X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Fbasic_topologies.ma;h=36a0d24c86a1593887c428a494b765b1c9ce2390;hb=c96d1f2066d37b84a34412f7c49fb3e4f54bd9a2;hp=92f4cdf4659106db599aa821a93979980d7e9909;hpb=f386b970edc5798769c96e44f5c9ab30efa06605;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/basic_topologies.ma b/helm/software/matita/library/formal_topology/basic_topologies.ma
index 92f4cdf46..36a0d24c8 100644
--- a/helm/software/matita/library/formal_topology/basic_topologies.ma
+++ b/helm/software/matita/library/formal_topology/basic_topologies.ma
@@ -14,39 +14,7 @@
 
 include "formal_topology/relations.ma".
 include "datatypes/categories.ma".
-
-definition is_saturation ≝
- λC:REL.λA:unary_morphism (Ω \sup C) (Ω \sup C).
-  ∀U,V. (U ⊆ A V) = (A U ⊆ A V).
-
-definition is_reduction ≝
- λC:REL.λJ:unary_morphism (Ω \sup C) (Ω \sup C).
-  ∀U,V. (J U ⊆ V) = (J U ⊆ J V).
-
-theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
- intros 4; assumption.
-qed.
-
-theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
- intros; apply transitive_subseteq_operator; [apply S2] assumption.
-qed.
-
-theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
- intros; apply (fi ?? (H ??)); apply subseteq_refl.
-qed.
-
-theorem saturation_monotone:
- ∀C,A. is_saturation C A →
-  ∀U,V. U ⊆ V → A U ⊆ A V.
- intros; apply (if ?? (H ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
- assumption.
-qed.
-
-theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
- intros; split;
-  [ apply (if ?? (H ??)); apply subseteq_refl
-  | apply saturation_expansive; assumption]
-qed.
+include "formal_topology/saturations_reductions.ma".
 
 record basic_topology: Type ≝
  { carrbt:> REL;
@@ -57,62 +25,6 @@ record basic_topology: Type ≝
    compatibility: ∀U,V. (A U ≬ J V) = (U ≬ J V)
  }.
 
-(* 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;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
-    intros; simplify; split; intro; cases H1; exists [1,3: apply w]
-     [ apply (. (#‡H)‡#); assumption
-     | apply (. (#‡H \sup -1)‡#); assumption]
-  | intros; split; simplify; intros; cases H2; exists [1,3: apply w]
-     [ apply (. #‡(#‡H1)); cases x; split; try assumption;
-       apply (if ?? (H ??)); assumption
-     | apply (. #‡(#‡H1 \sup -1)); cases x; split; try assumption;
-       apply (if ?? (H \sup -1 ??)); assumption]]
-qed.
-
-(* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
- intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
-    intros; simplify; split; intros; apply H1;
-     [ apply (. #‡H \sup -1); assumption
-     | apply (. #‡H); assumption]
-  | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
-    apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
-qed.
-
-(* minus_image is the same as ext *)
-
-theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
- intros; unfold image; simplify; split; simplify; intros;
-  [ change with (a ∈ U);
-    cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
-  | change in f with (a ∈ U);
-    exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
-qed.
-
-theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
- intros; unfold minus_star_image; simplify; split; simplify; intros;
-  [ change with (a ∈ U); apply H; change with (a=a); apply refl
-  | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
-qed.
-
-theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
- clear x; [ cases f; clear f; | cases f1; clear f1 ]
- exists; try assumption; cases x; clear x; split; try assumption;
- exists; try assumption; split; assumption.
-qed.
-
-theorem minus_star_image_comp:
- ∀A,B,C,r,s,X.
-  minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
- intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
-  [ apply H; exists; try assumption; split; assumption
-  | change with (x ∈ X); cases f; cases x1; apply H; assumption]
-qed.
-
 record continuous_relation (S,T: basic_topology) : Type ≝
  { cont_rel:> arrows1 ? S T;
    reduced: ∀U. U = J ? U → image ?? cont_rel U = J ? (image ?? cont_rel U);
@@ -137,46 +49,6 @@ definition cont_rel'': ∀S,T: basic_topology. continuous_relation_setoid S T 
 
 coercion cont_rel''.
 
