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[helm.git] / helm / software / matita / contribs / formal_topology / overlap / categories.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/categories.ma b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
index 1452652fa6e8c30a45e2c007f9b6d4a78b7d293a..75ca005c64b9ce03d24b487c22d103ee97499492 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/categories.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
@@ -12,34 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "logic/cprop_connectives.ma".
-
-definition Type0 := Type.
-definition Type1 := Type.
-definition Type2 := Type.
-definition Type3 := Type.
-definition Type0_lt_Type1 := (Type0 : Type1).
-definition Type1_lt_Type2 := (Type1 : Type2).
-definition Type2_lt_Type3 := (Type2 : Type3).
-
-definition Type_OF_Type0: Type0 → Type := λx.x.
-definition Type_OF_Type1: Type1 → Type := λx.x.
-definition Type_OF_Type2: Type2 → Type := λx.x.
-definition Type_OF_Type3: Type3 → Type := λx.x.
-coercion Type_OF_Type0.
-coercion Type_OF_Type1.
-coercion Type_OF_Type2.
-coercion Type_OF_Type3.
-
-definition CProp0 := CProp.
-definition CProp1 := CProp.
-definition CProp2 := CProp.
-definition CProp0_lt_CProp1 := (CProp0 : CProp1).
-definition CProp1_lt_CProp2 := (CProp1 : CProp2).
-
-definition CProp_OF_CProp0: CProp0 → CProp := λx.x.
-definition CProp_OF_CProp1: CProp1 → CProp := λx.x.
-definition CProp_OF_CProp2: CProp2 → CProp := λx.x.
+include "cprop_connectives.ma".
 
 record equivalence_relation (A:Type0) : Type1 ≝
  { eq_rel:2> A → A → CProp0;
@@ -53,9 +26,9 @@ record setoid : Type1 ≝
    eq: equivalence_relation carr
  }.
 
-definition reflexive1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
-definition symmetric1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
-definition transitive1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
+definition reflexive1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
+definition symmetric1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
+definition transitive1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
 
 record equivalence_relation1 (A:Type1) : Type2 ≝
  { eq_rel1:2> A → A → CProp1;
@@ -84,9 +57,9 @@ qed.
 (* questa coercion e' necessaria per problemi di unificazione *)
 coercion setoid1_of_setoid.
 
-definition reflexive2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
-definition symmetric2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
-definition transitive2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
+definition reflexive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
+definition symmetric2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
+definition transitive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
 
 record equivalence_relation2 (A:Type2) : Type3 ≝
  { eq_rel2:2> A → A → CProp2;
@@ -192,9 +165,9 @@ definition CPROP: setoid1.
   | constructor 1;
      [ apply Iff
      | intros 1; split; intro; assumption
-     | intros 3; cases H; split; assumption
-     | intros 5; cases H; cases H1; split; intro;
-        [ apply (H4 (H2 x1)) | apply (H3 (H5 z1))]]]
+     | intros 3; cases i; split; assumption
+     | intros 5; cases i; cases i1; split; intro;
+        [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
 qed.
 
 alias symbol "eq" = "setoid1 eq".
@@ -208,10 +181,10 @@ interpretation "if" 'if r = (if' __ r).
 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
  constructor 1;
   [ apply And
-  | intros; split; intro; cases H; split;
-     [ apply (if ?? e a1)
+  | intros; split; intro; cases a1; split;
+     [ apply (if ?? e a2)
      | apply (if ?? e1 b1)
-     | apply (fi ?? e a1)
+     | apply (fi ?? e a2)
      | apply (fi ?? e1 b1)]]
 qed.
 
@@ -220,7 +193,7 @@ interpretation "and_morphism" 'and a b = (fun21 ___ and_morphism a b).
 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
  constructor 1;
   [ apply Or
-  | intros; split; intro; cases H; [1,3:left |2,4: right]
+  | intros; split; intro; cases o; [1,3:left |2,4: right]
      [ apply (if ?? e a1)
      | apply (fi ?? e a1)
      | apply (if ?? e1 b1)
@@ -233,8 +206,8 @@ definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
  constructor 1;
   [ apply (λA,B. A → B)
   | intros; split; intros;
-     [ apply (if ?? e1); apply H; apply (fi ?? e); assumption
-     | apply (fi ?? e1); apply H; apply (if ?? e); assumption]]
+     [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
+     | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
 qed.
 
 (*
@@ -286,13 +259,11 @@ record category2 : Type3 ≝
  }.
 
 notation "'ASSOC'" with precedence 90 for @{'assoc}.
-notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
-notation "'ASSOC2'" with precedence 90 for @{'assoc2}.
 
 interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
-interpretation "category2 assoc" 'assoc1 = (comp_assoc2 ________).
+interpretation "category2 assoc" 'assoc = (comp_assoc2 ________).
 interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
-interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
+interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
 interpretation "category assoc" 'assoc = (comp_assoc ________).
 
@@ -305,8 +276,8 @@ definition unary_morphism_setoid: setoid → setoid → setoid1.
   | constructor 1;
      [ intros (f g); apply (∀a:s. eq ? (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]]]
+     | simplify; intros; apply sym; apply f;
+     | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
 qed.
 
 definition SET: category1.
@@ -356,8 +327,8 @@ definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid2.
        alias symbol "eq" = "setoid1 eq".
        apply (∀a: carr1 s. f a = g a);
      | intros 1; simplify; intros; apply refl1;
-     | simplify; intros; apply sym1; apply H;
-     | simplify; intros; apply trans1; [2: apply H; | skip | apply H1]]]
+     | simplify; intros; apply sym1; apply f;
+     | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
 qed.
 
 definition SET1: category2.
@@ -401,6 +372,12 @@ definition Type1_OF_SET1: Type_OF_category2 SET1 → Type1.
 qed.
 coercion Type1_OF_SET1.
 
+definition Type_OF_setoid1_of_carr: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
+ [ apply setoid1_of_SET; apply U
+ | intros; apply c;]
+qed.
+coercion Type_OF_setoid1_of_carr.
+
 interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
 interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
 interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).
@@ -408,4 +385,4 @@ interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).
 lemma unary_morphism1_of_arrows1_SET1: ∀S,T. (S ⇒ T) → unary_morphism1 S T.
  intros; apply t;
 qed.
-coercion unary_morphism1_of_arrows1_SET1.
\ No newline at end of file
+coercion unary_morphism1_of_arrows1_SET1.