+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "relations.ma".
-include "saturations.ma".
-
-record basic_topology: Type1 ≝
- { carrbt:> REL;
- A: Ω^carrbt ⇒_1 Ω^carrbt;
- J: Ω^carrbt ⇒_1 Ω^carrbt;
- A_is_saturation: is_saturation ? A;
- J_is_reduction: is_reduction ? J;
- compatibility: ∀U,V. (A U ≬ J V) =_1 (U ≬ J V)
- }.
-
-record continuous_relation (S,T: basic_topology) : Type1 ≝
- { cont_rel:> arrows1 ? S T;
- reduced: ∀U. U = J ? U → image ?? cont_rel U = J ? (image ?? cont_rel U);
- saturated: ∀U. U = A ? U → minus_star_image ?? cont_rel U = A ? (minus_star_image ?? cont_rel U)
- }.
-
-definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
- intros (S T); constructor 1;
- [ apply (continuous_relation S T)
- | constructor 1;
- [ apply (λr,s:continuous_relation S T.∀b. A ? (ext ?? r b) = A ? (ext ?? s b));
- | simplify; intros; apply refl1;
- | simplify; intros (x y H); apply sym1; apply H
- | simplify; intros; apply trans1; [2: apply f |3: apply f1; |1: skip]]]
-qed.
-
-definition continuos_relation_of_continuous_relation_setoid :
- ∀P,Q. continuous_relation_setoid P Q → continuous_relation P Q ≝ λP,Q,x.x.
-coercion continuos_relation_of_continuous_relation_setoid.
-
-axiom continuous_relation_eq':
- ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
- a = a' → ∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X).
-(*
- intros; split; intro; unfold minus_star_image; simplify; intros;
- [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
- lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
- cut (A ? (ext ?? a' a1) ⊆ A ? X); [2: apply (. (H ?)‡#); assumption]
- lapply (fi ?? (A_is_saturation ???) Hcut);
- apply (Hletin1 x); change with (x ∈ ext ?? a' a1); split; simplify;
- [ apply I | assumption ]
- | cut (ext ?? a' a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
- lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
- cut (A ? (ext ?? a a1) ⊆ A ? X); [2: apply (. (H ?)\sup -1‡#); assumption]
- lapply (fi ?? (A_is_saturation ???) Hcut);
- apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify;
- [ apply I | assumption ]]
-qed.*)
-
-axiom continuous_relation_eq_inv':
- ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
- (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'.
-(* intros 6;
- cut (∀a,a': continuous_relation_setoid o1 o2.
- (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) →
- ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
- [2: clear b H a' a; intros;
- lapply depth=0 (λV.saturation_expansive ??? (extS ?? a V)); [2: apply A_is_saturation;|skip]
- (* fundamental adjunction here! to be taken out *)
- cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V)));
- [2: intro; intros 2; unfold minus_star_image; simplify; intros;
- apply (Hletin V1 x); whd; split; [ exact I | exists; [apply a1] split; assumption]]
- clear Hletin;
- cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V)));
- [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut;
- (* second half of the fundamental adjunction here! to be taken out too *)
- intro; lapply (Hcut1 (singleton ? V)); clear Hcut1;
- unfold minus_star_image in Hletin; unfold singleton in Hletin; simplify in Hletin;
- whd in Hletin; whd in Hletin:(?→?→%); simplify in Hletin;
- apply (if ?? (A_is_saturation ???));
- intros 2 (x H); lapply (Hletin V ? x ?);
- [ apply refl | cases H; assumption; ]
- change with (x ∈ A ? (ext ?? a V));
- apply (. #‡(†(extS_singleton ????)));
- assumption;]
- 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)
- | intros;
- apply sym1;
- apply (.= †(image_comp ??????));
- apply (.= (reduced ?????)\sup -1);
- [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
- | apply (.= (image_comp ??????)\sup -1);
- apply refl1]
- | intros;
- apply sym1;
- apply (.= †(minus_star_image_comp ??????));
- apply (.= (saturated ?????)\sup -1);
- [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
- | apply (.= (minus_star_image_comp ??????)\sup -1);
- apply refl1]]
-qed.
-
-definition BTop: category1.
- constructor 1;
- [ apply basic_topology
- | apply continuous_relation_setoid
- | intro; constructor 1;
- [ apply id1
- | intros;
- apply (.= (image_id ??));
- apply sym1;
- apply (.= †(image_id ??));
- apply sym1;
- assumption
- | intros;
- apply (.= (minus_star_image_id ??));
- apply sym1;
- apply (.= †(minus_star_image_id ??));
- apply sym1;
- assumption]
- | intros; constructor 1;
- [ apply continuous_relation_comp;
- | intros; simplify; intro x; simplify;
- lapply depth=0 (continuous_relation_eq' ???? e) as H';
- lapply depth=0 (continuous_relation_eq' ???? e1) 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';
-alias symbol "compose" (instance 1) = "category1 composition".
-cut (∀X:Ω^o1.
- minus_star_image ?? (b ∘ a) (A o1 X) =_1 minus_star_image ?? (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;
- alias symbol "trans" (instance 1) = "trans1".
-alias symbol "refl" (instance 5) = "refl1".
-alias symbol "prop2" (instance 3) = "prop21".
-alias symbol "prop1" (instance 2) = "prop11".
-alias symbol "assoc" (instance 4) = "category1 assoc".
-apply (.= †(ASSOC‡#));
- 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]
-qed.
-
-(*
-(*CSC: unused! *)
-(* this proof is more logic-oriented than set/lattice oriented *)
-theorem continuous_relation_eqS:
- ∀o1,o2:basic_topology.∀a,a': continuous_relation_setoid o1 o2.
- a = a' → ∀X. A ? (extS ?? a X) = A ? (extS ?? a' X).
- intros;
- cut (∀a: arrows1 ? o1 ?.∀x. x ∈ extS ?? a X → ∃y:o2.y ∈ X ∧ x ∈ ext ?? a y);
- [2: intros; cases f; clear f; cases H1; exists [apply w] cases x1; split;
- try assumption; split; assumption]
- cut (∀a,a':continuous_relation_setoid o1 o2.eq1 ? a a' → ∀x. x ∈ extS ?? a X → ∃y:o2. y ∈ X ∧ x ∈ A ? (ext ?? a' y));
- [2: intros; cases (Hcut ?? f); exists; [apply w] cases x1; split; try assumption;
- apply (. #‡(H1 ?));
- apply (saturation_expansive ?? (A_is_saturation o1) (ext ?? a1 w) x);
- assumption;] clear Hcut;
- split; apply (if ?? (A_is_saturation ???)); intros 2;
- [lapply (Hcut1 a a' H a1 f) | lapply (Hcut1 a' a (H \sup -1) a1 f)]
- cases Hletin; clear Hletin; cases x; clear x;
- cut (∀a: arrows1 ? o1 ?. ext ?? a w ⊆ extS ?? a X);
- [2,4: intros 3; cases f3; clear f3; simplify in f5; split; try assumption;
- exists [1,3: apply w] split; assumption;]
- cut (∀a. A ? (ext o1 o2 a w) ⊆ A ? (extS o1 o2 a X));
- [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
- apply Hcut2; assumption.
-qed.
-*)
projects / helm.git / blobdiff