include "o-algebra.ma".
include "o-saturations.ma".
-record basic_topology: Type ≝
+record basic_topology: Type2 ≝
{ carrbt:> OA;
- A: arrows1 SET (oa_P carrbt) (oa_P carrbt);
- J: arrows1 SET (oa_P carrbt) (oa_P carrbt);
- A_is_saturation: is_saturation ? A;
- J_is_reduction: is_reduction ? J;
+ A: carrbt ⇒ carrbt;
+ J: carrbt ⇒ carrbt;
+ A_is_saturation: is_o_saturation ? A;
+ J_is_reduction: is_o_reduction ? J;
compatibility: ∀U,V. (A U >< J V) = (U >< J V)
}.
-record continuous_relation (S,T: basic_topology) : Type ≝
- { cont_rel:> arrows1 ? S T;
+lemma hint: OA → objs2 OA.
+ intro; apply t;
+qed.
+coercion hint.
+
+record continuous_relation (S,T: basic_topology) : Type2 ≝
+ { cont_rel:> arrows2 OA S T;
(* reduces uses eq1, saturated uses eq!!! *)
reduced: ∀U. U = J ? U → cont_rel U = J ? (cont_rel U);
saturated: ∀U. U = A ? U → cont_rel⎻* U = A ? (cont_rel⎻* U)
}.
-definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
+definition continuous_relation_setoid: basic_topology → basic_topology → setoid2.
intros (S T); constructor 1;
[ apply (continuous_relation S T)
| constructor 1;
- [ apply (λr,s:continuous_relation S T.∀b. eq1 (oa_P (carrbt S)) (A ? (r⎻ b)) (A ? (s⎻ b)));
- | simplify; intros; apply refl1;
- | simplify; intros; apply sym1; apply H
- | simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
+ [ (*apply (λr,s:continuous_relation S T.∀b. eq1 (oa_P (carrbt S)) (A ? (r⎻ b)) (A ? (s⎻ b)));*)
+ apply (λr,s:continuous_relation S T.r⎻* ∘ (A S) = s⎻* ∘ (A ?));
+ | simplify; intros; apply refl2;
+ | simplify; intros; apply sym2; apply e
+ | simplify; intros; apply trans2; [2: apply e |3: apply e1; |1: skip]]]
qed.
-definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → arrows1 ? S T ≝ cont_rel.
+definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → arrows2 ? S T ≝ cont_rel.
coercion cont_rel'.
definition cont_rel'':
∀S,T: basic_topology.
- continuous_relation_setoid S T → unary_morphism (oa_P (carrbt S)) (oa_P (carrbt T)).
+ carr2 (continuous_relation_setoid S T) → ORelation_setoid (carrbt S) (carrbt T).
intros; apply rule cont_rel; apply c;
qed.
coercion cont_rel''.
+
(*
theorem continuous_relation_eq':
∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
qed.
*)
+
definition continuous_relation_comp:
∀o1,o2,o3.
continuous_relation_setoid o1 o2 →
continuous_relation_setoid o2 o3 →
continuous_relation_setoid o1 o3.
intros (o1 o2 o3 r s); constructor 1;
- [ apply (s ∘ r)
+ [ apply (s ∘ r);
| intros;
apply sym1;
change in match ((s ∘ r) U) with (s (r U));
- (*BAD*) unfold FunClass_1_OF_carr1;
- apply (.= ((reduced : ?)\sup -1));
+ (*<BAD>*) unfold FunClass_1_OF_Type_OF_setoid2;
+ unfold objs2_OF_basic_topology1; unfold hint;
+ letin reduced := reduced; clearbody reduced;
+ unfold uncurry_arrows in reduced ⊢ %; (*</BAD>*)
+ apply (.= (reduced : ?)\sup -1);
[ (*BAD*) change with (eq1 ? (r U) (J ? (r U)));
(* BAD U *) apply (.= (reduced ??? U ?)); [ assumption | apply refl1 ]
| apply refl1]
| intros;
- apply sym;
+ apply sym1;
change in match ((s ∘ r)⎻* U) with (s⎻* (r⎻* U));
apply (.= (saturated : ?)\sup -1);
- [ apply (.= (saturated : ?)); [ assumption | apply refl ]
- | apply refl]]
+ [ apply (.= (saturated : ?)); [ assumption | apply refl1 ]
+ | apply refl1]]
qed.
-definition BTop: category1.
+definition BTop: category2.
constructor 1;
[ apply basic_topology
| apply continuous_relation_setoid
| intro; constructor 1;
- [ apply id1
- | intros; apply H;
- | intros; apply H;]
+ [ apply id2
+ | intros; apply e;
+ | intros; apply e;]
| intros; constructor 1;
[ apply continuous_relation_comp;
- | intros; simplify; intro x; simplify; (*
- lapply depth=0 (continuous_relation_eq' ???? H) as H';
- lapply depth=0 (continuous_relation_eq' ???? H1) as H1';
- letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
- cut (∀X:Ω \sup o1.
- minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X)))
- = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X))));
- [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);]
- clear K H' H1';
- cut (∀X:Ω \sup o1.
- minus_star_image o1 o3 (b ∘ a) (A o1 X) = minus_star_image o1 o3 (b'∘a') (A o1 X));
- [2: intro;
- apply (.= (minus_star_image_comp ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
- apply sym1;
- apply (.= (minus_star_image_comp ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
- apply ((Hcut X) \sup -1)]
- clear Hcut; generalize in match x; clear x;
- apply (continuous_relation_eq_inv');
- apply Hcut1;*)]
- | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
- (*apply (.= †(ASSOC1‡#));
- apply refl1*)
- | intros; simplify; intro; unfold continuous_relation_comp; simplify;
- (*apply (.= †((id_neutral_right1 ????)‡#));
- apply refl1*)
- | intros; simplify; intro; simplify;
- apply (.= †((id_neutral_left1 ????)‡#));
- apply refl1]
+ | intros; simplify;
+ change with ((b⎻* ∘ a⎻* ) ∘ A o1 = ((b'⎻* ∘ a'⎻* ) ∘ A o1));
+ change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
+ change in e with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
+ change in e1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
+ apply (.= e‡#);
+ intro x;
+ change with (b⎻* (a'⎻* (A o1 x)) = b'⎻*(a'⎻* (A o1 x)));
+ alias symbol "trans" = "trans1".
+ alias symbol "prop1" = "prop11".
+ alias symbol "invert" = "setoid1 symmetry".
+ lapply (.= †(saturated o1 o2 a' (A o1 x) : ?));
+ [3: apply (b⎻* ); | 5: apply Hletin; |1,2: skip;
+ |apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1); ]
+ change in e1 with (∀x.b⎻* (A o2 x) = b'⎻* (A o2 x));
+ apply (.= (e1 (a'⎻* (A o1 x))));
+ alias symbol "invert" = "setoid1 symmetry".
+ lapply (†((saturated ?? a' (A o1 x) : ?) ^ -1));
+ [2: apply (b'⎻* ); |4: apply Hletin; | skip;
+ |apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1);]]
+ | intros; simplify;
+ change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
+ apply rule (#‡ASSOC ^ -1);
+ | intros; simplify;
+ change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_right2 : ?));
+ | intros; simplify;
+ change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_left2 : ?));]
qed.
+definition btop_carr: BTop → Type1 ≝ λo:BTop. carr1 (oa_P (carrbt o)).
+coercion btop_carr.
+
(*
(*CSC: unused! *)
(* this proof is more logic-oriented than set/lattice oriented *)
[2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
apply Hcut2; assumption.
qed.
-*)
\ No newline at end of file
+*)
projects / helm.git / blobdiff