apply left_identity.
+  Qed.
+
+Lemma iso_comp2_left : forall `{C:Category}{a b}(i:a≅b) c (g:c~>b), (g >>> iso_backward i) >>> iso_forward i ~~ g.
+  intros.
+  setoid_rewrite associativity.
+  setoid_rewrite iso_comp2.
+  apply right_identity.
+  Qed.
+
+Lemma iso_comp1_right : forall `{C:Category}{a b}(i:a≅b) c (g:a~>c), iso_forward i >>> (iso_backward i >>> g) ~~ g.
+  intros.
+  setoid_rewrite <- associativity.
+  setoid_rewrite iso_comp1.
+  apply left_identity.
+  Qed.
+
+Lemma iso_comp1_left : forall `{C:Category}{a b}(i:a≅b) c (g:c~>a), (g >>> iso_forward i) >>> iso_backward i ~~ g.
+  intros.
+  setoid_rewrite associativity.
+  setoid_rewrite iso_comp1.
+  apply right_identity.
+  Qed.