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[helm.git] / helm / software / matita / contribs / formal_topology / overlap / o-basic_pairs_to_o-basic_topologies.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "o-basic_pairs.ma".
+include "o-basic_topologies.ma".
+
+lemma pippo: ∀S:OA.∀p,q,r:S. p ≤ q → p >< r → q >< r.
+ intros;
+ cut (r = binary_meet ? r r); (* la notazione non va ??? *)
+  [ apply (. (#‡Hcut) ^ -1);
+    apply oa_overlap_preservers_meet;
+  | 
+  ]
+
+(* Part of proposition 9.9 *)
+lemma lemmax: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R p ≤ R q.
+ intros;
+ apply oa_density; intros;
+ apply (. (or_prop3 : ?) ^ -1);
+ apply 
+
+(* Lemma 10.2, to be moved to OA *)
+lemma lemma_10_2_a: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R⎻* (R⎻ p).
+ intros;
+ apply (. (or_prop2 : ?));
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_b: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* p) ≤ p.
+ intros;
+ apply (. (or_prop2 : ?) ^ -1);
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_3: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* p)) = R⎻* p.
+ intros; apply oa_leq_antisym;
+  [ lapply (lemma_10_2_b ?? R p);
+    
+  | apply lemma_10_2_a;]
+qed.
+ 
+
+
+(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
+definition o_basic_topology_of_basic_pair: BP → BTop.
+ intro;
+ constructor 1;
+  [ apply (form t);
+  | apply (□_t ∘ Ext⎽t);
+  | apply (◊_t ∘ Rest⎽t);
+  | intros 2;
+    lapply depth=0 (or_prop1 ?? (rel t));
+    lapply depth=0 (or_prop2 ?? (rel t));
+    
+  |
+  |
+  ]
+qed.
+
+definition o_convergent_relation_pair_of_convergent_relation_pair:
+ ∀BP1,BP2.cic:/matita/formal_topology/concrete_spaces/convergent_relation_pair.ind#xpointer(1/1) BP1 BP2 →
+  convergent_relation_pair (o_concrete_space_of_concrete_space BP1) (o_concrete_space_of_concrete_space BP2).
+ intros;
+ constructor 1;
+  [ apply (orelation_of_relation ?? (r \sub \c));
+  | apply (orelation_of_relation ?? (r \sub \f));
+  | lapply (commute ?? r);
+    lapply (orelation_of_relation_preserves_equality ???? Hletin);
+    apply (.= (orelation_of_relation_preserves_composition (concr BP1) ??? (rel BP2)) ^ -1);
+    apply (.= (orelation_of_relation_preserves_equality ???? (commute ?? r)));
+    apply (orelation_of_relation_preserves_composition ?? (form BP2)  (rel BP1) ?); ]
+qed.