more ex and more notation

[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 ec32fca817fcc6723f51fc777d9c13a04b6b815b..3c03b4e667803df8815c98e4ef7806ae86b47715 100644 (file)
--- a/helm/software/matita/library/formal_topology/concrete_spaces.ma
+++ b/helm/software/matita/library/formal_topology/concrete_spaces.ma
@@ -14,28 +14,107 @@
 
 include "formal_topology/basic_pairs.ma".
 
-interpretation "REL carrier" 'card c = (carrier c).
+record concrete_space : Type ≝
+ { 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 comprehension: ∀b:REL. (b → CProp) → Ω \sup |b| ≝
- λb:REL.λP.{x | ∃p: x ∈ b. P (mk_ssigma ?? x p)}.
+definition bp': concrete_space → basic_pair ≝ λc.bp c.
 
-interpretation "subset comprehension" 'comprehension s p =
- (comprehension s p).
+coercion bp'.
 
-definition ext: ∀o: basic_pair. form o → Ω \sup |(concr o)| ≝
- λo,f.{x ∈ (concr o) | x ♮(rel o) f}.
+record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
+ { rp:> arrows1 ? CS1 CS2;
+   respects_converges:
+    ∀b,c.
+     extS ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
+     BPextS CS1 ((extS ?? rp \sub\f b) ↓ (extS ?? rp \sub\f c));
+   respects_all_covered:
+    extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)
+ }.
 
-definition fintersects ≝
- λo: basic_pair.λa,b: form o.
-  {c | ext ? c ⊆ ext ? a ∩ ext ? b }.
+definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
+ λCS1,CS2,c. rp CS1 CS2 c.
+ 
+coercion rp'.
 
-interpretation "fintersects" 'fintersects U V = (fintersects _ U V).
+definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
+ intros;
+ constructor 1;
+  [ apply (convergent_relation_pair c c1)
+  | constructor 1;
+     [ intros;
+       apply (relation_pair_equality c c1 c2 c3);
+     | intros 1; apply refl1;
+     | intros 2; apply sym1; 
+     | intros 3; apply trans1]]
+qed.
 
-definition relS: ∀o: basic_pair. concr o → Ω \sup (form o) → CProp ≝
- λo,x,S. ∃y. y ∈ S ∧ x ⊩ y.
+definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1 BP CS1 CS2 ≝
+ λCS1,CS2,c.rp ?? c.
 
-interpretation "basic pair relation for subsets" 'Vdash2 x y = (relS _ x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (relS _).
+coercion rp''.
 
-definition convergence ≝
- λo: basic_pair.∀a: concr o.∀U,V: form o. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V). 
\ No newline at end of file
+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 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 (.= (equalset_extS_id_X_X ??));
+       apply refl1]
+  | 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.