apply oa_leq_refl.
qed.
-lemma lemma_10_3_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* p)) = R⎻* p.
+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; ]
+ [ 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.
| 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).
+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 ]
+ lapply (†(lemma_10_3_a ?? R p)); [2: apply (R⎻* ); | skip | apply Hletin ]
qed.
(* VEERY 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.
+
(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_topology_of_basic_pair: BP → BTop.
+definition o_basic_topology_of_o_basic_pair: BP → BTop.
intro;
constructor 1;
[ apply (form 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).
+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;
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;
+ 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;
+ 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);
+ change in ⊢ (? ? ? % ?) with (((⊩) ∘ (⊩)* ) (((⊩) ∘ t \sub \c ∘ (⊩)* ) U));
+ apply (.= (lemma_10_3_c ?? (⊩) (t \sub \c ((⊩)* U))));
+ apply (.= Hcut ^ -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;
+ 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;
+ 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);
+ change in ⊢ (? ? ? % ?) with (((⊩)⎻* ∘ (⊩)⎻ ) (((⊩)⎻* ∘ t \sub \c⎻* ∘ (⊩)⎻ ) U));
+ apply (.= (lemma_10_3_d ?? (⊩) (t \sub \c⎻* ((⊩)⎻ U))));
+ apply (.= Hcut ^ -1);
+ change in ⊢ (? ? ? % ?) with (t \sub \f⎻* (((⊩)⎻* ∘ (⊩)⎻ ) U));
+ apply (prop11 ?? t \sub \f⎻* );
+ apply (e ^ -1); ]
+qed.
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