X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-basic_pairs_to_o-basic_topologies.ma;h=b78e7b037046bededec4973267647e8e115270bc;hb=3e094922bf3fec6975fdbe6feceb509eaafe0563;hp=f2f6af0208c5c914fa25ab41db74f806e7da9804;hpb=8c0ccf03dbefd83818bc3b6849634f422f8ec727;p=helm.git

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
index f2f6af020..b78e7b037 100644
--- 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
@@ -15,139 +15,13 @@
 include "o-basic_pairs.ma".
 include "o-basic_topologies.ma".
 
-(* qui la notazione non va *)
-lemma leq_to_eq_join: ∀S:OA.∀p,q:S. p ≤ q → q = binary_join ? p q.
- intros;
- apply oa_leq_antisym;
-  [ apply oa_density; intros;
-    apply oa_overlap_sym;
-    unfold binary_join; simplify;
-    apply (. (oa_join_split : ?));
-    exists; [ apply false ]
-    apply oa_overlap_sym;
-    assumption
-  | unfold binary_join; simplify;
-    apply (. (oa_join_sup : ?)); intro;
-    cases i; whd in ⊢ (? ? ? ? ? % ?);
-     [ assumption | apply oa_leq_refl ]]
-qed.
-
-lemma overlap_monotone_left: ∀S:OA.∀p,q,r:S. p ≤ q → p >< r → q >< r.
- intros;
- apply (. (leq_to_eq_join : ?)‡#);
-  [ apply f;
-  | skip
-  | apply oa_overlap_sym;
-    unfold binary_join; simplify;
-    apply (. (oa_join_split : ?));
-    exists [ apply true ]
-    apply oa_overlap_sym;
-    assumption; ]
-qed.
-
-(* Part of proposition 9.9 *)
-lemma f_minus_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻ p ≤ R⎻ q.
- intros;
- apply (. (or_prop2 : ?));
- apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop2 : ?)^ -1); apply oa_leq_refl;]
-qed.
- 
-(* Part of proposition 9.9 *)
-lemma f_minus_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻* p ≤ R⎻* q.
- intros;
- apply (. (or_prop2 : ?)^ -1);
- apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop2 : ?)); apply oa_leq_refl;]
-qed.
-
-(* Part of proposition 9.9 *)
-lemma f_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R p ≤ R q.
- intros;
- apply (. (or_prop1 : ?));
- apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop1 : ?)^ -1); apply oa_leq_refl;]
-qed.
-
-(* Part of proposition 9.9 *)
-lemma f_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R* p ≤ R* q.
- intros;
- apply (. (or_prop1 : ?)^ -1);
- apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop1 : ?)); apply oa_leq_refl;]
-qed.
-
-lemma lemma_10_2_a: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R⎻* (R⎻ p).
- intros;
- apply (. (or_prop2 : ?)^-1);
- 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 : ?));
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_2_c: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R* (R p).
- intros;
- apply (. (or_prop1 : ?)^-1);
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_2_d: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* p) ≤ p.
- intros;
- apply (. (or_prop1 : ?));
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_3_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* (R⎻ p)) = R⎻ p.
- intros; apply oa_leq_antisym;
-  [ apply lemma_10_2_b;
-  | apply f_minus_image_monotone;
-    apply lemma_10_2_a; ]
-qed.
-
-lemma lemma_10_3_b: ∀S,T.∀R:arrows2 OA S T.∀p. R* (R (R* p)) = R* p.
- intros; apply oa_leq_antisym;
-  [ apply f_star_image_monotone;
-    apply (lemma_10_2_d ?? R p);
-  | apply lemma_10_2_c; ]
-qed.
-
-lemma lemma_10_3_c: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R p)) = R p.
- intros; apply oa_leq_antisym;
-  [ apply lemma_10_2_d;
-  | apply f_image_monotone;
-    apply (lemma_10_2_c ?? R p); ]
-qed.
-
-lemma lemma_10_3_d: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* p)) = R⎻* p.
- intros; apply oa_leq_antisym;
-  [ apply f_minus_star_image_monotone;
-    apply (lemma_10_2_b ?? R p);
-  | apply lemma_10_2_a; ]
-qed.
-
-lemma lemma_10_4_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* (R⎻ p))) = R⎻* (R⎻ p).
- intros;
- (* BAD *)
- lapply (†(lemma_10_3_a ?? R p)); [2: apply (R⎻* ); | skip | apply Hletin ]
-qed.
-
-(* VEERY BAD! *)
-axiom lemma_10_4_b: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R (R* p))) = R (R* p).
-(*
- intros;
- (* BAD *)
- lapply (†(lemma_10_3_b ?? R p)); [2: apply rule R; | skip | apply Hletin ]
-qed. *)
-
-lemma oa_overlap_sym': ∀o:OA.∀U,V:o. (U >< V) = (V >< U).
- intros; split; intro; apply oa_overlap_sym; assumption.
-qed.
+alias symbol "eq" = "setoid1 eq".
 
