some work

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 9cadd514c24244d46c80ddaeb01e44d62c2f80c0..78372e72fc6e2394234451fd10f8e30b5728cdb7 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -22,23 +22,23 @@ constructor 1; [apply bool] constructor 1;
 [ intros (x y); apply (match x with [ true ⇒ match y with [ true ⇒ True | _ ⇒ False] | false ⇒ match y with [ true ⇒ False | false ⇒ True ]]);
 | whd; simplify; intros; cases x; apply I;
 | whd; simplify; intros 2; cases x; cases y; simplify; intros; assumption;
-| whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; try assumption; apply I]
+| whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; 
+  try assumption; apply I]
 qed.
 
-definition hint: objs1 SET → setoid.
- intros; apply o;
-qed.
+definition setoid_OF_SET: objs1 SET → setoid.
+ intros; apply o; qed.
 
-coercion hint.
+coercion setoid_OF_SET.
 
 lemma IF_THEN_ELSE_p :
   ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y → 
     (λm.match m with [ true ⇒ a | false ⇒ b ]) x =
     (λm.match m with [ true ⇒ a | false ⇒ b ]) y.
+whd in ⊢ (?→?→?→%→?);
 intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
 qed. 
 
-
 interpretation "unary morphism comprehension with no proof" 'comprehension T P = 
   (mk_unary_morphism T _ P _).
 
@@ -50,7 +50,6 @@ 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).
 
-
 record OAlgebra : Type := {
   oa_P :> SET;
   oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
@@ -70,8 +69,7 @@ record OAlgebra : Type := {
   oa_overlap_preservers_meet: 
       ∀p,q.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 });
-     (*(oa_meet BOOL (if_then_else oa_P p q));*)
-  oa_join_split: (* ha I → oa_P da castare a funX (ums A oa_P) *)
+  oa_join_split:
       ∀I:SET.∀p.∀q:arrows1 SET I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
   (*oa_base : setoid;
   oa_enum : ums oa_base oa_P;
@@ -122,33 +120,25 @@ interpretation "o-algebra join with explicit function" 'oa_join_mk f =
   (fun_1 __ (oa_join __) (mk_unary_morphism _ _ f _)).
 
 record ORelation (P,Q : OAlgebra) : Type ≝ {
-  or_f :> arrows1 SET P Q;
-  or_f_minus_star : arrows1 SET P Q;
-  or_f_star : arrows1 SET Q P;
-  or_f_minus : arrows1 SET Q P;
-  or_prop1 : ∀p,q. (or_f p ≤ q) = (p ≤ or_f_star q);
-  or_prop2 : ∀p,q. (or_f_minus p ≤ q) = (p ≤ or_f_minus_star q);
-  or_prop3 : ∀p,q. (or_f p >< q) = (p >< or_f_minus q)
+  or_f_ : arrows1 SET P Q;
+  or_f_minus_star_ : arrows1 SET P Q;
+  or_f_star_ : arrows1 SET Q P;
+  or_f_minus_ : arrows1 SET Q P;
+  or_prop1_ : ∀p,q. (or_f_ p ≤ q) = (p ≤ or_f_star_ q);
+  or_prop2_ : ∀p,q. (or_f_minus_ p ≤ q) = (p ≤ or_f_minus_star_ q);
+  or_prop3_ : ∀p,q. (or_f_ p >< q) = (p >< or_f_minus_ q)
 }.
 
-notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
-notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
-interpretation "o-relation f*" 'OR_f_star r = (or_f_star _ _ r).
-
-notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
-notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
-interpretation "o-relation f⎻*" 'OR_f_minus_star r = (or_f_minus_star _ _ r).
-
-notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
-notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
-interpretation "o-relation f⎻" 'OR_f_minus r = (or_f_minus _ _ r).
 
 definition ORelation_setoid : OAlgebra → OAlgebra → setoid1.
 intros (P Q);
 constructor 1;
 [ apply (ORelation P Q);
 | constructor 1;
-   [ apply (λp,q. And4 (eq1 ? p⎻* q⎻* ) (eq1 ? p⎻ q⎻) (eq1 ? p q) (eq1 ? p* q* )); 
+   [ apply (λp,q. And4 (eq1 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q)) 
+             (eq1 ? (or_f_minus_ ?? p) (or_f_minus_ ?? q)) 
+             (eq1 ? (or_f_ ?? p) (or_f_ ?? q)) 
+             (eq1 ? (or_f_star_ ?? p) (or_f_star_ ?? q))); 
    | whd; simplify; intros; repeat split; intros; apply refl1;
    | whd; simplify; intros; cases H; clear H; split; 
      intro a; apply sym; generalize in match a;assumption;
@@ -159,43 +149,87 @@ constructor 1;
      | apply (.= (H5 a)); apply H9;]]]
 qed.  
 
-lemma hint1 : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q. intros; apply (or_f ?? c);qed.
-coercion hint1.
-
-lemma hint3 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply c;qed.
-coercion hint3.
-
-lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
-coercion hint2.
-
-definition or_f_minus_star2: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+definition or_f_minus_star: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
  intros; constructor 1;
-  [ apply or_f_minus_star;
+  [ apply or_f_minus_star_;
   | intros; cases H; assumption]
 qed.
 
