WARNING: partial commit to try to understand something.

I have fixed the universes in categories, adding a new category (of level 2)
and trying to prove that powersets form a SET1.

Failing, so far.

diff --git a/helm/software/matita/contribs/formal_topology/overlap/categories.ma b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
new file mode 100644 (file)
index 0000000..5afda73
--- /dev/null
+++ b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
@@ -0,0 +1,376 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+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.
+
+record equivalence_relation (A:Type0) : Type1 ≝
+ { eq_rel:2> A → A → CProp0;
+   refl: reflexive ? eq_rel;
+   sym: symmetric ? eq_rel;
+   trans: transitive ? eq_rel
+ }.
+
+record setoid : Type1 ≝
+ { carr:> Type0;
+   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.
+
+record equivalence_relation1 (A:Type1) : Type1 ≝
+ { eq_rel1:2> A → A → CProp1;
+   refl1: reflexive1 ? eq_rel1;
+   sym1: symmetric1 ? eq_rel1;
+   trans1: transitive1 ? eq_rel1
+ }.
+
+record setoid1: Type2 ≝
+ { carr1:> Type1;
+   eq1: equivalence_relation1 carr1
+ }.
+
+definition setoid1_of_setoid: setoid → setoid1.
+ intro;
+ constructor 1;
+  [ apply (carr s)
+  | constructor 1;
+    [ apply (eq_rel s);
+      apply (eq s)
+    | apply (refl s)
+    | apply (sym s)
+    | apply (trans s)]]
+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.
+
+record equivalence_relation2 (A:Type2) : Type2 ≝
+ { eq_rel2:2> A → A → CProp2;
+   refl2: reflexive2 ? eq_rel2;
+   sym2: symmetric2 ? eq_rel2;
+   trans2: transitive2 ? eq_rel2
+ }.
+
+record setoid2: Type3 ≝
+ { carr2:> Type2;
+   eq2: equivalence_relation2 carr2
+ }.
+
+(*
+definition Leibniz: Type → setoid.
+ intro;
+ constructor 1;
+  [ apply T
+  | constructor 1;
+     [ apply (λx,y:T.cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y)
+     | alias id "refl_eq" = "cic:/matita/logic/equality/eq.ind#xpointer(1/1/1)".
+       apply refl_eq
+     | alias id "sym_eq" = "cic:/matita/logic/equality/sym_eq.con".
+       apply sym_eq
+     | alias id "trans_eq" = "cic:/matita/logic/equality/trans_eq.con".
+       apply trans_eq ]]
+qed.
+
+coercion Leibniz.
+*)
+
+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 "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 "trans2" 'trans r = (trans2 _____ r).
+interpretation "trans1" 'trans r = (trans1 _____ r).
+interpretation "trans" 'trans r = (trans _____ r).
+
+record unary_morphism (A,B: setoid) : Type0 ≝
+ { fun1:1> A → B;
+   prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
+ }.
+
+record unary_morphism1 (A,B: setoid1) : Type1 ≝
+ { fun11:1> A → B;
+   prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
+ }.
+
+record unary_morphism2 (A,B: setoid2) : Type2 ≝
+ { fun12:1> A → B;
+   prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 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')
+ }.
+
+record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
+ { fun21:2> A → B → C;
+   prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
+ }.
+
+record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
+ { fun22:2> A → B → C;
+   prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 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 "prop2" 'prop2 l r = (prop2 ________ l r).
+interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
+interpretation "refl" 'refl = (refl ___).
+interpretation "refl1" 'refl = (refl1 ___).
+interpretation "refl2" 'refl = (refl2 ___).
+
+definition CPROP: setoid1.
+ constructor 1;
+  [ apply CProp0
+  | 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))]]]
+qed.
+
+definition if': ∀A,B:CPROP. A = B → A → B.
+ intros; apply (if ?? e); assumption.
+qed.
+
+notation ". r" with precedence 50 for @{'if $r}.
+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)
+     | apply (if ?? e1 b1)
+     | apply (fi ?? e a1)
+     | apply (fi ?? e1 b1)]]
+qed.
+
+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]
+     [ apply (if ?? e a1)
+     | apply (fi ?? e a1)
+     | apply (if ?? e1 b1)
+     | apply (fi ?? e1 b1)]]
+qed.
+
+interpretation "or_morphism" 'or a b = (fun21 ___ or_morphism a b).
+
+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]]
+qed.
+
+(*
+definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
+ intro;
+ constructor 1;
+  [ apply (eq_rel ? (eq S))
+  | intros; split; intro;
+     [ apply (.= H \sup -1);
+       apply (.= H2);
+       assumption
+     | apply (.= H);
+       apply (.= H2);
+       apply (H1 \sup -1)]]
