CAT2

diff --git a/helm/software/matita/contribs/formal_topology/overlap/categories.ma b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
index c5db6ad6082e91d03461ffba6f3341b924849b0b..7ac1b0b3d6d2f3b66a46c0fc3b4e49465d7aa4f2 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/categories.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
@@ -26,10 +26,6 @@ record setoid : Type1 ≝
    eq: equivalence_relation carr
  }.
 
-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;
    refl1: reflexive1 ? eq_rel1;
@@ -57,10 +53,6 @@ qed.
 coercion setoid1_of_setoid.
 prefer coercion Type_OF_setoid.
 
-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;
    refl2: reflexive2 ? eq_rel2;
@@ -91,13 +83,29 @@ prefer coercion Type_OF_setoid.
 prefer coercion Type_OF_setoid1. 
 (* we prefer 0 < 1 < 2 *)
 
+record equivalence_relation3 (A:Type3) : Type4 ≝
+ { eq_rel3:2> A → A → CProp3;
+   refl3: reflexive3 ? eq_rel3;
+   sym3: symmetric3 ? eq_rel3;
+   trans3: transitive3 ? eq_rel3
+ }.
+
+record setoid3: Type4 ≝
+ { carr3:> Type3;
+   eq3: equivalence_relation3 carr3
+ }.
+
+
+interpretation "setoid3 eq" 'eq x y = (eq_rel3 _ (eq3 _) x y).
 interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
 interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
 interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
+interpretation "setoid3 symmetry" 'invert r = (sym3 ____ r).
 interpretation "setoid2 symmetry" 'invert r = (sym2 ____ r).
 interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
 interpretation "setoid symmetry" 'invert r = (sym ____ r).
 notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans3" 'trans r = (trans3 _____ r).
 interpretation "trans2" 'trans r = (trans2 _____ r).
 interpretation "trans1" 'trans r = (trans1 _____ r).
 interpretation "trans" 'trans r = (trans _____ r).
@@ -117,6 +125,11 @@ record unary_morphism2 (A,B: setoid2) : Type2 ≝
    prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
  }.
 
+record unary_morphism3 (A,B: setoid3) : Type3 ≝
+ { fun13:1> A → B;
+   prop13: ∀a,a'. eq3 ? a a' → eq3 ? (fun13 a) (fun13 a')
+ }.
+
 record binary_morphism (A,B,C:setoid) : Type0 ≝
  { fun2:2> A → B → C;
    prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
@@ -132,18 +145,33 @@ record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
    prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
  }.
 
+record binary_morphism3 (A,B,C:setoid3) : Type3 ≝
+ { fun23:2> A → B → C;
+   prop23: ∀a,a',b,b'. eq3 ? a a' → eq3 ? b b' → eq3 ? (fun23 a b) (fun23 a' b')
+ }.
+
 notation "† c" with precedence 90 for @{'prop1 $c }.
 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
 notation "#" with precedence 90 for @{'refl}.
 interpretation "prop1" 'prop1 c  = (prop1 _____ c).
 interpretation "prop11" 'prop1 c = (prop11 _____ c).
 interpretation "prop12" 'prop1 c = (prop12 _____ c).
+interpretation "prop13" 'prop1 c = (prop13 _____ c).
 interpretation "prop2" 'prop2 l r = (prop2 ________ l r).
 interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
 interpretation "prop22" 'prop2 l r = (prop22 ________ l r).
+interpretation "prop23" 'prop2 l r = (prop23 ________ l r).
 interpretation "refl" 'refl = (refl ___).
 interpretation "refl1" 'refl = (refl1 ___).
 interpretation "refl2" 'refl = (refl2 ___).
+interpretation "refl3" 'refl = (refl3 ___).
+
+definition unary_morphism2_of_unary_morphism1: ∀S,T.unary_morphism1 S T → unary_morphism2 S T.
+ intros;
+ constructor 1;
+  [ apply (fun11 ?? u);
+  | apply (prop11 ?? u); ]
+qed.
 
 definition CPROP: setoid1.
  constructor 1;
@@ -233,6 +261,17 @@ record category2 : Type3 ≝
    id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
  }.
 