-definition or_f2: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+definition or_f: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
  intros; constructor 1;
-  [ apply or_f;
+  [ apply or_f_;
   | intros; cases H; assumption]
 qed.
 
-definition or_f_minus2: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
+coercion or_f.
+
+definition or_f_minus: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
  intros; constructor 1;
-  [ apply or_f_minus;
+  [ apply or_f_minus_;
   | intros; cases H; assumption]
 qed.
 
-definition or_f_star2: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
+definition or_f_star: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
  intros; constructor 1;
-  [ apply or_f_star;
+  [ apply or_f_star_;
   | intros; cases H; assumption]
 qed.
 
-interpretation "o-relation f⎻* 2" 'OR_f_minus_star r = (fun_1 __ (or_f_minus_star2 _ _) r).
-interpretation "o-relation f⎻ 2" 'OR_f_minus r = (fun_1 __ (or_f_minus2 _ _) r).
-interpretation "o-relation f* 2" 'OR_f_star r = (fun_1 __ (or_f_star2 _ _) r).
-coercion or_f2.
+lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q.
+intros; apply (or_f ?? c);
+qed.
+
+coercion arrows1_OF_ORelation_setoid nocomposites.
+
+lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → P ⇒ Q.
+intros; apply (or_f ?? c);
+qed.
+
+coercion umorphism_OF_ORelation_setoid.
+
+
+lemma uncurry_arrows : ∀B,C. arrows1 SET B C → B → C. 
+intros; apply ((fun_1 ?? c) t);
+qed.
+
+coercion uncurry_arrows 1.
+
+lemma hint3 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply c;qed.
+coercion hint3 nocomposites.
+
+(*
+lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
+coercion hint2 nocomposites.
+*)
+
+
+notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
+notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
+
+notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+
+notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
+notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
+
+interpretation "o-relation f⎻*" 'OR_f_minus_star r = (fun_1 __ (or_f_minus_star _ _) r).
+interpretation "o-relation f⎻" 'OR_f_minus r = (fun_1 __ (or_f_minus _ _) r).
+interpretation "o-relation f*" 'OR_f_star r = (fun_1 __ (or_f_star _ _) r).
+
+definition or_prop1 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+   (F p ≤ q) = (p ≤ F* q).
+intros; apply (or_prop1_ ?? F p q);
+qed.
+
+definition or_prop2 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+   (F⎻ p ≤ q) = (p ≤ F⎻* q).
+intros; apply (or_prop2_ ?? F p q);
+qed.
+
+definition or_prop3 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+   (F p >< q) = (p >< F⎻ q).
+intros; apply (or_prop3_ ?? F p q);
+qed.
 
 definition ORelation_composition : ∀P,Q,R. 
   binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
@@ -203,29 +237,26 @@ intros;
 constructor 1;
 [ intros (F G);
   constructor 1;
-  [ apply (or_f2 ?? G ∘ or_f2 ?? F);
-  | alias symbol "compose" = "category1 composition".
-    apply (G⎻* ∘ F⎻* );
+  [ lapply (G ∘ F);
+    apply (G ∘ F);
+  | apply (G⎻* ∘ F⎻* );
   | apply (F* ∘ G* );
   | apply (F⎻ ∘ G⎻);
-  | intros;
-    alias symbol "eq" = "setoid1 eq".
+  | intros; 
     change with ((G (F p) ≤ q) = (p ≤ (F* (G* q))));
-    apply (.= or_prop1 ??? (F p) ?);
-    apply (.= or_prop1 ??? p ?);
-    apply refl1
-  | intros; alias symbol "eq" = "setoid1 eq".
+    apply (.= (or_prop1 :?));
+    apply (or_prop1 :?);
+  | intros;
     change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
-    alias symbol "trans" = "trans1".
-    apply (.= or_prop2 ?? F ??);
-    apply (.= or_prop2 ?? G ??);
-    apply refl1;
+    apply (.= (or_prop2 :?));
+    apply or_prop2 ; 
   | intros; change with ((G (F (p)) >< q) = (p >< (F⎻ (G⎻ q))));
-    apply (.= or_prop3 ??? (F p) ?);
-    apply (.= or_prop3 ??? p ?);
-    apply refl1
+    apply (.= (or_prop3 :?));
+    apply or_prop3;
   ]
-| intros; split; simplify; [1,3: apply ((†H)‡(†H1)); | 2,4: apply ((†H1)‡(†H));]]
+| intros; split; simplify; 
+   [1,3: unfold arrows1_OF_ORelation_setoid; apply ((†H)‡(†H1));
+   |2,4: apply ((†H1)‡(†H));]]
 qed.
 
