X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Fconcrete_spaces.ma;h=3c03b4e667803df8815c98e4ef7806ae86b47715;hb=e189bde1e0a562e8689ff41d55d392431609e749;hp=703facfa564f0b00d082f891eb6776ea9e270beb;hpb=6c49bf832dd39cb170f6bcdbc6940e2e81abc650;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/concrete_spaces.ma b/helm/software/matita/library/formal_topology/concrete_spaces.ma
index 703facfa5..3c03b4e66 100644
--- a/helm/software/matita/library/formal_topology/concrete_spaces.ma
+++ b/helm/software/matita/library/formal_topology/concrete_spaces.ma
@@ -14,87 +14,18 @@
 
 include "formal_topology/basic_pairs.ma".
 
-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.
-qed.
-
-interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism __ p _)).
-
-definition ext: ∀X,S:REL. ∀r: arrows1 ? X S. S ⇒ Ω \sup X.
- apply (λX,S,r.mk_unary_morphism ?? (λf.{x ∈ X | x ♮r f}) ?);
-  [ intros; simplify; apply (.= (H‡#)); apply refl1
-  | intros; simplify; split; intros; simplify; intros;
-     [ apply (. #‡(#‡H)); assumption
-     | 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;
-  [ intro F; constructor 1; constructor 1;
-    [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
-    | intros; split; intro; cases f (H1 H2); clear f; split;
-       [ apply (. (H‡#)); assumption
-       |3: apply (. (H\sup -1‡#)); assumption
-       |2,4: cases H2 (w H3); exists; [1,3: apply w]
-         [ apply (. (#‡(H‡#))); assumption
-         | apply (. (#‡(H \sup -1‡#))); assumption]]]
-  | intros; split; simplify; intros; cases f; cases H1; split;
-     [1,3: assumption
-     |2,4: exists; [1,3: apply w]
-      [ apply (. (#‡H)‡#); assumption
-      | apply (. (#‡H\sup -1)‡#); assumption]]]
-qed.
-
-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 = (fun1 ___ (fintersects _) U V).
-
-definition fintersectsS:
- ∀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 = (fun1 ___ (fintersectsS _) U V).
-
-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 = (fun1 (concr _) __ (relS _) x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
-
 record concrete_space : Type ≝
- { bp:> basic_pair;
+ { bp:> BP;
    converges: ∀a: concr bp.∀U,V: form bp. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V);
    all_covered: ∀x: concr bp. x ⊩ form bp
  }.
 
+definition bp': concrete_space → basic_pair ≝ λc.bp c.
+
+coercion bp'.
+
 record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
- { rp:> relation_pair CS1 CS2;
+ { rp:> arrows1 ? CS1 CS2;
    respects_converges:
     ∀b,c.
      extS ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
@@ -103,6 +34,11 @@ record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
     extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)
  }.
 
+definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
+ λCS1,CS2,c. rp CS1 CS2 c.
+ 
+coercion rp'.
+
 definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
  intros;
  constructor 1;
@@ -115,65 +51,70 @@ definition convergent_relation_space_setoid: concrete_space → concrete_space 
      | intros 3; apply trans1]]
 qed.
 
-lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
- intros;
- unfold extS; simplify;
- split; simplify;
-  [ intros 2; change with (a ∈ X);
-    cases f; clear f;
-    cases H; clear H;
-    cases x; clear x;
-    change in f2 with (eq1 ? a w);
-    apply (. (f2\sup -1‡#));
-    assumption
-  | intros 2; change in f with (a ∈ X);
-    split;
-     [ whd; exact I 
-     | exists; [ apply a ]
-       split;
-        [ assumption
-        | change with (a = a); apply refl]]]
-qed.
+definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1 BP CS1 CS2 ≝
+ λCS1,CS2,c.rp ?? c.
 
-lemma extS_id: ∀o:basic_pair.∀X.extS (concr o) (concr o) (id o) \sub \c X = X.
- intros;
- unfold extS; simplify;
- split; simplify; intros;
-  [ change with (a ∈ X);
-    cases f; cases H; cases x; change in f3 with (eq1 ? a w);
-    apply (. (f3\sup -1‡#));
-    assumption
-  | change in f with (a ∈ X);
-    split;
-     [ apply I
-     | exists; [apply a]
-       split; [ assumption | change with (a = a); apply refl]]]
+coercion rp''.
+
+definition convergent_relation_space_composition:
+ ∀o1,o2,o3: concrete_space.
+  binary_morphism1
+   (convergent_relation_space_setoid o1 o2)
+   (convergent_relation_space_setoid o2 o3)
+   (convergent_relation_space_setoid o1 o3).
+ intros; constructor 1;
+     [ intros; whd in c c1 ⊢ %;
+       constructor 1;
+        [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
+        | intros;
+          change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
+          change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
+            with (c1 \sub \f ∘ c \sub \f);
+          change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
+            with (c1 \sub \f ∘ c \sub \f);
+          apply (.= (extS_com ??????));
+          apply (.= (†(respects_converges ?????)));
+          apply (.= (respects_converges ?????));
+          apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
+          apply refl1;
+        | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
+          apply (.= (extS_com ??????));
+          apply (.= (†(respects_all_covered ???)));
+          apply (.= respects_all_covered ???);
+          apply refl1]
+     | intros;
+       change with (b ∘ a = b' ∘ a');
+       change in H with (rp'' ?? a = rp'' ?? a');
+       change in H1 with (rp'' ?? b = rp ?? b');
+       apply (.= (H‡H1));
+       apply refl1]
 qed.
 
-(*
 definition CSPA: category1.
  constructor 1;
   [ apply concrete_space
   | apply convergent_relation_space_setoid
   | intro; constructor 1;
-     [ apply id
+     [ apply id1
      | intros;
        unfold id; simplify;
        apply (.= (equalset_extS_id_X_X ??));
        apply (.= (†((equalset_extS_id_X_X ??)\sup -1‡
                     (equalset_extS_id_X_X ??)\sup -1)));
        apply refl1;
-     | apply (.= (extS_id ??));
+     | apply (.= (equalset_extS_id_X_X ??));
        apply refl1]
-  | intros; constructor 1;
-     [ intros; whd in c c1 ⊢ %;
-       constructor 1;
-        [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
-        | intros;
-        |
-        ]
-     | intros; intros 2; simplify;
-       letin xxx ≝ (comp BP); clearbody xxx; unfold BP in xxx:(?→?→?→?→?→%); simplify in xxx;
-       unfold basic_pair in xxx; simplify in xxx;
-     ]
-*)
\ No newline at end of file
+  | apply convergent_relation_space_composition
+  | intros; simplify;
+    change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
+    apply (.= ASSOC1);
+    apply refl1
+  | intros; simplify;
+    change with (a ∘ id1 ? o1 = a);
+    apply (.= id_neutral_right1 ????);
+    apply refl1
+  | intros; simplify;
+    change with (id1 ? o2 ∘ a = a);
+    apply (.= id_neutral_left1 ????);
+    apply refl1]
+qed.