-theorem ext_comp:
- ∀o1,o2,o3: REL.
-  ∀a: arrows1 ? o1 o2.
-   ∀b: arrows1 ? o2 o3.
-    ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
- intros;
- unfold ext; unfold extS; simplify; split; intro; simplify; intros;
- cases f; clear f; split; try assumption;
-  [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
-     [1: split] assumption;
-  | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
-     [2: cases f] assumption]
-qed.
-
-(*
-(* this proof is more logic-oriented than set/lattice oriented *)
-theorem continuous_relation_eqS:
- ∀o1,o2:basic_topology.∀a,a': continuous_relation_setoid o1 o2.
-  a = a' → ∀X. A ? (extS ?? a X) = A ? (extS ?? a' X).
- intros;
- cut (∀a: arrows1 ? o1 ?.∀x. x ∈ extS ?? a X → ∃y:o2.y ∈ X ∧ x ∈ ext ?? a y);
-  [2: intros; cases f; clear f; cases H1; exists [apply w] cases x1; split;
-      try assumption; split; assumption]
- cut (∀a,a':continuous_relation_setoid o1 o2.eq1 ? a a' → ∀x. x ∈ extS ?? a X → ∃y:o2. y ∈ X ∧ x ∈ A ? (ext ?? a' y));
-  [2: intros; cases (Hcut ?? f); exists; [apply w] cases x1; split; try assumption;
-      apply (. #‡(H1 ?));
-      apply (saturation_expansive ?? (A_is_saturation o1) (ext ?? a1 w) x);
-      assumption;] clear Hcut;
- split; apply (if ?? (A_is_saturation ???)); intros 2;
-  [lapply (Hcut1 a a' H a1 f) | lapply (Hcut1 a' a (H \sup -1) a1 f)]
-  cases Hletin; clear Hletin; cases x; clear x;
- cut (∀a: arrows1 ? o1 ?. ext ?? a w ⊆ extS ?? a X);
-  [2,4: intros 3; cases f3; clear f3; simplify in f5; split; try assumption;
-      exists [1,3: apply w] split; assumption;]
- cut (∀a. A ? (ext o1 o2 a w) ⊆ A ? (extS o1 o2 a X));
-  [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
- apply Hcut2; assumption.
-qed.
-*)
-
 theorem continuous_relation_eq':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
   a = a' → ∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X).
@@ -195,15 +67,6 @@ theorem continuous_relation_eq':
      [ apply I | assumption ]]
 qed.
 
-theorem extS_singleton:
- ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
- intros; unfold extS; unfold ext; unfold singleton; simplify;
- split; intros 2; simplify; cases f; split; try assumption;
-  [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
-    assumption
-  | exists; try assumption; split; try assumption; change with (x = x); apply refl]
-qed.
-
 theorem continuous_relation_eq_inv':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
   (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'.
@@ -308,4 +171,29 @@ definition BTop: category1.
   | intros; simplify; intro; simplify;
     apply (.= †((id_neutral_left1 ????)‡#));
     apply refl1]
-qed.
\ No newline at end of file
+qed.
+
+(*CSC: unused! *)
+(* this proof is more logic-oriented than set/lattice oriented *)
+theorem continuous_relation_eqS:
+ ∀o1,o2:basic_topology.∀a,a': continuous_relation_setoid o1 o2.
+  a = a' → ∀X. A ? (extS ?? a X) = A ? (extS ?? a' X).
+ intros;
+ cut (∀a: arrows1 ? o1 ?.∀x. x ∈ extS ?? a X → ∃y:o2.y ∈ X ∧ x ∈ ext ?? a y);
+  [2: intros; cases f; clear f; cases H1; exists [apply w] cases x1; split;
+      try assumption; split; assumption]
+ cut (∀a,a':continuous_relation_setoid o1 o2.eq1 ? a a' → ∀x. x ∈ extS ?? a X → ∃y:o2. y ∈ X ∧ x ∈ A ? (ext ?? a' y));
+  [2: intros; cases (Hcut ?? f); exists; [apply w] cases x1; split; try assumption;
+      apply (. #‡(H1 ?));
+      apply (saturation_expansive ?? (A_is_saturation o1) (ext ?? a1 w) x);
+      assumption;] clear Hcut;
+ split; apply (if ?? (A_is_saturation ???)); intros 2;
+  [lapply (Hcut1 a a' H a1 f) | lapply (Hcut1 a' a (H \sup -1) a1 f)]
+  cases Hletin; clear Hletin; cases x; clear x;
+ cut (∀a: arrows1 ? o1 ?. ext ?? a w ⊆ extS ?? a X);
+  [2,4: intros 3; cases f3; clear f3; simplify in f5; split; try assumption;
+      exists [1,3: apply w] split; assumption;]
+ cut (∀a. A ? (ext o1 o2 a w) ⊆ A ? (extS o1 o2 a X));
+  [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
+ apply Hcut2; assumption.
+qed.