update in groud_2 and models

[helm.git] / helm / software / matita / library / datatypes / categories.ma

diff --git a/helm/software/matita/library/datatypes/categories.ma b/helm/software/matita/library/datatypes/categories.ma
index 6dac9b044bb537175b1f4dc33708356563836fbc..b9e365c7e53d0e98ed01ac6f12ddea099d577081 100644 (file)
--- a/helm/software/matita/library/datatypes/categories.ma
+++ b/helm/software/matita/library/datatypes/categories.ma
@@ -80,20 +80,13 @@ qed.
 coercion Leibniz.
 *)
 
-interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
-interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
-interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
-interpretation "setoid symmetry" 'invert r = (sym ____ r).
+interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
 notation ".= r" with precedence 50 for @{'trans $r}.
-interpretation "trans1" 'trans r = (trans1 _____ r).
-interpretation "trans" 'trans r = (trans _____ r).
-
-(*
-record unary_morphism0 (A,B: setoid) : Type0 ≝
- { fun_0:1> A → B;
-   prop_0: ∀a,a'. eq ? a a' → eq ? (fun_0 a) (fun_0 a')
- }.
-*)
+interpretation "trans1" 'trans r = (trans1 ????? r).
+interpretation "trans" 'trans r = (trans ????? r).
 
 record unary_morphism (A,B: setoid1) : Type0 ≝
  { fun_1:1> A → B;
@@ -110,18 +103,14 @@ 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}.
-interpretation "prop_1" 'prop1 c = (prop_1 _____ c).
-(* interpretation "prop_0" 'prop1 c = (prop_0 _____ c). *)
-interpretation "prop1" 'prop l r = (prop1 ________ l r).
-interpretation "prop" 'prop l r = (prop ________ l r).
-interpretation "refl1" 'refl = (refl1 ___).
-interpretation "refl" 'refl = (refl ___).
+interpretation "prop_1" 'prop1 c = (prop_1 ????? c).
+interpretation "prop1" 'prop l r = (prop1 ???????? l r).
+interpretation "prop" 'prop l r = (prop ???????? l r).
+interpretation "refl1" 'refl = (refl1 ???).
+interpretation "refl" 'refl = (refl ???).
 
 definition CPROP: setoid1.
  constructor 1;
@@ -139,7 +128,7 @@ definition if': ∀A,B:CPROP. A = B → A → B.
 qed.
 
 notation ". r" with precedence 50 for @{'if $r}.
-interpretation "if" 'if r = (if' __ r).
+interpretation "if" 'if r = (if' ?? r).
 
 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
  constructor 1;
@@ -151,7 +140,7 @@ definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
      | apply (fi ?? H1 b1)]]
 qed.
 
-interpretation "and_morphism" 'and a b = (fun1 ___ and_morphism a b).
+interpretation "and_morphism" 'and a b = (fun1 ??? and_morphism a b).
 
 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
  constructor 1;
@@ -163,7 +152,7 @@ definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
      | apply (fi ?? H1 b1)]]
 qed.
 
-interpretation "or_morphism" 'or a b = (fun1 ___ or_morphism a b).
+interpretation "or_morphism" 'or a b = (fun1 ??? or_morphism a b).
 
 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
  constructor 1;
@@ -213,7 +202,51 @@ record category1 : Type2 ≝
 notation "'ASSOC'" with precedence 90 for @{'assoc}.
 notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
 
-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 ________).
+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.
+
+definition setoid_OF_SET: objs1 SET → setoid.
+ intros; apply o; qed.
+
+coercion setoid_OF_SET.
+
+
+definition prop_1_SET : 
+ ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:A.eq1 ? a b→eq1 ? (w a) (w b).
+intros; apply (prop_1 A B w a b H);
+qed.
+
+interpretation "SET dagger" 'prop1 h = (prop_1_SET ? ? ? ? ? h).