+qed.
+*)
+
+record category : Type1 ≝
+ { objs:> Type0;
+   arrows: objs → objs → setoid;
+   id: ∀o:objs. arrows o o;
+   comp: ∀o1,o2,o3. binary_morphism1 (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
+   comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+    comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
+   id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
+   id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
+ }.
+
+record category1 : Type2 ≝
+ { objs1:> Type1;
+   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_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
+ }.
+
+record category2 : Type3 ≝
+ { objs2:> Type2;
+   arrows2: objs2 → objs2 → setoid2;
+   id2: ∀o:objs2. arrows2 o o;
+   comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
+   comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+    comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 = comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
+   id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a = a;
+   id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
+ }.
+
+notation "'ASSOC'" with precedence 90 for @{'assoc}.
+notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
+notation "'ASSOC2'" with precedence 90 for @{'assoc2}.
+
+interpretation "category1 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
+interpretation "category1 assoc" 'assoc1 = (comp_assoc2 ________).
+interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
+interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
+interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
+interpretation "category assoc" 'assoc = (comp_assoc ________).
+
+(* bug grande come una casa?
+   Ma come fa a passare la quantificazione larga??? *)
+definition unary_morphism_setoid: setoid → setoid → setoid.
+ intros;
+ constructor 1;
+  [ apply (unary_morphism s s1);
+  | constructor 1;
+     [ intros (f g); apply (∀a:s. 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.
+
+definition SET: category1.
+ constructor 1;
+  [ apply setoid;
+  | apply rule (λS,T:setoid.unary_morphism_setoid S T);
+  | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
+  | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
+     apply († (†e));]
+  | intros; whd; intros; simplify; whd in H1; whd in H;
+    apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
+     [ apply Hletin | apply (e a1); ]  | apply e1; ]]
+  | 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 setoid1_of_SET: SET → setoid1.
+ intro; whd in t; apply setoid1_of_setoid; apply t.
+qed.
+coercion setoid1_of_SET.
+
+definition eq': ∀w:SET.equivalence_relation ? := λw.eq w.
+
+definition prop1_SET : 
+ ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:Type_OF_objs1 A.eq' ? a b→eq' ? (w a) (w b).
+intros; apply (prop1 A B w a b e);
+qed.
+
+
+interpretation "SET dagger" 'prop1 h = (prop1_SET _ _ _ _ _ h).
+notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
+interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
+interpretation "SET eq" 'eq x y = (eq_rel _ (eq' _) x y).
+
+definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid2.
+ intros;
+ constructor 1;
+  [ apply (unary_morphism1 s s1);
+  | constructor 1;
+     [ intros (f g); 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]]]
+qed.
+
+definition SET1: category2.
+ constructor 1;
+  [ apply setoid1;
+  | apply rule (λS,T.unary_morphism1_setoid1 S T);
+  | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
+  | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
+     apply († (†e));]
+  | intros; whd; intros; simplify; whd in H1; whd in H;
+    apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
+     [ apply Hletin | apply (e a1); ]  | apply e1; ]]
+  | intros; whd; intros; simplify; apply refl1;
+  | intros; simplify; whd; intros; simplify; apply refl1;
+  | intros; simplify; whd; intros; simplify; apply refl1;
+  ]
+qed.
+
+definition setoid1_OF_SET1: objs2 SET1 → setoid1.
+ intros; apply o; qed.
+
+coercion setoid1_OF_SET1.
+
+definition eq'': ∀w:SET1.equivalence_relation1 ? := λw.eq1 w.
+
+definition prop11_SET1 : 
+ ∀A,B:SET1.∀w:arrows2 SET1 A B.∀a,b:Type_OF_objs2 A.eq'' ? a b→eq'' ? (w a) (w b).
+intros; apply (prop11 A B w a b e);
+qed.
+
+interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
+notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
+interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
+interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).
\ No newline at end of file