+record category3 : Type4 ≝
+ { objs3:> Type3;
+   arrows3: objs3 → objs3 → setoid3;
+   id3: ∀o:objs3. arrows3 o o;
+   comp3: ∀o1,o2,o3. binary_morphism3 (arrows3 o1 o2) (arrows3 o2 o3) (arrows3 o1 o3);
+   comp_assoc3: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+    comp3 o1 o3 o4 (comp3 o1 o2 o3 a12 a23) a34 = comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
+   id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a = a;
+   id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) = a
+ }.
+
 notation "'ASSOC'" with precedence 90 for @{'assoc}.
 
 interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
@@ -242,6 +281,85 @@ interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
 interpretation "category assoc" 'assoc = (comp_assoc ________).
 
+definition category2_of_category1: category1 → category2.
+ intro;
+ constructor 1;
+  [ apply (objs1 c);
+  | intros; apply (setoid2_of_setoid1 (arrows1 c o o1));
+  | apply (id1 c);
+  | intros;
+    constructor 1;
+     [ intros; apply (comp1 c o1 o2 o3 c1 c2);
+     | intros; whd in e e1 a a' b b'; change with (eq1 ? (b∘a) (b'∘a')); apply (e‡e1); ]
+  | intros; simplify; whd in a12 a23 a34; whd; apply rule (ASSOC);
+  | intros; simplify; whd in a; whd; apply id_neutral_right1;
+  | intros; simplify; whd in a; whd; apply id_neutral_left1; ]
+qed.
+(*coercion category2_of_category1.*)
+
+record functor2 (C1: category2) (C2: category2) : Type3 ≝
+ { map_objs2:1> C1 → C2;
+   map_arrows2: ∀S,T. unary_morphism2 (arrows2 ? S T) (arrows2 ? (map_objs2 S) (map_objs2 T));
+   respects_id2: ∀o:C1. map_arrows2 ?? (id2 ? o) = id2 ? (map_objs2 o);
+   respects_comp2:
+     ∀o1,o2,o3,o4.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.∀f3:arrows2 ? o3 o4.
+     map_arrows2 ?? (f3 ∘ f2 ∘ f1) =
+      map_arrows2 ?? f3 ∘ map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
+
+definition functor2_setoid: category2 → category2 → setoid3.
+ intros (C1 C2);
+ constructor 1;
+  [ apply (functor2 C1 C2);
+  | constructor 1;
+     [ intros (f g);
+       apply (∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
+     | simplify; intros; apply cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
+     | simplify; intros; apply cic:/matita/logic/equality/sym_eq.con; apply H;
+     | simplify; intros; apply cic:/matita/logic/equality/trans_eq.con;
+        [2: apply H; | skip | apply H1;]]]
+qed.
+
+definition functor2_of_functor2_setoid: ∀S,T. functor2_setoid S T → functor2 S T ≝ λS,T,x.x.
+coercion functor2_of_functor2_setoid.
+
+definition CAT2: category3.
+ constructor 1;
+  [ apply category2;
+  | apply functor2_setoid;
+  | intros; constructor 1;
+     [ apply (λx.x);
+     | intros; constructor 1;
+        [ apply (λx.x);
+        | intros; assumption;]
+     | intros; apply rule #;
+     | intros; apply rule #; ]
+  | intros; constructor 1;
+     [ intros; constructor 1;
+        [ intros; apply (c1 (c o));
+        | intros; constructor 1;
+           [ intro; apply (map_arrows2 ?? c1 ?? (map_arrows2 ?? c ?? c2));
+           | intros; apply (††e); ]
+        | intros; simplify;
+          apply (.= †(respects_id2 : ?));
+          apply (respects_id2 : ?);
+        | intros; simplify;
+          apply (.= †(respects_comp2 : ?));
+          apply (respects_comp2 : ?); ]
+        | intros; intro; simplify;
+          apply (cic:/matita/logic/equality/eq_ind.con ????? (e ?));
+          apply (cic:/matita/logic/equality/eq_ind.con ????? (e1 ?));
+          constructor 1; ]
+        | intros; intro; simplify; constructor 1;
+        | intros; intro; simplify; constructor 1;
+        | intros; intro; simplify; constructor 1; ]
+qed.
+
+definition category2_of_objs3_CAT2: objs3 CAT2 → category2 ≝ λx.x.
+coercion category2_of_objs3_CAT2.
+
+definition functor2_setoid_of_arrows3_CAT2: ∀S,T. arrows3 CAT2 S T → functor2_setoid S T ≝ λS,T,x.x.
+coercion functor2_setoid_of_arrows3_CAT2.
+
 definition unary_morphism_setoid: setoid → setoid → setoid.
  intros;
  constructor 1;
@@ -296,7 +414,7 @@ definition unary_morphism1_of_unary_morphism1_setoid1 :
   ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
 coercion unary_morphism1_of_unary_morphism1_setoid1.
 
