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[helm.git] / helm / software / matita / contribs / formal_topology / overlap / o-algebra.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 9f1c0fada66347786d940103f4bf7879885a7d89..c1d766f7e1c65375a3cb4869d7b86714c76420c5 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -52,6 +52,10 @@ interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \e
    lattices, Definizione 0.9 *)
 (* USARE L'ESISTENZIALE DEBOLE *)
 
+definition if_then_else ≝ λT:Type.λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
+notation > "'if' term 19 e 'then' term 19 t 'else' term 90 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
 
 notation > "hvbox(a break ≤ b)" non associative with precedence 45 for @{oa_leq $a $b}.
 notation > "a >< b" non associative with precedence 45 for @{oa_overlap $a $b}.
@@ -76,7 +80,7 @@ record OAlgebra : Type2 := {
   oa_zero_bot: ∀p:oa_P.𝟘 ≤ p;
   oa_one_top: ∀p:oa_P.p ≤ 𝟙;
   oa_overlap_preserves_meet_: ∀p,q:oa_P.p >< q → 
-        p >< (⋀ { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
+        p >< (⋀ { x ∈ BOOL | if x then p else q(*match x with [ true ⇒ p | false ⇒ q ]*) | IF_THEN_ELSE_p oa_P p q });
   oa_join_split: ∀I:SET.∀p.∀q:I ⇒_2 oa_P.p >< (⋁ q) = (∃i:I.p >< (q i));
   (*oa_base : setoid;
   1) enum non e' il nome giusto perche' non e' suriettiva
@@ -105,7 +109,7 @@ for @{ 'oa_meet $f }.
 interpretation "o-algebra meet" 'oa_meet f = 
   (fun12 ?? (oa_meet ??) f).
 interpretation "o-algebra meet with explicit function" 'oa_meet_mk f = 
-  (fun12 ?? (oa_meet ??) (mk_unary_morphism ?? f ?)).
+  (fun12 ?? (oa_meet ??) (mk_unary_morphism1 ?? f ?)).
 
 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)" 
 non associative with precedence 50 for @{ 'oa_join $p }.
@@ -150,11 +154,9 @@ interpretation "o-algebra binary join" 'or a b =
   (fun21 ??? (binary_join ?) a b).
 
 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;
+intros;  lapply (oa_overlap_preserves_meet_ O p q f) as H; clear f;
+(** screenshot "screenoa". *)
+assumption;
 qed.
 
 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)" 
@@ -175,10 +177,10 @@ interpretation "o-algebra join with explicit function" 'oa_join_mk f =
   (fun12 ?? (oa_join ??) (mk_unary_morphism ?? f ?)).
 
 record ORelation (P,Q : OAlgebra) : Type2 ≝ {
-  or_f_ : carr2 (P ⇒_2 Q);
-  or_f_minus_star_ : carr2(P ⇒_2 Q);
-  or_f_star_ : carr2(Q ⇒_2 P);
-  or_f_minus_ : carr2(Q ⇒_2 P);
+  or_f_ : P ⇒_2 Q;
+  or_f_minus_star_ : P ⇒_2 Q;
+  or_f_star_ : Q ⇒_2 P;
+  or_f_minus_ : Q ⇒_2 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)
@@ -210,8 +212,7 @@ definition ORelation_of_ORelation_setoid :
   ∀P,Q.ORelation_setoid P Q → ORelation P Q ≝ λP,Q,x.x.
 coercion ORelation_of_ORelation_setoid.
 
-definition or_f_minus_star:
- ∀P,Q:OAlgebra.(ORelation_setoid P Q) ⇒_2 (P ⇒_2 Q).
+definition or_f_minus_star: ∀P,Q:OAlgebra.(ORelation_setoid P Q) ⇒_2 (P ⇒_2 Q).
  intros; constructor 1;
   [ apply or_f_minus_star_;
   | intros; cases e; assumption]
@@ -269,7 +270,7 @@ intros; apply (or_prop3_ ?? F p q);
 qed.
 
 definition ORelation_composition : ∀P,Q,R. 
-  binary_morphism2 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
+  (ORelation_setoid P Q) × (ORelation_setoid Q R) ⇒_2 (ORelation_setoid P R).
 intros;
 constructor 1;
 [ intros (F G);
@@ -323,8 +324,6 @@ coercion ORelation_setoid_of_arrows2_OA.
 
 prefer coercion Type_OF_objs2.
 
-(* alias symbol "eq" = "setoid1 eq". *)
-
 (* qui la notazione non va *)
 lemma leq_to_eq_join: ∀S:OA.∀p,q:S. p ≤ q → q = (binary_join ? p q).
  intros;