Using the new category SET.

Rewriting is no longer working as expected.

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 877887ad95ff9833d8680e886d0aaa0bcc8f75bc..4dcf99b8bd2b1b0e720d0a9b9a5bd13e8db4c0d5 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -17,18 +17,7 @@ include "logic/cprop_connectives.ma".
 
 inductive bool : Type := true : bool | false : bool.
 
-lemma ums : setoid → setoid → setoid.
-intros (S T);
-constructor 1;
-[ apply (unary_morphism S T);
-| constructor 1;
-  [ intros (f1 f2); apply (∀a,b:S.eq1 ? a b → eq1 ? (f1 a) (f2 b));
-  | whd; simplify; intros; apply (.= (†H)); apply refl1;
-  | whd; simplify; intros; apply (.= (†H1)); apply sym1; apply H; apply refl1;
-  | whd; simplify; intros; apply (.= (†H2)); apply (.= (H ?? #)); apply (.= (H1 ?? #)); apply rule #;]] 
-qed.
-
-lemma BOOL : setoid.
+lemma BOOL : objs1 SET.
 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;
@@ -36,16 +25,22 @@ constructor 1; [apply bool] constructor 1;
 | 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.
+
+coercion hint.
+
 lemma IF_THEN_ELSE_p :
   ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y → 
   let f ≝ λm.match m with [ true ⇒ a | false ⇒ b ] in f x = f y.
 intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
 qed. 
 
-lemma if_then_else : ∀T:setoid. ∀a,b:T. ums BOOL T.
+lemma if_then_else : ∀T:SET. ∀a,b:T. arrows1 SET BOOL T.
 intros; constructor 1; intros; 
-[ apply (match c2 with [ true ⇒ c | false ⇒ c1 ]);
-| apply (IF_THEN_ELSE_p T c c1 a a' H);]
+[ apply (match c with [ true ⇒ t | false ⇒ t1 ]);
+| apply (IF_THEN_ELSE_p T t t1 a a' H);]
 qed.
 
 interpretation "mk " 'comprehension T P = 
@@ -57,16 +52,16 @@ for @{ 'comprehension_by $s (\lambda ${ident i}. $p) $by}.
 interpretation "unary morphism comprehension with proof" 'comprehension_by s f p = 
   (mk_unary_morphism s _ f p).
 
-definition A : ∀S:setoid.∀a,b:S.ums BOOL S.
+definition A : ∀S:SET.∀a,b:S.arrows1 SET BOOL S.
 apply (λS,a,b.{ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b] | IF_THEN_ELSE_p S a b}).
 qed.
 
 record OAlgebra : Type := {
-  oa_P :> setoid;
+  oa_P :> SET;
   oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
   oa_overlap: binary_morphism1 oa_P oa_P CPROP;
-  oa_meet: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
-  oa_join: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
+  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_one: oa_P;
   oa_zero: oa_P;
   oa_leq_refl: ∀a:oa_P. oa_leq a a; 
@@ -82,7 +77,7 @@ record OAlgebra : Type := {
        (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) *)
-      ∀I:setoid.∀p.∀q:ums I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+      ∀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;
   oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q*)
@@ -127,13 +122,13 @@ interpretation "o-algebra join" 'oa_join \eta.f = (oa_join _ _ f).
 *)
 
 record ORelation (P,Q : OAlgebra) : Type ≝ {
-  or_f :> P ⇒ Q;
-  or_f_minus_star : P ⇒ Q;
-  or_f_star : Q ⇒ P;
-  or_f_minus : 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}.
@@ -155,21 +150,29 @@ intros (P Q);
 constructor 1;
 [ apply (ORelation P Q);
 | constructor 1;
-   [ 
-     alias symbol "and" = "constructive and".
-     apply (λp,q.
-            (∀a.p⎻* a = q⎻* a) ∧ 
-            (∀a.p⎻ a  = q⎻ a) ∧
-            (∀a.p a   = q a) ∧ 
-            (∀a.p* a  = q* a)); 
-   | whd; simplify; intros; repeat split; intros; apply refl;
+   [ apply (λp,q. eq1 ? p⎻* q⎻* ∧ eq1 ? p⎻ q⎻ ∧ eq1 ? p q ∧ eq1 ? p* q* ); 
+   | whd; simplify; intros; repeat split; intros; apply refl1;
    | whd; simplify; intros; cases H; cases H1; cases H3; clear H H3 H1;
-     repeat split; intros; apply sym; generalize in match a;assumption;
-   | whd; simplify; intros; elim DAEMON;]]
-qed.  
+     repeat split; intros; apply sym1; assumption;
+   | whd; simplify; intros; cases H; cases H1; cases H2; cases H4; cases H6; cases H8;
+     repeat split; intros; clear H H1 H2 H4 H6 H8; apply trans1;
+      [2: apply H10;
+      |5: apply H11;
+      |8: apply H7;
+      |11: apply H3;
+      |1,4,7,10: skip
+      |*: assumption
+      ]]]
+qed.
 
-lemma hint : ∀P,Q. ORelation_setoid P Q → P ⇒ Q. intros; apply (or_f ?? c);qed.
-coercion hint.
+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 composition : ∀P,Q,R. 
   binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
@@ -177,21 +180,42 @@ intros;
 constructor 1;
 [ intros (F G);
   constructor 1;
-  [ constructor 1; [apply (λx. G (F x)); | intros; apply (†(†H));]
-  |2,3,4,5,6,7: cases DAEMON;]
-| intros; cases DAEMON;]
+  [ apply (G ∘ F);
+  | apply (G⎻* ∘ F⎻* );
+  | apply (F* ∘ G* );
+  | apply (F⎻ ∘ G⎻);
+  | intros; change with ((G (F p) ≤ q) = (p ≤ (F* (G* q))));
+    apply (.= or_prop1 ??? (F p) ?);
+    apply (.= or_prop1 ??? p ?);
+    apply refl1
+  | intros; change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
+    apply (.= or_prop2 ??? (G⎻ p) ?);
+    apply (.= or_prop2 ??? p ?);
+    apply refl1;
+  | intros; change with ((G (F (p)) >< q) = (p >< (F⎻ (G⎻ q))));
+    apply (.= or_prop3 ??? (F p) ?);
+    apply (.= or_prop3 ??? p ?);
+    apply refl1
+  ]
+| intros; repeat split; simplify; cases DAEMON (*
+   [ apply trans1; [2: apply prop1; [3: apply rule #; | skip | 4:
+     apply rule (†?);
+   
+     lapply (.= ((†H1)‡#)); [8: apply Hletin;
+   [ apply trans1; [2: lapply (prop1); [apply Hletin;
+*)]
 qed.
 
-definition OA : category1. (* category2 *)
+definition OA : category1.
 split;
 [ apply (OAlgebra);
 | intros; apply (ORelation_setoid o o1);
 | intro O; split;
-  [1,2,3,4: constructor 1; [1,3,5,7:apply (λx.x);|*:intros;assumption]
-  |5,6,7:intros;split;intros; assumption; ] 
-|4: apply composition;
-|*: elim DAEMON;]
-qed. 
-
-
-
+  [1,2,3,4: apply id1;
+  |5,6,7:intros; apply refl1;] 
+| apply composition;
+| intros; repeat split; unfold composition; simplify;
+  [1,3: apply (comp_assoc1); | 2,4: apply ((comp_assoc1 ????????) \sup -1);]
+| intros; repeat split; unfold composition; simplify; apply id_neutral_left1;
+| intros; repeat split; unfold composition; simplify; apply id_neutral_right1;]
+qed.
\ No newline at end of file