From 7c9b20db66af78579a5312e4a6a5a42471d6312b Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Mon, 15 Sep 2008 20:26:34 +0000
Subject: [PATCH] BTop is a category.

---
 .../formal_topology/basic_topologies.ma       | 196 ++++++++++++++++--
 1 file changed, 174 insertions(+), 22 deletions(-)

diff --git a/helm/software/matita/library/formal_topology/basic_topologies.ma b/helm/software/matita/library/formal_topology/basic_topologies.ma
index 749089a89..92f4cdf46 100644
--- a/helm/software/matita/library/formal_topology/basic_topologies.ma
+++ b/helm/software/matita/library/formal_topology/basic_topologies.ma
@@ -23,6 +23,31 @@ 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.
+
 record basic_topology: Type ≝
  { carrbt:> REL;
    A: unary_morphism (Ω \sup carrbt) (Ω \sup carrbt);
@@ -108,7 +133,129 @@ definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T →
 
 coercion cont_rel'.
 
+definition cont_rel'': ∀S,T: basic_topology. continuous_relation_setoid S T → binary_relation S T ≝ cont_rel.
+
+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).
+ intros; split; intro; unfold minus_star_image; simplify; intros;
+  [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
+    lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
+    cut (A ? (ext ?? a' a1) ⊆ A ? X); [2: apply (. (H ?)‡#); assumption]
+    lapply (fi ?? (A_is_saturation ???) Hcut);
+    apply (Hletin1 x); change with (x ∈ ext ?? a' a1); split; simplify;
+     [ apply I | assumption ]
+  | cut (ext ?? a' a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
+    lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
+    cut (A ? (ext ?? a a1) ⊆ A ? X); [2: apply (. (H ?)\sup -1‡#); assumption]
+    lapply (fi ?? (A_is_saturation ???) Hcut);
+    apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify;
+     [ 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'.
+ intros 6;
+ cut (∀a,a': continuous_relation_setoid o1 o2.
+  (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → 
+   ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
+  [2: clear b H a' a; intros;
+      lapply depth=0 (λV.saturation_expansive ??? (extS ?? a V)); [2: apply A_is_saturation;|skip]
+       (* fundamental adjunction here! to be taken out *)
+       cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V)));
+        [2: intro; intros 2; unfold minus_star_image; simplify; intros;
+            apply (Hletin V1 x); whd; split; [ exact I | exists; [apply a1] split; assumption]]
+       clear Hletin;
+       cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V)));
+        [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut;
+       (* second half of the fundamental adjunction here! to be taken out too *)
+      intro; lapply (Hcut1 (singleton ? V)); clear Hcut1;
+      unfold minus_star_image in Hletin; unfold singleton in Hletin; simplify in Hletin;
+      whd in Hletin; whd in Hletin:(?→?→%); simplify in Hletin;
+      apply (if ?? (A_is_saturation ???));
+      intros 2 (x H); lapply (Hletin V ? x ?);
+       [ apply refl | cases H; assumption; ]
+      change with (x ∈ A ? (ext ?? a V));
+      apply (. #‡(†(extS_singleton ????)));
+      assumption;]
+ split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
+qed.
+
+definition continuous_relation_comp:
+ ∀o1,o2,o3.
+  continuous_relation_setoid o1 o2 →
+   continuous_relation_setoid o2 o3 →
+    continuous_relation_setoid o1 o3.
+ intros (o1 o2 o3 r s); constructor 1;
+  [ apply (s ∘ r)
+  | intros;
+    apply sym1;
+    apply (.= †(image_comp ??????));
+    apply (.= (reduced ?????)\sup -1);
+     [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
+     | apply (.= (image_comp ??????)\sup -1);
+       apply refl1]
+     | intros;
+       apply sym1;
+       apply (.= †(minus_star_image_comp ??????));
+       apply (.= (saturated ?????)\sup -1);
+        [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
+        | apply (.= (minus_star_image_comp ??????)\sup -1);
+          apply refl1]]
+qed.
+
 definition BTop: category1.
  constructor 1;
   [ apply basic_topology
@@ -128,32 +275,37 @@ definition BTop: category1.
        apply sym1;
        assumption]
   | intros; constructor 1;
-     [ intros (r s); constructor 1;
-        [ apply (s ∘ r)
-        | intros;
-          apply sym1;
-          apply (.= †(image_comp ??????));
-          apply (.= (reduced ?????)\sup -1);
-           [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
-           | apply (.= (image_comp ??????)\sup -1);
-             apply refl1]
-        | intros;
-          apply sym1;
-          apply (.= †(minus_star_image_comp ??????));
-          apply (.= (saturated ?????)\sup -1);
-           [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
-           | apply (.= (minus_star_image_comp ??????)\sup -1);
-             apply refl1]]
-     | intros; simplify; intro; simplify; whd in H H1; 
-       apply (.= †(ext_comp ???));
-     ]
-  | intros; simplify; intro; simplify;
+     [ apply continuous_relation_comp;
+     | intros; simplify; intro x; simplify;
+       lapply depth=0 (continuous_relation_eq' ???? H) as H';
+       lapply depth=0 (continuous_relation_eq' ???? H1) as H1';
+       letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
+       cut (∀X:Ω \sup o1.
+              minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X)))
+            = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X))));
+        [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);]
+       clear K H' H1';
+       cut (∀X:Ω \sup o1.
+              minus_star_image o1 o3 (b ∘ a) (A o1 X) = minus_star_image o1 o3 (b'∘a') (A o1 X));
+        [2: intro;
+            apply (.= (minus_star_image_comp ??????));
+            apply (.= #‡(saturated ?????));
+             [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
+            apply sym1; 
+            apply (.= (minus_star_image_comp ??????));
+            apply (.= #‡(saturated ?????));
+             [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
+           apply ((Hcut X) \sup -1)]
+       clear Hcut; generalize in match x; clear x;
+       apply (continuous_relation_eq_inv');
+       apply Hcut1;]
+  | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
     apply (.= †(ASSOC1‡#));
     apply refl1
-  | intros; simplify; intro; simplify;
+  | intros; simplify; intro; unfold continuous_relation_comp; simplify;
     apply (.= †((id_neutral_right1 ????)‡#));
     apply refl1
   | intros; simplify; intro; simplify;
     apply (.= †((id_neutral_left1 ????)‡#));
     apply refl1]
-qed.*)
\ No newline at end of file
+qed.
\ No newline at end of file
-- 
2.39.2