(* *)
(**************************************************************************)
+include "notation.ma".
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 → OBTop.
+ intro t;
constructor 1;
- [ apply (form t);
+ [ apply (Oform t);
| apply (□_t ∘ Ext⎽t);
| apply (◊_t ∘ Rest⎽t);
- | intros 2; split; intro;
+ | 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;
| change with (U ≤ (⊩)⎻* ((⊩)⎻ U));
apply (. (or_prop2 : ?) ^ -1);
apply oa_leq_refl; ]]
- | intros 2; split; intro;
+ | 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));
| change with ((⊩) ((⊩)* V) ≤ V);
apply (. (or_prop1 : ?));
apply oa_leq_refl; ]]
- | intros;
+ | apply hide; intros;
apply (.= (oa_overlap_sym' : ?));
change with ((◊_t ((⊩)* V) >< (⊩)⎻* ((⊩)⎻ U)) = (U >< (◊_t ((⊩)* V))));
apply (.= (or_prop3 ?? (⊩) ((⊩)* V) ?));
qed.
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.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);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ | apply hide; 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);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ 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 (.= †?); [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.
+
+
+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.
+
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