From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Sun, 21 Dec 2008 17:56:31 +0000 (+0000)
Subject: New category SET (whose objects are setoids).
X-Git-Tag: make_still_working~4342
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=87ff605d8efbc7d5ca7b89cdb6bf04f0a2ddf489;p=helm.git

New category SET (whose objects are setoids).
---

diff --git a/helm/software/matita/library/datatypes/categories.ma b/helm/software/matita/library/datatypes/categories.ma
index 3a8a6c759..ca49cbdd3 100644
--- a/helm/software/matita/library/datatypes/categories.ma
+++ b/helm/software/matita/library/datatypes/categories.ma
@@ -103,9 +103,6 @@ record binary_morphism1 (A,B,C:setoid1) : Type0 ≝
    prop1: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun1 a b) (fun1 a' b')
  }.
 
-notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
-interpretation "unary morphism" 'Imply a b = (unary_morphism a b).
-
 notation "† c" with precedence 90 for @{'prop1 $c }.
 notation "l ‡ r" with precedence 90 for @{'prop $l $r }.
 notation "#" with precedence 90 for @{'refl}.
@@ -209,3 +206,34 @@ interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) y x
 interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
 interpretation "category composition" 'compose x y = (fun ___ (comp ____) y x).
 interpretation "category assoc" 'assoc = (comp_assoc ________).
+
+definition unary_morphism_setoid: setoid → setoid → setoid.
+ intros;
+ constructor 1;
+  [ apply (unary_morphism s s1);
+  | constructor 1;
+     [ intros (f g); apply (∀a. f a = g a);
+     | intros 1; simplify; intros; apply refl;
+     | simplify; intros; apply sym; apply H;
+     | simplify; intros; apply trans; [2: apply H; | skip | apply H1]]]
+qed.
+
+notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
+interpretation "unary morphism" 'Imply a b = (unary_morphism_setoid a b).
+interpretation "unary morphism" 'Imply a b = (unary_morphism a b).
+
+definition SET: category1.
+ constructor 1;
+  [ apply setoid;
+  | apply rule (λS,T.unary_morphism_setoid S T);
+  | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
+  | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
+     apply († (†H));]
+  | intros; whd; intros; simplify; whd in H1; whd in H;
+    apply trans; [ apply (b (a' a1)); | lapply (prop_1 ?? b (a a1) (a' a1));
+     [ apply Hletin | apply (H a1); ]  | apply H1; ]]
+  | intros; whd; intros; simplify; apply refl;
+  | intros; simplify; whd; intros; simplify; apply refl;
+  | intros; simplify; whd; intros; simplify; apply refl;
+  ]
+qed.
\ No newline at end of file