(**************************************************************************)
include "formal_topology/relations.ma".
-include "datatypes/categories.ma".
+include "formal_topology/saturations.ma".
-definition is_saturation ≝
- λC:REL.λA:unary_morphism (Ω \sup C) (Ω \sup C).
- ∀U,V. (U ⊆ A V) = (A U ⊆ A V).
-
-definition is_reduction ≝
- λC:REL.λJ:unary_morphism (Ω \sup C) (Ω \sup C).
- ∀U,V. (J U ⊆ V) = (J U ⊆ J V).
-
-record basic_topology: Type ≝
+record basic_topology: Type1 ≝
{ carrbt:> REL;
- A: unary_morphism (Ω \sup carrbt) (Ω \sup carrbt);
- J: unary_morphism (Ω \sup carrbt) (Ω \sup carrbt);
+ 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) = (U ≬ J V)
+ compatibility: ∀U,V. (A U ≬ J V) =_1 (U ≬ J V)
}.
-(* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
- intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
- intros; simplify; split; intro; cases H1; exists [1,3: apply w]
- [ apply (. (#‡H)‡#); assumption
- | apply (. (#‡H \sup -1)‡#); assumption]
- | intros; split; simplify; intros; cases H2; exists [1,3: apply w]
- [ apply (. #‡(#‡H1)); cases x; split; try assumption;
- apply (if ?? (H ??)); assumption
- | apply (. #‡(#‡H1 \sup -1)); cases x; split; try assumption;
- apply (if ?? (H \sup -1 ??)); assumption]]
-qed.
-
-(* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
- intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
- intros; simplify; split; intros; apply H1;
- [ apply (. #‡H \sup -1); assumption
- | apply (. #‡H); assumption]
- | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
- apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
-qed.
-
-(* minus_image is the same as ext *)
-
-theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
- intros; unfold image; simplify; split; simplify; intros;
- [ change with (a ∈ U);
- cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
- | change in f with (a ∈ U);
- exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
-qed.
-
-theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
- intros; unfold minus_star_image; simplify; split; simplify; intros;
- [ change with (a ∈ U); apply H; change with (a=a); apply refl
- | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
-qed.
-
-theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
- clear x; [ cases f; clear f; | cases f1; clear f1 ]
- exists; try assumption; cases x; clear x; split; try assumption;
- exists; try assumption; split; assumption.
-qed.
-
-theorem minus_star_image_comp:
- ∀A,B,C,r,s,X.
- minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
- intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
- [ apply H; exists; try assumption; split; assumption
- | change with (x ∈ X); cases f; cases x1; apply H; assumption]
-qed.
-
-record continuous_relation (S,T: basic_topology) : Type ≝
- { 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)
+record continuous_relation (S,T: basic_topology) : Type1 ≝
+ { cont_rel:> S ⇒_\r1 T;
+ reduced: ∀U. U =_1 J ? U → image_coercion ?? cont_rel U =_1 J ? (image_coercion ?? cont_rel U);
+ saturated: ∀U. U =_1 A ? U → (cont_rel)⎻* U = _1A ? ((cont_rel)⎻* U)
}.
definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
| constructor 1;
[ apply (λr,s:continuous_relation S T.∀b. A ? (ext ?? r b) = A ? (ext ?? s b));
| simplify; intros; apply refl1;
- | simplify; intros; apply sym1; apply H
- | simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
+ | simplify; intros (x y H); apply sym1; apply H
+ | simplify; intros; apply trans1; [2: apply f |3: apply f1; |1: skip]]]
qed.
-definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → arrows1 ? S T ≝ cont_rel.
+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.
-coercion cont_rel'.
+axiom continuous_relation_eq':
+ ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
+ a = a' → ∀X.a⎻* (A o1 X) = 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.*)
+
+lemma continuous_relation_eq_inv':
+ ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
+ (∀X.a⎻* (A o1 X) = a'⎻* (A o1 X)) → a=a'.
+ intros 6;
+ cut (∀a,a': continuous_relation_setoid o1 o2.
+ (∀X.a⎻* (A o1 X) = a'⎻* (A o1 X)) →
+ ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
+ [2: clear b f 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^o2.V ⊆ 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^o2.V ⊆ a'⎻* (A ? (extS ?? a V)));
+ [2: intro; apply (. #‡(f ?)^-1); apply Hcut] clear f Hcut;
+ (* second half of the fundamental adjunction here! to be taken out too *)
+ intro; lapply (Hcut1 {(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 | unfold foo; apply H; ]
+ change with (x ∈ A ? (ext ?? a V));
+ apply (. #‡(†(extS_singleton ????)^-1));
+ assumption;]
+ split; apply Hcut; [2: assumption | intro; apply sym1; apply f]
+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;
+ [ alias symbol "compose" (instance 1) = "category1 composition".
+apply (s ∘ r)
+ | intros;
+ apply sym1;
+ (*change in ⊢ (? ? ? (? ? ? ? %) ?) with (image_coercion ?? (s ∘ r) U);*)
+ apply (.= †(image_comp ??????));
+ apply (.= (reduced ?? s (image_coercion ?? r U) ?)^-1);
+ [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
+ | change in ⊢ (? ? ? % ?) with ((image_coercion ?? s ∘ image_coercion ?? r) U);
+ apply (.= (image_comp ??????)^-1);
+ apply refl1]
+ | intros;
+ apply sym1;
+ apply (.= †(minus_star_image_comp ??? s r ?));
+ apply (.= (saturated ?? s ((r)⎻* U) ?)^-1);
+ [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
+ | change in ⊢ (? ? ? % ?) with ((s⎻* ∘ r⎻* ) U);
+ apply (.= (minus_star_image_comp ??????)^-1);
+ apply refl1]]
+qed.
+
definition BTop: category1.
constructor 1;
[ apply basic_topology
apply sym1;
assumption]
| intros; constructor 1;
- [ intros (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]]]
- | intros; simplify; intro; simplify;
- | intros; simplify; intro; simplify;
+ [ 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' ((a)⎻* (A ? X))); clearbody K;
+ cut (∀X:Ω \sup o1.
+ (b)⎻* (A o2 ((a)⎻* (A o1 X)))
+ =_1 (b')⎻* (A o2 ((a')⎻* (A o1 X))));
+ [2: intro; apply sym1;
+ apply (.= (†(†((H' X)^-1)))); apply sym1; apply (K X);]
+ clear K H' H1';
+alias symbol "powerset" (instance 5) = "powerset low".
+alias symbol "compose" (instance 2) = "category1 composition".
+cut (∀X:Ω^o1.
+ ((b ∘ a))⎻* (A o1 X) =_1 ((b'∘a'))⎻* (A o1 X));
+ [2: intro; unfold foo;
+ apply (.= (minus_star_image_comp ??????));
+ change in ⊢ (? ? ? % ?) with ((b)⎻* ((a)⎻* (A o1 X)));
+ apply (.= †(saturated ?????));
+ [ apply ((saturation_idempotent ????)^-1); apply A_is_saturation ]
+ apply sym1;
+ apply (.= (minus_star_image_comp ??????));
+ change in ⊢ (? ? ? % ?) with ((b')⎻* ((a')⎻* (A o1 X)));
+ apply (.= †(saturated ?????));
+ [ apply ((saturation_idempotent ????)^-1); apply A_is_saturation ]
+ apply ((Hcut X)^-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.
-*)
\ No newline at end of file
+*)
projects / helm.git / blobdiff