diff --git a/helm/software/matita/contribs/formal_topology/overlap/depends b/helm/software/matita/contribs/formal_topology/overlap/depends
index 0032b3151b0fefd9c3d8946b602610bac65ab56e..377b1d24327157451666edd9690929bf39894db6 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/depends
+++ b/helm/software/matita/contribs/formal_topology/overlap/depends
@@ -2,7 +2,8 @@ o-basic_pairs.ma datatypes/categories.ma o-algebra.ma
 o-concrete_spaces.ma o-basic_pairs.ma o-saturations.ma
 o-saturations.ma o-algebra.ma
 o-algebra.ma datatypes/categories.ma logic/cprop_connectives.ma
-o-formal_topologies.ma o-concrete_spaces.ma
+o-formal_topologies.ma o-basic_topologies.ma
+categories.ma logic/cprop_connectives.ma
 o-basic_topologies.ma o-algebra.ma o-saturations.ma
 datatypes/categories.ma 
 logic/cprop_connectives.ma 

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
index cb211341427ed8cc934f3e390d76b611af124f80..617c58b602c12c06431457c4fad0d984bb162dbc 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -12,10 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "datatypes/categories.ma".
+include "categories.ma".
 include "logic/cprop_connectives.ma". 
 
-inductive bool : Type := true : bool | false : bool.
+inductive bool : Type0 := true : bool | false : bool.
 
 lemma BOOL : objs1 SET.
 constructor 1; [apply bool] constructor 1;
@@ -36,6 +36,8 @@ qed.
 
 interpretation "unary morphism comprehension with no proof" 'comprehension T P = 
   (mk_unary_morphism T _ P _).
+interpretation "unary morphism1 comprehension with no proof" 'comprehension T P = 
+  (mk_unary_morphism1 T _ P _).
 
 notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
 for @{ 'comprehension_by $s (λ${ident i}. $p) $by}.
@@ -44,17 +46,23 @@ for @{ 'comprehension_by $s (λ${ident i}:$_. $p) $by}.
 
 interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p = 
   (mk_unary_morphism s _ f p).
+interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \eta.f p = 
+  (mk_unary_morphism1 s _ f p).
+
+definition Type_of_SET: SET → Type0 := λS.carr S.
+coercion Type_of_SET.
 
 (* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
    lattices, Definizione 0.9 *)
 (* USARE L'ESISTENZIALE DEBOLE *)
 (* Far salire SET usando setoidi1 *)
-record OAlgebra : Type := {
+alias symbol "comprehension_by" = "unary morphism comprehension with proof".
+record OAlgebra : Type1 := {
   oa_P :> SET;
-  oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1, CPROP importante che sia small *)
-  oa_overlap: binary_morphism1 oa_P oa_P CPROP;
-  oa_meet: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
-  oa_join: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
+  oa_leq : binary_morphism1 (setoid1_of_setoid oa_P) (setoid1_of_setoid oa_P) CPROP; (* CPROP is setoid1, CPROP importante che sia small *)
+  oa_overlap: binary_morphism1 (setoid1_of_setoid oa_P) (setoid1_of_setoid oa_P) CPROP;
+  oa_meet: ∀I:SET.unary_morphism1 (arrows1 SET I oa_P) (setoid1_of_setoid oa_P);
+  oa_join: ∀I:SET.unary_morphism1 (arrows1 SET I oa_P) (setoid1_of_setoid oa_P);
   oa_one: oa_P;
   oa_zero: oa_P;
   oa_leq_refl: ∀a:oa_P. oa_leq a a; 
@@ -68,7 +76,7 @@ record OAlgebra : Type := {
   oa_one_top: ∀p:oa_P.oa_leq p oa_one;
   (* preservers!! (typo) *)
   oa_overlap_preservers_meet_: 
-      ∀p,q.oa_overlap p q → oa_overlap p 
+      ∀p,q:oa_P.oa_overlap p q → oa_overlap p 
        (oa_meet ? { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
   (* ⇔ deve essere =, l'esiste debole *)
   oa_join_split:
@@ -105,9 +113,9 @@ notation > "hovbox(a ∧ b)" left associative with precedence 50
 for @{ 'oa_meet (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
 *)
 interpretation "o-algebra meet" 'oa_meet f = 
-  (fun_1 __ (oa_meet __) f).
+  (fun11 __ (oa_meet __) f).
 interpretation "o-algebra meet with explicit function" 'oa_meet_mk f = 
-  (fun_1 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
+  (fun11 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
 
