- transcript: bugfix

[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 e90e1457d222d23702ec37792f252ac67a270b6a..f5b23dfdd3a4fcbfc832edc89908eca62c3dacf1 100644 (file)
--- a/helm/software/matita/library/datatypes/categories.ma
+++ b/helm/software/matita/library/datatypes/categories.ma
@@ -14,6 +14,12 @@
 
 include "logic/cprop_connectives.ma".
 
+definition Type0 := Type.
+definition Type1 := Type.
+definition Type2 := Type.
+definition Type0_lt_Type1 := (Type0 : Type1).
+definition Type1_lt_Type2 := (Type1 : Type2).
+
 record equivalence_relation (A:Type) : Type ≝
  { eq_rel:2> A → A → CProp;
    refl: reflexive ? eq_rel;
@@ -21,7 +27,7 @@ record equivalence_relation (A:Type) : Type ≝
    trans: transitive ? eq_rel
  }.
 
-record setoid : Type ≝
+record setoid : Type1 ≝
  { carr:> Type;
    eq: equivalence_relation carr
  }.
@@ -30,7 +36,7 @@ definition reflexive1 ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
 definition symmetric1 ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
 definition transitive1 ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.
 
-record equivalence_relation1 (A:Type) : Type ≝
+record equivalence_relation1 (A:Type) : Type2 ≝
  { eq_rel1:2> A → A → CProp;
    refl1: reflexive1 ? eq_rel1;
    sym1: symmetric1 ? eq_rel1;
@@ -74,32 +80,29 @@ 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 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_morphism (A,B: setoid1) : Type ≝
+record unary_morphism (A,B: setoid1) : Type0 ≝
  { fun_1:1> A → B;
    prop_1: ∀a,a'. eq1 ? a a' → eq1 ? (fun_1 a) (fun_1 a')
  }.
 
-record binary_morphism (A,B,C:setoid) : Type ≝
+record binary_morphism (A,B,C:setoid) : Type0 ≝
  { fun:2> A → B → C;
    prop: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun a b) (fun a' b')
  }.
 
-record binary_morphism1 (A,B,C:setoid1) : Type ≝
+record binary_morphism1 (A,B,C:setoid1) : Type0 ≝
  { fun1:2> A → B → C;
    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}.
@@ -174,7 +177,7 @@ definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
 qed.
 *)
 
-record category : Type ≝
+record category : Type1 ≝
  { objs:> Type;
    arrows: objs → objs → setoid;
    id: ∀o:objs. arrows o o;
@@ -185,21 +188,65 @@ record category : Type ≝
    id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
  }.
 
-record category1 : Type ≝
+record category1 : Type2 ≝
  { objs1:> Type;
    arrows1: objs1 → objs1 → setoid1;
    id1: ∀o:objs1. arrows1 o o;
    comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
    comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
     comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
-   id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
-   id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
+   id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
+   id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
  }.
 
 notation "'ASSOC'" with precedence 90 for @{'assoc}.
 notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
 
-interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) x y).
+interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) y x).
 interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
-interpretation "category composition" 'compose x y = (fun ___ (comp ____) x y).
+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).