universe inconsistency fixed

[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
index 324e51df7a2fcc651b0af478255db26f1364ab6a..bdadaf0fe376a197d162194faefb3dac21e69166 100644 (file)
--- 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
@@ -15,69 +15,80 @@
 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.
- 
-
+alias symbol "eq" = "setoid1 eq".
 
 (* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_topology_of_basic_pair: BP → BTop.
- intro;
+definition o_basic_topology_of_o_basic_pair: BP → BTop.
+ intro t;
  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));
-    
-  |
-  |
-  ]
+  | 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; ]]
+  | 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; ]]
+  | 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.
 
-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;
+definition o_continuous_relation_of_o_relation_pair:
+ ∀BP1,BP2.arrows2 BP BP1 BP2 →
+  arrows2 BTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair BP2).
+ intros (BP1 BP2 t);
  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.
+  [ apply (t \sub \f);
+  | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+    apply sym1;
+    unfold in ⊢ (? ? ? (? ? ? ? %) ?);
+    apply (.= †(†e));
+    change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
+    cut ((t \sub \f ∘ (⊩)) ((⊩)* U) = ((⊩) ∘ t \sub \c) ((⊩)* U)) as COM;[2:
+      cases (commute ?? 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);
+    unfold in ⊢ (? ? ? % %); apply (†e^-1);
+  | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+    apply sym1;
+    unfold in ⊢ (? ? ? (? ? ? ? %) ?);
+    apply (.= †(†e));
+    change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U));
+    cut ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U) = ((⊩)⎻* ∘ t \sub \c⎻* ) ((⊩)⎻ U)) as COM;[2:
+      cases (commute ?? 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);
+    unfold in ⊢ (? ? ? % %); apply (†e^-1);]
+qed.
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