}.
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)
+ { 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.
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;]
[ apply I | assumption ]]
qed.*)
-axiom 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'.
-(* intros 6;
+ (∀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:
∀o1,o2,o3.
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; 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;
+ 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';
-alias symbol "compose" (instance 1) = "category1 composition".
+alias symbol "powerset" (instance 5) = "powerset low".
+alias symbol "compose" (instance 2) = "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;
+ ((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;]
projects / helm.git / blobdiff