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;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-basic_pairs_to_o-basic_topologies.ma;h=0000000000000000000000000000000000000000;hb=1ed4fe0f28d3b0b915387330cd722bfb80fb1063;hp=80fec034864530c78435b77e40daa82118e6ba10;hpb=05cfeb82d2624860e66941421a937f308d66cf33;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
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--- a/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 "notation.ma".
-include "o-basic_pairs.ma".
-include "o-basic_topologies.ma".
-
-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: 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.
-
-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 (U e);
-    apply sym1;
-    apply (.= †(†e)); 
-    change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩\sub BP1)* U));
-    cut ((t \sub \f ∘ (⊩)) ((⊩\sub BP1)* U) = ((⊩) ∘ t \sub \c) ((⊩\sub BP1)* U)) as COM;[2:
-      cases (Ocommute ?? t); apply (e3 ^ -1 ((⊩\sub BP1)* U));]
-    apply (.= †COM);
-    change in ⊢ (? ? ? % ?) with (((⊩) ∘ (⊩)* ) (((⊩) ∘ t \sub \c ∘ (⊩)* ) U));
-    apply (.= (lemma_10_3_c ?? (⊩) (t \sub \c ((⊩\sub BP1)* U))));
-    apply (.= COM ^ -1);
-    change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩\sub BP1)* ) 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⎻* ∘ (⊩)⎻* ) ((⊩\sub BP1)⎻ U));
-    cut ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩\sub BP1)⎻ U) = ((⊩)⎻* ∘ t \sub \c⎻* ) ((⊩\sub BP1)⎻ U)) as COM;[2:
-      cases (Ocommute ?? t); apply (e1 ^ -1 ((⊩\sub BP1)⎻ U));]
-    apply (.= †COM);
-    change in ⊢ (? ? ? % ?) with (((⊩)⎻* ∘ (⊩)⎻ ) (((⊩)⎻* ∘ t \sub \c⎻* ∘ (⊩)⎻ ) U));
-    apply (.= (lemma_10_3_d ?? (⊩) (t \sub \c⎻* ((⊩\sub BP1)⎻ U))));
-    apply (.= COM ^ -1);
-    change in ⊢ (? ? ? % ?) with (t \sub \f⎻* (((⊩)⎻* ∘ (⊩\sub BP1)⎻ ) U));
-    change in e with (U=((⊩)⎻* ∘(⊩ \sub BP1)⎻ ) U);
-    apply (†e^-1);]
-qed.
-
-
-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.
-