+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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.
-