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 f_minus_star_image_monotone;
- apply (lemma_10_2_b ?? R p);
- | 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_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. *)
+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; split; intro;
[ change with ((⊩) \sup ⎻* ((⊩) \sup ⎻ U) ≤ (⊩) \sup ⎻* ((⊩) \sup ⎻ V));
- (* apply (.= #‡
- (* BAD *)
- whd in t;
- apply oa_leq_antisym;
- lapply depth=0 (or_prop1 ?? (rel t));
- lapply depth=0 (or_prop2 ?? (rel t));
- *)
- |
- ]
+ 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 (concr t) (form t) (⊩) U)^-1)‡#);*)
- |
- ]
- |
- ]
+ 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.
\ No newline at end of file
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