From 79ce67a7a7502462e827de098b1056516092c0a7 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Sun, 4 Jan 2009 17:55:20 +0000
Subject: [PATCH] Snapshot to try to understand something.

---
 .../formal_topology/overlap/basic_pairs.ma    | 192 ++++++++++++++++++
 .../contribs/formal_topology/overlap/depends  |   1 +
 .../formal_topology/overlap/relations.ma      |  40 +++-
 3 files changed, 223 insertions(+), 10 deletions(-)
 create mode 100644 helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
new file mode 100644
index 000000000..b13317c67
--- /dev/null
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
@@ -0,0 +1,192 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "relations.ma".
+
+record basic_pair: Type1 ≝
+ { concr: REL;
+   form: REL;
+   rel: arrows1 ? concr form
+ }.
+
+notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
+notation "⊩" with precedence 60 for @{'Vdash}.
+
+interpretation "basic pair relation" 'Vdash2 x y = (rel _ x y).
+interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
+
+alias symbol "eq" = "setoid1 eq".
+alias symbol "compose" = "category1 composition".
+record relation_pair (BP1,BP2: basic_pair): Type1 ≝
+ { concr_rel: arrows1 ? (concr BP1) (concr BP2);
+   form_rel: arrows1 ? (form BP1) (form BP2);
+   commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
+ }.
+
+notation "hvbox (r \sub \c)"  with precedence 90 for @{'concr_rel $r}.
+notation "hvbox (r \sub \f)"  with precedence 90 for @{'form_rel $r}.
+
+interpretation "concrete relation" 'concr_rel r = (concr_rel __ r). 
+interpretation "formal relation" 'form_rel r = (form_rel __ r).
+
+include "o-basic_pairs.ma".
+
+definition o_basic_pair_of_basic_pair: cic:/matita/formal_topology/basic_pairs/basic_pair.ind#xpointer(1/1) → basic_pair.
+ intro;
+ constructor 1;
+  [ apply (SUBSETS (concr b));
+  | apply (SUBSETS (form b));
+  | constructor 1;
+  ]
+qed.
+ 
+
+definition relation_pair_equality:
+ ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
+ intros;
+ constructor 1;
+  [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+  | simplify;
+    intros;
+    apply refl1;
+  | simplify;
+    intros 2;
+    apply sym1;
+  | simplify;
+    intros 3;
+    apply trans1;
+  ]      
+qed.
+
+definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
+ intros;
+ constructor 1;
+  [ apply (relation_pair b b1)
+  | apply relation_pair_equality
+  ]
+qed.
+
+lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+ intros 7 (o1 o2 r r' H c1 f2);
+ split; intro H1;
+  [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
+    lapply (if ?? (H c1 f2) H2) as H3;
+    apply (if ?? (commute ?? r' c1 f2) H3);
+  | lapply (fi ?? (commute ?? r' c1 f2) H1) as H2;
+    lapply (fi ?? (H c1 f2) H2) as H3;
+    apply (if ?? (commute ?? r c1 f2) H3);
+  ]
+qed.
+
+definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
+ intro;
+ constructor 1;
+  [1,2: apply id1;
+  | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
+    lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
+    apply (.= H);
+    apply (H1 \sup -1);]
+qed.
+
+definition relation_pair_composition:
+ ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ intros;
+ constructor 1;
+  [ intros (r r1);
+    constructor 1;
+     [ apply (r1 \sub\c ∘ r \sub\c) 
+     | apply (r1 \sub\f ∘ r \sub\f)
+     | lapply (commute ?? r) as H;
+       lapply (commute ?? r1) as H1;
+       apply (.= ASSOC1);
+       apply (.= #‡H1);
+       apply (.= ASSOC1\sup -1);
+       apply (.= H‡#);
+       apply ASSOC1]
+  | intros;
+    change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));  
+    change in H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+    change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
+    apply (.= ASSOC1);
+    apply (.= #‡H1);
+    apply (.= #‡(commute ?? b'));
+    apply (.= ASSOC1 \sup -1);
+    apply (.= H‡#);
+    apply (.= ASSOC1);
+    apply (.= #‡(commute ?? b')\sup -1);
+    apply (ASSOC1 \sup -1)]
+qed.
+    
+definition BP: category1.
+ constructor 1;
+  [ apply basic_pair
+  | apply relation_pair_setoid
+  | apply id_relation_pair
+  | apply relation_pair_composition
+  | intros;
+    change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
+                 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
+    apply (ASSOC1‡#);
+  | intros;
+    change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
+    apply ((id_neutral_right1 ????)‡#);
+  | intros;
+    change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+    apply ((id_neutral_left1 ????)‡#);]
+qed.
+
+definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
+ intros; constructor 1;
+  [ apply (ext ? ? (rel o));
+  | intros;
+    apply (.= #‡H);
+    apply refl1]
+qed.
+
+definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
+ λo.extS ?? (rel o).
+
+definition fintersects: ∀o: BP. 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:BP. 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: BP. 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 _)).
diff --git a/helm/software/matita/contribs/formal_topology/overlap/depends b/helm/software/matita/contribs/formal_topology/overlap/depends
index fb3dc8f87..4962a0dbc 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/depends
+++ b/helm/software/matita/contribs/formal_topology/overlap/depends
@@ -1,6 +1,7 @@
 o-basic_pairs.ma o-algebra.ma
 o-concrete_spaces.ma o-basic_pairs.ma o-saturations.ma
 o-saturations.ma o-algebra.ma
+basic_pairs.ma relations.ma
 o-algebra.ma categories.ma logic/cprop_connectives.ma
 o-formal_topologies.ma o-basic_topologies.ma
 categories.ma logic/cprop_connectives.ma
diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index 88af29263..c99239beb 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -96,6 +96,7 @@ definition REL: category1.
           first [apply refl | assumption]]]
 qed.
 
+(*
 definition full_subset: ∀s:REL. Ω \sup s.
  apply (λs.{x | True});
  intros; simplify; split; intro; assumption.
@@ -172,21 +173,24 @@ lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (e
     cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
     assumption]
 qed.
-
+*)
+axiom daemon: False.
 (* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
+definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) ?(*(Ω \sup V)*).
+cases daemon; qed.
  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]]
+    intros; simplify; split; intro; cases H; exists [1,3: apply w]
+     [ apply (. (#‡e)‡#); assumption
+     | apply (. (#‡e ^ -1)‡#); assumption]
+  | intros; split; simplify; intros; cases H; exists [1,3: apply w]
+     [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+       apply (if ?? (e ??)); assumption
+     | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+       apply (if ?? (e ^ -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;
@@ -251,4 +255,20 @@ theorem extS_singleton:
   [ 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.
+*)
+
+include "o-algebra.ma".
+
+definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (SUBSETS o1) (SUBSETS o2).
+ intros;
+ constructor 1;
+  [ 
+  |
+  |
+  |
+  |
+  |
+  |
+  ]
 qed.
\ No newline at end of file
-- 
2.39.2