X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-algebra.ma;h=6be19d92c31ac4518d402650c362e85dc4b3fd9b;hb=c78cbede35ed85575e274864e6b6b9c635c6956d;hp=ce9583da36098ffd0861c9b517aa3c84561be823;hpb=b93b2e4f499c30b01b838f75a1e95df43920ffcc;p=helm.git

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 ce9583da3..6be19d92c 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -13,7 +13,6 @@
 (**************************************************************************)
 
 include "categories.ma".
-include "logic/cprop_connectives.ma". 
 
 inductive bool : Type0 := true : bool | false : bool.
 
@@ -49,14 +48,6 @@ interpretation "unary morphism comprehension with proof" 'comprehension_by s \et
 interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \eta.f p = 
   (mk_unary_morphism1 s _ f p).
 
-definition hint: Type_OF_category2 SET1 → setoid2.
- intro; apply (setoid2_of_setoid1 t); qed.
-coercion hint.
-
-definition hint2: Type_OF_category1 SET → objs2 SET1.
- intro; apply (setoid1_of_setoid t); qed.
-coercion hint2.
-
 (* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
    lattices, Definizione 0.9 *)
 (* USARE L'ESISTENZIALE DEBOLE *)
@@ -84,7 +75,7 @@ record OAlgebra : Type2 := {
   (* ⇔ deve essere =, l'esiste debole *)
   oa_join_split:
       ∀I:SET.∀p.∀q:arrows2 SET1 I oa_P.
-       oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+       oa_overlap p (oa_join I q) ⇔ ∃i:carr I.oa_overlap p (q i);
   (*oa_base : setoid;
   1) enum non e' il nome giusto perche' non e' suriettiva
   2) manca (vedere altro capitolo) la "suriettivita'" come immagine di insiemi di oa_base
@@ -142,12 +133,14 @@ intros; split;
   intro x; simplify; cases x; simplify; assumption;]
 qed.
 
-notation "hovbox(a ∧ b)" left associative with precedence 35
-for @{ 'oa_meet_bin $a $b }.
-interpretation "o-algebra binary meet" 'oa_meet_bin a b = 
+interpretation "o-algebra binary meet" 'and a b = 
   (fun21 ___ (binary_meet _) a b).
 
+coercion Type1_OF_OAlgebra nocomposites.
+
 lemma oa_overlap_preservers_meet: ∀O:OAlgebra.∀p,q:O.p >< q → p >< (p ∧ q).
+(* next change to avoid universe inconsistency *)
+change in ⊢ (?→%→%→?) with (Type1_OF_OAlgebra O);
 intros;  lapply (oa_overlap_preserves_meet_ O p q f);
 lapply (prop21 O O CPROP (oa_overlap O) p p ? (p ∧ q) # ?);
 [3: apply (if ?? (Hletin1)); apply Hletin;|skip] apply refl1;
@@ -192,14 +185,14 @@ constructor 1;
 | constructor 1;
    (* tenere solo una uguaglianza e usare la proposizione 9.9 per
       le altre (unicita' degli aggiunti e del simmetrico) *)
-   [ apply (λp,q. And4 (eq2 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q)) 
+   [ apply (λp,q. And42 (eq2 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q)) 
              (eq2 ? (or_f_minus_ ?? p) (or_f_minus_ ?? q)) 
              (eq2 ? (or_f_ ?? p) (or_f_ ?? q)) 
              (eq2 ? (or_f_star_ ?? p) (or_f_star_ ?? q))); 
    | whd; simplify; intros; repeat split; intros; apply refl2;
-   | whd; simplify; intros; cases H; clear H; split; 
+   | whd; simplify; intros; cases a; clear a; split; 
      intro a; apply sym1; generalize in match a;assumption;
-   | whd; simplify; intros; cases H; cases H1; clear H H1; split; intro a;
+   | whd; simplify; intros; cases a; cases a1; clear a a1; split; intro a;
      [ apply (.= (e a)); apply e4;
      | apply (.= (e1 a)); apply e5;
      | apply (.= (e2 a)); apply e6;
@@ -239,28 +232,29 @@ qed.
 
 coercion arrows1_OF_ORelation_setoid.
 
-lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → P ⇒ Q.
+lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1 P Q.
 intros; apply (or_f ?? t);
 qed.
 
 coercion umorphism_OF_ORelation_setoid.
 
+lemma umorphism_setoid_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1_setoid1 P Q.
+intros; apply (or_f ?? t);
+qed.
+
+coercion umorphism_setoid_OF_ORelation_setoid.
 
-lemma uncurry_arrows : ∀B,C. arrows1 SET B C → B → C. 
-intros; apply ((fun1 ?? t) t1);
+lemma uncurry_arrows : ∀B,C. ORelation_setoid B C → B → C. 
+intros; apply ((fun11 ?? t) t1);
 qed.
 
 coercion uncurry_arrows 1.
 
-lemma hint6 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply t;qed.
+lemma hint6: ∀P,Q. Type_OF_setoid2 (hint5 P ⇒ hint5 Q) → unary_morphism1 P Q.
+ intros; apply t;
+qed.
 coercion hint6.
 
-(*
-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}.
 
@@ -290,13 +284,13 @@ 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).
+  binary_morphism2 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
 intros;
 constructor 1;
 [ intros (F G);
   constructor 1;
   [ apply (G ∘ F);
-  | apply (G⎻* ∘ F⎻* );
+  | apply rule (G⎻* ∘ F⎻* );
   | apply (F* ∘ G* );
   | apply (F⎻ ∘ G⎻);
   | intros; 
@@ -312,24 +306,44 @@ constructor 1;
     apply or_prop3;
   ]
 | intros; split; simplify; 
-   [1,3: unfold arrows1_OF_ORelation_setoid; apply ((†H)‡(†H1));
-   |2,4: apply ((†H1)‡(†H));]]
+   [1,3: unfold umorphism_setoid_OF_ORelation_setoid; unfold arrows1_OF_ORelation_setoid; apply ((†e)‡(†e1));
+   |2,4: apply ((†e1)‡(†e));]]
 qed.
 
-definition OA : category1.
+definition OA : category2.
 split;
 [ apply (OAlgebra);
 | intros; apply (ORelation_setoid o o1);
 | intro O; split;
-  [1,2,3,4: apply id1;
+  [1,2,3,4: apply id2;
   |5,6,7:intros; apply refl1;] 
 | apply ORelation_composition;
 | 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
+     apply (comp_assoc2 ????? (F⎻* ) (G⎻* ) (H⎻* ));
+   | apply ((comp_assoc2 ????? (H⎻) (G⎻) (F⎻))^-1);
+   | apply ((comp_assoc2 ????? F G H)^-1);
+   | apply ((comp_assoc2 ????? H* G* F* ));]
+| intros; split; unfold ORelation_composition; simplify; apply id_neutral_left2;
+| intros; split; unfold ORelation_composition; simplify; apply id_neutral_right2;]
+qed.
+
+lemma setoid1_of_OA: OA → setoid1.
+ intro; apply (oa_P t);
+qed.
+coercion setoid1_of_OA.
+
+lemma SET1_of_OA: OA → SET1.
+ intro; whd; apply (setoid1_of_OA t);
+qed.
+coercion SET1_of_OA.
+
+lemma objs2_SET1_OF_OA: OA → objs2 SET1.
+ intro; whd; apply (setoid1_of_OA t);
+qed.
+coercion objs2_SET1_OF_OA.
+
+lemma Type_OF_category2_OF_SET1_OF_OA: OA → Type_OF_category2 SET1.
+ intro; apply (oa_P t);
+qed.
+coercion Type_OF_category2_OF_SET1_OF_OA.