 definition OA : category1.
@@ -236,12 +267,12 @@ split;
   [1,2,3,4: apply id1;
   |5,6,7:intros; apply refl1;] 
 | apply ORelation_composition;
-| intros; split;
-   [ apply (comp_assoc1 ????? (a12⎻* ) (a23⎻* ) (a34⎻* ));
-   | alias symbol "invert" = "setoid1 symmetry".
-     apply ((comp_assoc1 ????? (a34⎻) (a23⎻) (a12⎻)) \sup -1);
-   | apply (comp_assoc1 ????? a12 a23 a34);
-   | apply ((comp_assoc1 ????? (a34* ) (a23* ) (a12* )) \sup -1);]
+| intros (P Q R S F G H); split;
+   [ change with (H⎻* ∘ G⎻* ∘ F⎻* = H⎻* ∘ (G⎻* ∘ F⎻* ));
+     apply (comp_assoc1 ????? (F⎻* ) (G⎻* ) (H⎻* ));
+   | apply ((comp_assoc1 ????? (H⎻) (G⎻) (F⎻))^-1);
+   | apply ((comp_assoc1 ????? F G H)^-1);
+   | apply ((comp_assoc1 ????? H* G* F* ));]
 | intros; split; unfold ORelation_composition; simplify; apply id_neutral_left1;
 | intros; split; unfold ORelation_composition; simplify; apply id_neutral_right1;]
 qed.
\ No newline at end of file

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma b/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
index cbdf68fdd99c93c3620ee020e63e28a15bea8ef2..633f0b89e01591db6d5c73c75800fa608ed50385 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
@@ -17,34 +17,37 @@ include "o-saturations.ma".
 
 notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
 notation > "□_term 90 b" non associative with precedence 90 for @{'box $b}.
-interpretation "Universal image ⊩⎻*" 'box x = (or_f_minus_star _ _ (rel x)).
+interpretation "Universal image ⊩⎻*" 'box x = (fun_1 _ _ (or_f_minus_star _ _) (rel x)).
  
 notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
 notation > "◊_term 90 b" non associative with precedence 90 for @{'diamond $b}.
-interpretation "Existential image ⊩" 'diamond x = (or_f _ _ (rel x)).
+interpretation "Existential image ⊩" 'diamond x = (fun_1 _ _ (or_f _ _) (rel x)).
 
 notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
 notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
-interpretation "Universal pre-image ⊩*" 'rest x = (or_f_star _ _ (rel x)).
+interpretation "Universal pre-image ⊩*" 'rest x = (fun_1 _ _ (or_f_star _ _) (rel x)).
 
 notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
 notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
-interpretation "Existential pre-image ⊩⎻" 'ext x = (or_f_minus _ _ (rel x)).
+interpretation "Existential pre-image ⊩⎻" 'ext x = (fun_1 _ _ (or_f_minus _ _) (rel x)).
+
+lemma hint : ∀p,q.arrows1 OA p q → ORelation_setoid p q.
+intros; assumption;
+qed.
+
+coercion hint nocomposites.
 
 definition A : ∀b:BP. unary_morphism (oa_P (form b)) (oa_P (form b)).
 intros; constructor 1;
  [ apply (λx.□_b (Ext⎽b x));
- | do 2 unfold FunClass_1_OF_carr1; intros; apply  (†(†H));]
+ | do 2 unfold uncurry_arrows; intros; apply  (†(†H));]
 qed.
 
 lemma xxx : ∀x.carr x → carr1 (setoid1_of_setoid x). intros; assumption; qed.
-coercion xxx.
+coercion xxx nocomposites.
 
-definition d_p_i : 
-  ∀S,I:SET.∀d:unary_morphism S S.∀p:arrows1 SET I S.arrows1 SET I S.
-intros; constructor 1;
- [ apply (λi:I. u (c i));
- | unfold FunClass_1_OF_carr1; intros; apply (†(†H));].
+lemma down_p : ∀S,I:SET.∀u:S⇒S.∀c:arrows1 SET I S.∀a:I.∀a':I.a=a'→u (c a)=u (c a').
+intros; unfold uncurry_arrows; apply (†(†H));
 qed.
 
 alias symbol "eq" = "setoid eq".
@@ -58,12 +61,13 @@ record concrete_space : Type ≝
      (Ext⎽bp q1 ∧ (Ext⎽bp q2)) = (Ext⎽bp ((downarrow q1) ∧ (downarrow q2)));
    all_covered: Ext⎽bp (oa_one (form bp)) = oa_one (concr bp);
    il2: ∀I:SET.∀p:arrows1 SET I (oa_P (form bp)).
-     downarrow (oa_join ? I (d_p_i ?? downarrow p)) =
-     oa_join ? I (d_p_i ?? downarrow p);
+     downarrow (∨ { x ∈ I | downarrow (p x) | down_p ???? }) =
+     ∨ { x ∈ I | downarrow (p x) | down_p ???? };
    il1: ∀q.downarrow (A ? q) = A ? q
  }.
 
-interpretation "o-concrete space downarrow" 'downarrow x = (fun_1 __ (downarrow _) x).
+interpretation "o-concrete space downarrow" 'downarrow x = 
+  (fun_1 __ (downarrow _) x).
 
 definition bp': concrete_space → basic_pair ≝ λc.bp c.
 coercion bp'.