record basic_topology: Type1 ≝
{ carrbt:> REL;
- A: unary_morphism1 (Ω \sup carrbt) (Ω \sup carrbt);
- J: unary_morphism1 (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)
}.
-lemma hint: basic_topology → objs1 REL.
- intro; apply (carrbt b);
-qed.
-coercion hint.
-
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; 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 cont_rel'': ∀S,T: basic_topology. continuous_relation_setoid S T → binary_relation 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''.
-
-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).
+(*
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 continuous_relation_eq_inv':
+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;
+(* 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));
assumption;]
split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
qed.
+*)
definition continuous_relation_comp:
∀o1,o2,o3.
| 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';
+ 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';
- 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));
+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 (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 ????)‡#));
apply Hcut2; assumption.
qed.
*)
-*)
\ No newline at end of file