 definition binary_meet : ∀O:OAlgebra. binary_morphism1 O O O.
 intros; split;

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-formal_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/o-formal_topologies.ma
index 1693b55f61eb62677243c92d2e0a61380477adac..e136821af74f5a5c025ce0923e9f39deb3116880 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-formal_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-formal_topologies.ma
@@ -12,13 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "o-concrete_spaces.ma".
-
-definition btop_carr: BTop → Type ≝ λo:BTop. carr (carrbt o).
+include "o-basic_topologies.ma".
 
+definition btop_carr: BTop → Type ≝ λo:BTop. carr1 (oa_P (carrbt o)).
 coercion btop_carr.
 
-definition btop_carr': BTop → setoid ≝ λo:BTop. carrbt o.
+(*
+definition btop_carr': BTop → setoid1 ≝ λo:BTop. carrbt o.
 
 coercion btop_carr'.
 
@@ -40,12 +40,14 @@ definition ffintersects: ∀S:BTop. binary_morphism1 (Ω \sup S) (Ω \sup S) (Ω
 qed.
 
 interpretation "ffintersects" 'fintersects U V = (fun1 ___ (ffintersects _) U V).
+*)
 
 record formal_topology: Type ≝
  { bt:> BTop;
-   converges: ∀U,V: Ω \sup bt. A ? (U ↓ V) = A ? U ∩ A ? V
+   converges: ∀U,V: bt. A ? (U ↓ V) = (A ? U ∧ A ? V)
  }.
 
+(*
 definition bt': formal_topology → basic_topology ≝ λo:formal_topology.bt o.
 
 coercion bt'.
@@ -57,7 +59,7 @@ definition ffintersects': ∀S:BTop. binary_morphism1 S S (Ω \sup S).
 qed.
 
 interpretation "ffintersects'" 'fintersects U V = (fun1 ___ (ffintersects' _) U V).
-
+*)
 record formal_map (S,T: formal_topology) : Type ≝
  { cr:> continuous_relation_setoid S T;
    C1: ∀b,c. extS ?? cr (b ↓ c) = ext ?? cr b ↓ ext ?? cr c;