 (* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_topology_of_o_basic_pair: BP → BTop.
- intro;
+definition o_basic_topology_of_o_basic_pair: OBP → BTop.
+ intro t;
  constructor 1;
-  [ apply (form t);
+  [ apply (Oform t);
   | apply (□_t ∘ Ext⎽t);
   | apply (◊_t ∘ Rest⎽t);
   | intros 2; split; intro;
@@ -184,44 +58,35 @@ definition o_basic_topology_of_o_basic_pair: BP → BTop.
 qed.
 
 definition o_continuous_relation_of_o_relation_pair:
- ∀BP1,BP2.arrows2 BP BP1 BP2 →
+ ∀BP1,BP2.arrows2 OBP BP1 BP2 →
   arrows2 BTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair BP2).
- intros;
+ intros (BP1 BP2 t);
  constructor 1;
   [ apply (t \sub \f);
   | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
     apply sym1;
-    alias symbol "refl" = "refl1".
-    apply (.= †?); [1: apply (t \sub \f (((◊_BP1∘(⊩)* ) U))); |
-    lapply (†e); [2: apply rule t \sub \f; | skip | apply Hletin]]
-    change in ⊢ (? ? ? % ?) with ((◊_BP2 ∘(⊩)* ) ((t \sub \f ∘ (◊_BP1∘(⊩)* )) U));
-    lapply (comp_assoc2 ????? (⊩)* (⊩) t \sub \f);
-    apply (.= †(Hletin ?)); clear Hletin;
+    apply (.= †(†e));
     change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
-    cut ?;
-     [3: apply CProp1; |5: cases (commute ?? t); [2: apply (e3 ^ -1 ((⊩)* U));] | 2,4: skip]
-    apply (.= †Hcut);
+    cut ((t \sub \f ∘ (⊩)) ((⊩)* U) = ((⊩) ∘ t \sub \c) ((⊩)* U)) as COM;[2:
+      cases (Ocommute ?? t); apply (e3 ^ -1 ((⊩)* U));]
+    apply (.= †COM);
     change in ⊢ (? ? ? % ?) with (((⊩) ∘ (⊩)* ) (((⊩) ∘ t \sub \c ∘ (⊩)* ) U));
     apply (.= (lemma_10_3_c ?? (⊩) (t \sub \c ((⊩)* U))));
-    apply (.= Hcut ^ -1);
+    apply (.= COM ^ -1);
     change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩)* ) U));
-    apply (prop11 ?? t \sub \f);
-    apply (e ^ -1);
+    change in e with (U=((⊩)∘(⊩ \sub BP1) \sup * ) U);
+    apply (†e^-1);
   | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
     apply sym1;
-    apply (.= †?); [1: apply (t \sub \f⎻* ((((⊩)⎻* ∘ (⊩)⎻) U))); |
-    lapply (†e); [2: apply rule (t \sub \f⎻* ); | skip | apply Hletin]]
-    change in ⊢ (? ? ? % ?) with (((⊩)⎻* ∘(⊩)⎻ ) ((t \sub \f⎻* ∘ ((⊩)⎻*∘(⊩)⎻ )) U));
-    lapply (comp_assoc2 ????? (⊩)⎻ (⊩)⎻* t \sub \f⎻* );
-    apply (.= †(Hletin ?)); clear Hletin;
+    apply (.= †(†e));
     change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U));
-    cut ?;
-     [3: apply CProp1; |5: cases (commute ?? t); [2: apply (e1 ^ -1 ((⊩)⎻ U));] | 2,4: skip]
-    apply (.= †Hcut);
+    cut ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U) = ((⊩)⎻* ∘ t \sub \c⎻* ) ((⊩)⎻ U)) as COM;[2:
+      cases (Ocommute ?? t); apply (e1 ^ -1 ((⊩)⎻ U));]
+    apply (.= †COM);
     change in ⊢ (? ? ? % ?) with (((⊩)⎻* ∘ (⊩)⎻ ) (((⊩)⎻* ∘ t \sub \c⎻* ∘ (⊩)⎻ ) U));
     apply (.= (lemma_10_3_d ?? (⊩) (t \sub \c⎻* ((⊩)⎻ U))));
-    apply (.= Hcut ^ -1);
+    apply (.= COM ^ -1);
     change in ⊢ (? ? ? % ?) with (t \sub \f⎻* (((⊩)⎻* ∘ (⊩)⎻ ) U));
-    apply (prop11 ?? t \sub \f⎻* );
-    apply (e ^ -1); ]
-qed.
\ No newline at end of file
+    change in e with (U=((⊩)⎻* ∘(⊩ \sub BP1)⎻ ) U);
+    apply (†e^-1);]
+qed.