More work on concrete spaces.

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / relations.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index eb60386fe3d1abc0831f7f747d7a5093062845c9..c4502f3d080b61874360cfb4e650bde1f3b99694 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -96,7 +96,6 @@ definition REL: category1.
           first [apply refl | assumption]]]
 qed.
 
-(*
 definition full_subset: ∀s:REL. Ω \sup s.
  apply (λs.{x | True});
  intros; simplify; split; intro; assumption.
@@ -105,25 +104,35 @@ qed.
 coercion full_subset.
 
 definition setoid1_of_REL: REL → setoid ≝ λS. S.
-
 coercion setoid1_of_REL.
 
-definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
- apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
- intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
+lemma Type_OF_setoid1_of_REL: ∀o1:Type_OF_category1 REL. Type_OF_objs1 o1 → Type_OF_setoid1 ?(*(setoid1_of_SET o1)*).
+ [ apply (setoid1_of_SET o1);
+ | intros; apply t;]
+qed.
+coercion Type_OF_setoid1_of_REL.
+
+definition comprehension: ∀b:REL. (unary_morphism1 b CPROP) → Ω \sup b.
+ apply (λb:REL. λP: b ⇒ CPROP. {x | P x});
+ intros; simplify;
+ alias symbol "trans" = "trans1".
+ alias symbol "prop1" = "prop11".
+ apply (.= †e); apply refl1.
 qed.
 
 interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism __ p _)).
+ (comprehension s (mk_unary_morphism1 __ p _)).
 
 definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
- apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
-  [ intros; simplify; apply (.= (H‡#)); apply refl1
-  | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
-     [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
-     | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
+ apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 REL X S.λf:S.{x ∈ X | x ♮r f}) ?);
+  [ intros; simplify; apply (.= (e‡#)); apply refl1
+  | intros; simplify; split; intros; simplify;
+     [ change with (∀x. x ♮a b → x ♮a' b'); intros;
+       apply (. (#‡e1)); whd in e; apply (if ?? (e ??)); assumption
+     | change with (∀x. x ♮a' b' → x ♮a b); intros;
+       apply (. (#‡e1\sup -1)); whd in e; apply (fi ?? (e ??));assumption]]
 qed.
-
+(*
 definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
  (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
  intros (X S r); constructor 1;
@@ -175,12 +184,6 @@ lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (e
 qed.
 *)
 
-lemma hint: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
- [ apply setoid1_of_SET; apply U
- | intros; apply c;]
-qed.
-coercion hint.
-
 (* the same as ⋄ for a basic pair *)
 definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
  intros; constructor 1;
@@ -288,43 +291,3 @@ theorem extS_singleton:
   | exists; try assumption; split; try assumption; change with (x = x); apply refl]
 qed.
 *)
-
-include "o-algebra.ma".
-
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (SUBSETS o1) (SUBSETS o2).
- intros;
- constructor 1;
-  [ constructor 1; 
-     [ apply (λU.image ?? t U);
-     | intros; apply (#‡e); ]
-  | constructor 1;
-     [ apply (λU.minus_star_image ?? t U);
-     | intros; apply (#‡e); ]
-  | constructor 1;
-     [ apply (λU.star_image ?? t U);
-     | intros; apply (#‡e); ]
-  | constructor 1;
-     [ apply (λU.minus_image ?? t U);
-     | intros; apply (#‡e); ]
-  | intros; split; intro;
-     [ change in f with (∀a. a ∈ image ?? t p → a ∈ q);
-       change with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
-       intros 4; apply f; exists; [apply a] split; assumption;
-     | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
-       change with (∀a. a ∈ image ?? t p → a ∈ q);
-       intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
-  | intros; split; intro;
-     [ change in f with (∀a. a ∈ minus_image ?? t p → a ∈ q);
-       change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
-       intros 4; apply f; exists; [apply a] split; assumption;
-     | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
-       change with (∀a. a ∈ minus_image ?? t p → a ∈ q);
-       intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
-  | intros; split; intro; cases f; clear f;
-     [ cases x; cases x2; clear x x2; exists; [apply w1]
-        [ assumption;
-        | exists; [apply w] split; assumption]
-     | cases x1; cases x2; clear x1 x2; exists; [apply w1]
-        [ exists; [apply w] split; assumption;
-        | assumption; ]]]
-qed. (*sistemare anche l'hint da un'altra parte *)