(* *)
(**************************************************************************)
+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 → BTop.
+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);
- | intros 2; split; intro;
+ | 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;
| 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 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;
| 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))));
+ change with ((◊⎽t ((⊩)* V) >< (⊩)⎻* ((⊩)⎻ U)) = (U >< (◊⎽t ((⊩)* V))));
apply (.= (or_prop3 ?? (⊩) ((⊩)* V) ?));
apply (.= #‡(lemma_10_3_a : ?));
apply (.= (or_prop3 : ?)^-1);
definition o_continuous_relation_of_o_relation_pair:
∀BP1,BP2.arrows2 OBP BP1 BP2 →
- arrows2 BTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair 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;
apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩)* ) U));
change in e with (U=((⊩)∘(⊩ \sub BP1) \sup * ) U);
apply (†e^-1);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
apply sym1;
apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ 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.
+