(**************************************************************************)
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).
-
-theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
- intros 4; assumption.
-qed.
-
-theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
- intros; apply transitive_subseteq_operator; [apply S2] assumption.
-qed.
-
-theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
- intros; apply (fi ?? (H ??)); apply subseteq_refl.
-qed.
-
-theorem saturation_monotone:
- ∀C,A. is_saturation C A →
- ∀U,V. U ⊆ V → A U ⊆ A V.
- intros; apply (if ?? (H ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
- assumption.
-qed.
-
-theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
- intros; split;
- [ apply (if ?? (H ??)); apply subseteq_refl
- | apply saturation_expansive; assumption]
-qed.
-
-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.
-
-coercion 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.
-definition cont_rel'': ∀S,T: basic_topology. continuous_relation_setoid S T → binary_relation S T ≝ cont_rel.
-
-coercion cont_rel''.
-
-theorem ext_comp:
- ∀o1,o2,o3: REL.
- ∀a: arrows1 ? o1 o2.
- ∀b: arrows1 ? o2 o3.
- ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
- intros;
- unfold ext; unfold extS; simplify; split; intro; simplify; intros;
- cases f; clear f; split; try assumption;
- [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
- [1: split] assumption;
- | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
- [2: cases f] assumption]
-qed.
-
-(*
-(* 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.
-*)
-
-theorem continuous_relation_eq':
+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).
+ 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;
lapply (fi ?? (A_is_saturation ???) Hcut);
apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify;
[ apply I | assumption ]]
-qed.
+qed.*)
-theorem extS_singleton:
- ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
- intros; unfold extS; unfold ext; unfold singleton; simplify;
- split; intros 2; simplify; cases f; split; try assumption;
- [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
- assumption
- | exists; try assumption; split; try assumption; change with (x = x); apply refl]
-qed.
-
-theorem continuous_relation_eq_inv':
+lemma 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'.
+ (∀X.a⎻* (A o1 X) = 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)) →
+ (∀X.a⎻* (A o1 X) = a'⎻* (A o1 X)) →
∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
- [2: clear b H a' a; intros;
+ [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 \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V)));
+ 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 \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V)));
- [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut;
+ 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 (singleton ? V)); clear Hcut1;
+ 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 | cases H; assumption; ]
+ [ apply refl | unfold foo; apply H; ]
change with (x ∈ A ? (ext ?? a V));
- apply (. #‡(†(extS_singleton ????)));
+ apply (. #‡(†(extS_singleton ????)^-1));
assumption;]
- split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
+ split; apply Hcut; [2: assumption | intro; apply sym1; apply f]
qed.
definition continuous_relation_comp:
continuous_relation_setoid o2 o3 →
continuous_relation_setoid o1 o3.
intros (o1 o2 o3 r s); constructor 1;
- [ apply (s ∘ r)
+ [ alias symbol "compose" (instance 1) = "category1 composition".
+apply (s ∘ r)
| intros;
- apply sym1;
+ apply sym1;
+ (*change in ⊢ (? ? ? (? ? ? ? %) ?) with (image_coercion ?? (s ∘ r) U);*)
apply (.= †(image_comp ??????));
- apply (.= (reduced ?????)\sup -1);
+ apply (.= (reduced ?? s (image_coercion ?? r U) ?)^-1);
[ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
- | apply (.= (image_comp ??????)\sup -1);
+ | change in ⊢ (? ? ? % ?) with ((image_coercion ?? s ∘ image_coercion ?? r) U);
+ apply (.= (image_comp ??????)^-1);
apply refl1]
| intros;
apply sym1;
- apply (.= †(minus_star_image_comp ??????));
- apply (.= (saturated ?????)\sup -1);
+ apply (.= †(minus_star_image_comp ??? s r ?));
+ apply (.= (saturated ?? s ((r)⎻* U) ?)^-1);
[ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
- | apply (.= (minus_star_image_comp ??????)\sup -1);
+ | change in ⊢ (? ? ? % ?) with ((s⎻* ∘ r⎻* ) U);
+ apply (.= (minus_star_image_comp ??????)^-1);
apply refl1]]
qed.
| 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;
+ 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.
- 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);]
+ (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';
- 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;
+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 ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
+ 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 ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
- apply ((Hcut X) \sup -1)]
+ 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;
- apply (.= †(ASSOC1‡#));
+ 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 ????)‡#));
| intros; simplify; intro; simplify;
apply (.= †((id_neutral_left1 ????)‡#));
apply refl1]
-qed.
\ No newline at end of file
+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