Isomorphisms: add alternative forms, useful for rewriting

[coq-categories.git] / src / Isomorphisms_ch1_5.v

diff --git a/src/Isomorphisms_ch1_5.v b/src/Isomorphisms_ch1_5.v
index 372ad43..422d4bb 100644 (file)
--- a/src/Isomorphisms_ch1_5.v
+++ b/src/Isomorphisms_ch1_5.v
@@ -1,5 +1,5 @@
 Generalizable All Variables.
 Generalizable All Variables.

-Require Import Preamble.
+Require Import Notations.

 Require Import Categories_ch1_3.
 Require Import Functors_ch1_4.
 
 Require Import Categories_ch1_3.
 Require Import Functors_ch1_4.
 

@@ -52,6 +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.
     [ 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).
 
 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).

@@ -117,3 +118,32 @@ Lemma iso_inv_inv `{C:Category}{a}{b}(i:a ≅ b) : #(i⁻¹)⁻¹ ~~ #i.
   unfold iso_inv; simpl.
   reflexivity.
   Qed.
   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.