diff --git a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
new file mode 100644 (file)
index 0000000..17128ae
--- /dev/null
+++ b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
@@ -0,0 +1,139 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "logic/cprop_connectives.ma".
+include "categories.ma".
+
+record powerset_carrier (A: SET) : Type1 ≝ { mem_operator: unary_morphism1 A CPROP }.
+
+definition subseteq_operator: ∀A: SET. powerset_carrier A → powerset_carrier A → CProp2 ≝
+ λA:SET.λU,V.∀a:A. mem_operator ? U a → mem_operator ? V a.
+
+theorem transitive_subseteq_operator: ∀A. transitive2 ? (subseteq_operator A).
+ intros 6; intros 2;
+ apply s1; apply s;
+ assumption.
+qed.
+
+definition powerset_setoid1: SET → SET1.
+ intros (T);
+ constructor 1;
+  [ apply (powerset_carrier T)
+  | constructor 1;
+     [ apply (λU,V. subseteq_operator ? U V ∧ subseteq_operator ? V U)
+     | simplify; intros; split; intros 2; assumption
+     | simplify; intros (x y H); cases H; split; assumption
+     | simplify; intros (x y z H H1); cases H; cases H1; split;
+       apply transitive_subseteq_operator; [1,4: apply y ]
+       assumption ]]
+qed.
+
+interpretation "powerset" 'powerset A = (powerset_setoid1 A).
+
+interpretation "subset construction" 'subset \eta.x =
+ (mk_powerset_carrier _ (mk_unary_morphism1 _ CPROP x _)).
+
+definition mem: ∀A. binary_morphism1 A (Ω \sup A) CPROP.
+ intros;
+ constructor 1;
+  [ apply (λx,S. mem_operator ? S x)
+  | intros 5;
+    cases b; clear b; cases b'; clear b'; simplify; intros;
+    apply (trans1 ????? (prop11 ?? u ?? e));
+    cases e1; whd in s s1;
+    split; intro;
+     [ apply s; assumption
+     | apply s1; assumption]]
+qed.
+
+interpretation "mem" 'mem a S = (fun21 ___ (mem _) a S).
+
+definition subseteq: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+ intros;
+ constructor 1;
+  [ apply (λU,V. subseteq_operator ? U V)
+  | intros;
+    cases e; cases e1;
+    split; intros 1;
+    [ apply (transitive_subseteq_operator ????? s2);
+      apply (transitive_subseteq_operator ???? s1 s4)
+    | apply (transitive_subseteq_operator ????? s3);
+      apply (transitive_subseteq_operator ???? s s4) ]]
+qed.
+
+interpretation "subseteq" 'subseteq U V = (fun21 ___ (subseteq _) U V).
+
+theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
+ intros 4; assumption.
+qed.
+
+theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
+ intros; apply transitive_subseteq_operator; [apply S2] assumption.
+qed.
+
+definition overlaps: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+ intros;
+ constructor 1;
+  [ apply (λA.λU,V:Ω \sup A.exT2 ? (λx:A.x ∈ U) (λx:A.x ∈ V))
+  | intros;
+    constructor 1; intro; cases H; exists; [1,4: apply w]
+     [ apply (. #‡e); assumption
+     | apply (. #‡e1); assumption
+     | apply (. #‡(e \sup -1)); assumption;
+     | apply (. #‡(e1 \sup -1)); assumption]]
+qed.
+
+interpretation "overlaps" 'overlaps U V = (fun21 ___ (overlaps _) U V).
+
+definition intersects:
+ ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+ intros;
+ constructor 1;
+  [ apply rule (λU,V. {x | x ∈ U ∧ x ∈ V });
+    intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1;
+  | intros;
+    split; intros 2; simplify in f ⊢ %;
+    [ apply (. (#‡e)‡(#‡e1)); assumption
+    | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+qed.
+
+interpretation "intersects" 'intersects U V = (fun21 ___ (intersects _) U V).
+
+definition union:
+ ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+ intros;
+ constructor 1;
+  [ apply (λU,V. {x | x ∈ U ∨ x ∈ V });
+    intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1
+  | intros;
+    split; intros 2; simplify in f ⊢ %;
+    [ apply (. (#‡e)‡(#‡e1)); assumption
+    | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+qed.
+
+interpretation "union" 'union U V = (fun21 ___ (union _) U V).
+
+definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
+ intros; constructor 1;
+  [ apply (λa:A.{b | eq ? a b}); unfold setoid1_of_setoid; simplify;
+    intros; simplify;
+    split; intro;
+    apply (.= e1);
+     [ apply e | apply (e \sup -1) ]
+  | unfold setoid1_of_setoid; simplify;
+    intros; split; intros 2; simplify in f ⊢ %; apply trans;
+     [ apply a |4: apply a'] try assumption; apply sym; assumption]
+qed.
+
+interpretation "singleton" 'singl a = (fun11 __ (singleton _) a).
\ No newline at end of file