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=b745652569c9458b389903f2a220c93c03c3d6f9;hb=3e5c359c75874748cfed8a9046031b62396e0e6d;hp=324e51df7a2fcc651b0af478255db26f1364ab6a;hpb=05669a3646cbf8219bf772718b4b2c5bfd1ddd82;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 324e51df7..b74565256 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
@@ -12,72 +12,108 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "notation.ma".
 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;
-  | 
-  ]
+alias symbol "eq" = "setoid1 eq".
 
-(* 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.
+(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
+definition o_basic_topology_of_o_basic_pair: OBP → OBTop.
+ intro t;
+ constructor 1;
+  [ apply (Oform t);
+  | apply (□⎽t ∘ Ext⎽t);
+  | apply (◊⎽t ∘ Rest⎽t);
+  | apply hide; intros 2; split; intro;
+     [ change with ((⊩) \sup ⎻* ((⊩) \sup ⎻ U) ≤ (⊩) \sup ⎻* ((⊩) \sup ⎻ V));
+       apply (. (#‡(lemma_10_4_a ?? (⊩) V)^-1));
+       apply f_minus_star_image_monotone;
+       apply f_minus_image_monotone;
+       assumption
+     | apply oa_leq_trans;
+        [3: apply f;
+        | skip
+        | change with (U ≤ (⊩)⎻* ((⊩)⎻ U));
+          apply (. (or_prop2 : ?) ^ -1);
+          apply oa_leq_refl; ]]
+  | apply hide; intros 2; split; intro;
+     [ change with (◊⎽t ((⊩) \sup * U) ≤ ◊⎽t ((⊩) \sup * V));
+       apply (. ((lemma_10_4_b ?? (⊩) U)^-1)‡#);
+       apply (f_image_monotone ?? (⊩) ? ((⊩)* V));
+       apply f_star_image_monotone;
+       assumption;
+     | apply oa_leq_trans;
+        [2: apply f;
+        | skip
+        | change with ((⊩) ((⊩)* V) ≤ V);
+          apply (. (or_prop1 : ?));
+          apply oa_leq_refl; ]]
+  | apply hide; intros;
+    apply (.= (oa_overlap_sym' : ?));
+    change with ((◊⎽t ((⊩)* V) >< (⊩)⎻* ((⊩)⎻ U)) = (U >< (◊⎽t ((⊩)* V))));
+    apply (.= (or_prop3 ?? (⊩) ((⊩)* V) ?));
+    apply (.= #‡(lemma_10_3_a : ?));
+    apply (.= (or_prop3 : ?)^-1);
+    apply (oa_overlap_sym' ? ((⊩) ((⊩)* V)) U); ]
 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;]
+definition o_continuous_relation_of_o_relation_pair:
+ ∀BP1,BP2.arrows2 OBP BP1 BP2 →
+  arrows2 OBTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair BP2).
+ intros (BP1 BP2 t);
+ constructor 1;
+  [ apply (t \sub \f);
+  | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+    apply sym1;
+    apply (.= †(†e));
+    change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
+    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 (.= COM ^ -1);
+    change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩)* ) U));
+    change in e with (U=((⊩)∘(⊩ \sub BP1) \sup * ) U);
+    apply (†e^-1);
+  | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+    apply sym1;
+    apply (.= †(†e));
+    change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U));
+    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 (.= COM ^ -1);
+    change in ⊢ (? ? ? % ?) with (t \sub \f⎻* (((⊩)⎻* ∘ (⊩)⎻ ) U));
+    change in e with (U=((⊩)⎻* ∘(⊩ \sub BP1)⎻ ) U);
+    apply (†e^-1);]
 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));
-    
-  |
-  |
-  ]
+definition OR : carr3 (arrows3 CAT2 OBP OBTop).
+constructor 1;
+[ apply o_basic_topology_of_o_basic_pair;
+| intros; constructor 1;
+  [ apply o_continuous_relation_of_o_relation_pair;
+  | apply hide; 
+    intros; whd; unfold o_continuous_relation_of_o_relation_pair; simplify;;
+    change with ((a \sub \f ⎻* ∘ oA (o_basic_topology_of_o_basic_pair S)) =
+                 (a' \sub \f ⎻*∘ oA (o_basic_topology_of_o_basic_pair S)));
+    whd in e; cases e; clear e e2 e3 e4;
+    change in ⊢ (? ? ? (? ? ? ? ? % ?) ?) with ((⊩\sub S)⎻* ∘ (⊩\sub S)⎻);
+    apply (.= (comp_assoc2 ? ???? ?? a\sub\f⎻* ));
+    change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (a\sub\f ∘ ⊩\sub S)⎻*;
+    apply (.= #‡†(Ocommute:?)^-1);
+    apply (.= #‡e1);
+    change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (⊩\sub T ∘ a'\sub\c)⎻*;
+    apply (.= #‡†(Ocommute:?));    
+    change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (a'\sub\f⎻* ∘ (⊩\sub S)⎻* );    
+    apply (.= (comp_assoc2 ? ???? ?? a'\sub\f⎻* )^-1);
+    apply refl2;]
+| intros 2 (o a); apply refl1;
+| intros 6; apply refl1;]
 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.