-definition SET1: category2.
+definition SET1: objs3 CAT2.
  constructor 1;
   [ apply setoid1;
   | apply rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
@@ -325,4 +443,4 @@ coercion objs2_of_category1.
 prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
 prefer coercion Type_OF_objs1.
 
-interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
+interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
\ No newline at end of file

diff --git a/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma b/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma
index 167c33317c8d4810e7cd0533e5b3aab7b0bf7b6f..644acc2183a3d813951252fd8921a175f4b5c340 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma
@@ -14,7 +14,8 @@
 
 include "logic/connectives.ma".
 
-definition Type3 : Type := Type.
+definition Type4 : Type := Type.
+definition Type3 : Type4 := Type.
 definition Type2 : Type3 := Type.
 definition Type1 : Type2 := Type.
 definition Type0 : Type1 := Type.
@@ -23,20 +24,25 @@ 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.
+definition Type_of_Type4: Type4 → Type := λx.x.
 coercion Type_of_Type0.
 coercion Type_of_Type1.
 coercion Type_of_Type2.
 coercion Type_of_Type3.
+coercion Type_of_Type4.
 
 definition CProp0 : Type1 := Type0.
 definition CProp1 : Type2 := Type1.
 definition CProp2 : Type3 := Type2.
+definition CProp3 : Type4 := Type3.
 definition CProp_of_CProp0: CProp0 → CProp ≝ λx.x.
 definition CProp_of_CProp1: CProp1 → CProp ≝ λx.x.
 definition CProp_of_CProp2: CProp2 → CProp ≝ λx.x.
+definition CProp_of_CProp3: CProp3 → CProp ≝ λx.x.
 coercion CProp_of_CProp0.
 coercion CProp_of_CProp1.
 coercion CProp_of_CProp2.
+coercion CProp_of_CProp3.
 
 inductive Or (A,B:CProp0) : CProp0 ≝
  | Left : A → Or A B
@@ -155,3 +161,15 @@ definition antisymmetric: ∀A:Type0. ∀R:A→A→CProp0. ∀eq:A→A→Prop.CP
 definition reflexive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝ λA:Type0.λR:A→A→CProp0.∀x:A.R x x.
 
 definition transitive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝ λA:Type0.λR:A→A→CProp0.∀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.
+
+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.
+
+definition reflexive3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λA:Type3.λR:A→A→CProp3.∀x:A.R x x.
+definition symmetric3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λC:Type3.λlt:C→C→CProp3. ∀x,y:C.lt x y → lt y x.
+definition transitive3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λA:Type3.λR:A→A→CProp3.∀x,y,z:A.R x y → R y z → R x z.

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
index 3317c0e64672b220864f8c975903cdfe61163286..a7041357af3a0cffd16f257a87008e4be21aac7f 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
@@ -157,3 +157,17 @@ lemma orelation_of_relation_preserves_composition:
   | whd; intros; apply f; exists; [ apply y] split; assumption;
   | cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
 qed.
+
+definition SUBSETS': carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
+ constructor 1;
+  [ apply SUBSETS;
+  | intros; constructor 1;
+     [ apply (orelation_of_relation S T);
+     | intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
+  | apply orelation_of_relation_preserves_identity;
+  | simplify; intros;
+    apply (.= (orelation_of_relation_preserves_composition o1 o2 o4 f1 (f3∘f2)));
+    apply (#‡(orelation_of_relation_preserves_composition o2 o3 o4 f2 f3)); ]
+qed.
+    
+    
\ No newline at end of file