the new coercion behaviour (variants + composition with ID) and the new

[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 ca3d0379b2dcb0fe6bd99ce24656b146e0ce86c1..41f9bfd0e0d1e5765f89e285696057b6a3048119 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -63,7 +63,9 @@ record OAlgebra : Type2 := {
   oa_leq_antisym: ∀a,b:oa_P.oa_leq a b → oa_leq b a → a = b;
   oa_leq_trans: ∀a,b,c:oa_P.oa_leq a b → oa_leq b c → oa_leq a c;
   oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
-  oa_meet_inf: ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.oa_leq p (oa_meet I p_i) = ∀i:I.oa_leq p (p_i i);
+  oa_meet_inf: 
+    ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.
+      oa_leq p (oa_meet I p_i) = ∀i:I.oa_leq p (p_i i);
   oa_join_sup: ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.oa_leq (oa_join I p_i) p = ∀i:I.oa_leq (p_i i) p;
   oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
   oa_one_top: ∀p:oa_P.oa_leq p oa_one;
@@ -113,16 +115,6 @@ interpretation "o-algebra join" 'oa_join f =
 interpretation "o-algebra join with explicit function" 'oa_join_mk f = 
   (fun12 __ (oa_join __) (mk_unary_morphism _ _ f _)).
 
-definition hint3: OAlgebra → setoid1.
- intro; apply (oa_P o);
-qed.
-coercion hint3.
-
-definition hint4: ∀A. setoid2_OF_OAlgebra A → hint3 A.
- intros; apply t;
-qed.
-coercion hint4.
-
 definition binary_meet : ∀O:OAlgebra. binary_morphism1 O O O.
 intros; split;
 [ intros (p q); 
@@ -178,16 +170,11 @@ interpretation "o-algebra join" 'oa_join f =
 interpretation "o-algebra join with explicit function" 'oa_join_mk f = 
   (fun12 __ (oa_join __) (mk_unary_morphism _ _ f _)).
 
-definition hint5: OAlgebra → objs2 SET1.
- intro; apply (oa_P o);
-qed.
-coercion hint5.
-
 record ORelation (P,Q : OAlgebra) : Type2 ≝ {
-  or_f_ : P ⇒ Q;
-  or_f_minus_star_ : P ⇒ Q;
-  or_f_star_ : Q ⇒ P;
-  or_f_minus_ : Q ⇒ P;
+  or_f_ : carr2 (P ⇒ Q);
+  or_f_minus_star_ : carr2(P ⇒ Q);
+  or_f_star_ : carr2(Q ⇒ P);
+  or_f_minus_ : carr2(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)
@@ -214,6 +201,10 @@ constructor 1;
      | apply (.= (e3 a)); apply e7;]]]
 qed.
 
+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.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
  intros; constructor 1;
@@ -227,8 +218,6 @@ definition or_f: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q
   | intros; cases e; assumption]
 qed.
 
-coercion or_f.
-
 definition or_f_minus: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
  intros; constructor 1;
   [ apply or_f_minus_;
@@ -241,36 +230,10 @@ definition or_f_star: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q
   | intros; cases e; assumption]
 qed.
 
-lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q).
-intros; apply (or_f ?? t);
-qed.
-
-coercion arrows1_OF_ORelation_setoid.
-
-lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1 P Q.
-intros; apply (or_f ?? t);
+lemma arrows1_of_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q). 
+intros; apply (or_f ?? c);
 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. ORelation_setoid B C → B → C. 
-intros; apply (t t1);
-qed.
-
-coercion uncurry_arrows 1.
-
-(*
-lemma hint6: ∀P,Q. Type_OF_setoid2 (hint5 P ⇒ hint5 Q) → unary_morphism1 P Q.
- intros; apply t;
-qed.
-coercion hint6.
-*)
+coercion arrows1_of_ORelation_setoid.
 
 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}.
@@ -323,7 +286,9 @@ constructor 1;
     apply or_prop3;
   ]
 | intros; split; simplify; 
-   [1,3: unfold umorphism_setoid_OF_ORelation_setoid; unfold arrows1_OF_ORelation_setoid; apply ((†e)‡(†e1));
+   [3: unfold arrows1_of_ORelation_setoid;
+         apply ((†e)‡(†e1));
+   |1: apply ((†e)‡(†e1));
    |2,4: apply ((†e1)‡(†e));]]
 qed.
 
@@ -345,22 +310,11 @@ split;
 | 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.
+definition OAlgebra_of_objs2_OA: objs2 OA → OAlgebra ≝ λx.x.
+coercion OAlgebra_of_objs2_OA.
 
-lemma objs2_SET1_OF_OA: OA → objs2 SET1.
- intro; whd; apply (setoid1_of_OA t);
-qed.
-coercion objs2_SET1_OF_OA.
+definition ORelation_setoid_of_arrows2_OA: 
+  ∀P,Q. arrows2 OA P Q → ORelation_setoid P Q ≝ λP,Q,c.c.
+coercion ORelation_setoid_of_arrows2_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.
+prefer coercion Type_OF_objs2.
\ No newline at end of file