Concrete spaces now defined.

[helm.git] / helm / software / matita / library / formal_topology / concrete_spaces.ma

diff --git a/helm/software/matita/library/formal_topology/concrete_spaces.ma b/helm/software/matita/library/formal_topology/concrete_spaces.ma
index 568f3c0528488ec22c45a10378ebb0e5861f8435..5c39beddeee5d2895078095ad53c129ee05cc876 100644 (file)
--- a/helm/software/matita/library/formal_topology/concrete_spaces.ma
+++ b/helm/software/matita/library/formal_topology/concrete_spaces.ma
@@ -30,6 +30,8 @@ definition ext: ∀X,S:REL. ∀r: arrows1 ? X S. S ⇒ Ω \sup X.
      | apply (. #‡(#‡H\sup -1)); assumption]]
 qed.
 
+definition BPext: ∀o: basic_pair. form o ⇒ Ω \sup (concr o) ≝ λo.ext ? ? (rel o).
+
 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;
@@ -48,30 +50,42 @@ definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
       | apply (. (#‡H\sup -1)‡#); assumption]]]
 qed.
 
-definition fintersects: ∀o: basic_pair. form o → form o → Ω \sup (form o).
- apply
-  (λo: basic_pair.λa,b: form o.
-    {c | ext ?? (rel o) c ⊆ ext ?? (rel o) a ∩ ext ?? (rel o) b });
- intros; simplify; apply (.= (†H)‡#); apply refl1.
+definition BPextS: ∀o: basic_pair. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
+ λo.extS ?? (rel o).
+
+definition fintersects: ∀o: basic_pair. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
+ intros (o); constructor 1;
+  [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
+    intros; simplify; apply (.= (†H)‡#); apply refl1
+  | intros; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
 qed.
 
-interpretation "fintersects" 'fintersects U V = (fintersects _ U V).
+interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
 
 definition fintersectsS:
- ∀o:basic_pair. Ω \sup (form o) → Ω \sup (form o) → Ω \sup (form o).
- apply (λo: basic_pair.λa,b: Ω \sup (form o).
-  {c | ext ?? (rel o) c ⊆ extS ?? (rel o) a ∩ extS ?? (rel o) b });
- intros; simplify; apply (.= (†H)‡#); apply refl1.
+ ∀o:basic_pair. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
+ intros (o); constructor 1;
+  [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
+    intros; simplify; apply (.= (†H)‡#); apply refl1
+  | intros; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
 qed.
 
-interpretation "fintersectsS" 'fintersects U V = (fintersectsS _ U V).
+interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
 
-(*
-definition relS: ∀o: basic_pair. concr o → Ω \sup (form o) → CProp.
- apply (λo:basic_pair.λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
+definition relS: ∀o: basic_pair. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
+ intros (o); constructor 1;
+  [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
+  | intros; split; intros; cases H2; exists [1,3: apply w]
+     [ apply (. (#‡H1)‡(H‡#)); assumption
+     | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
+qed.
 
-interpretation "basic pair relation for subsets" 'Vdash2 x y = (relS _ x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (relS _).
+interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
+interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
 
 record concrete_space : Type ≝
  { bp:> basic_pair;
@@ -79,6 +93,7 @@ record concrete_space : Type ≝
    all_covered: ∀x: concr bp. x ⊩ form bp
  }.
 
+(*
 record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
  { rp:> relation_pair CS1 CS2;
    respects_converges: