record continuous_relation (S,T: basic_topology) : Type ≝
{ cont_rel:> arrows1 ? S T;
+ (* reduces uses eq1, saturated uses eq!!! *)
reduced: ∀U. U = J ? U → cont_rel U = J ? (cont_rel U);
saturated: ∀U. U = A ? U → cont_rel⎻* U = A ? (cont_rel⎻* U)
}.
intros (S T); constructor 1;
[ apply (continuous_relation S T)
| constructor 1;
- [ apply (λr,s:continuous_relation S T.∀b. eq1 (oa_P (carrbt S)) (A ? (r⎻ b)) (A ? (s⎻ b)));
+ [ (*apply (λr,s:continuous_relation S T.∀b. eq1 (oa_P (carrbt S)) (A ? (r⎻ b)) (A ? (s⎻ b)));*)
+ apply (λr,s:continuous_relation S T.r⎻* ∘ (A S) = s⎻* ∘ (A ?));
| simplify; intros; apply refl1;
| simplify; intros; apply sym1; apply H
| simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
qed.
coercion cont_rel''.
-
+(*
theorem continuous_relation_eq':
∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
a = a' → ∀X.a⎻* (A o1 X) = a'⎻* (A o1 X).
- intros;
- lapply (prop_1_SET ??? H);
-
- split; intro; unfold minus_star_image; simplify; intros;
+ intros; apply oa_leq_antisym; 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]
theorem 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)) →
- ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
+ (∀X.a⎻* (A o1 X) = a'⎻* (A o1 X)) →
+ ∀V:(oa_P (carrbt o2)). A o1 (a'⎻ V) ≤ A o1 (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 *)
assumption;]
split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
qed.
-
+*)
definition continuous_relation_comp:
∀o1,o2,o3.
continuous_relation_setoid o1 o2 →
[ 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]]
+ change in match ((s ∘ r) U) with (s (r U));
+ (*BAD*) unfold FunClass_1_OF_carr1;
+ apply (.= ((reduced : ?)\sup -1));
+ [ (*BAD*) change with (eq1 ? (r U) (J ? (r U)));
+ (* BAD U *) apply (.= (reduced ??? U ?)); [ assumption | apply refl1 ]
+ | apply refl1]
+ | intros;
+ apply sym;
+ change in match ((s ∘ r)⎻* U) with (s⎻* (r⎻* U));
+ apply (.= (saturated : ?)\sup -1);
+ [ apply (.= (saturated : ?)); [ assumption | apply refl ]
+ | apply refl]]
qed.
definition BTop: category1.
| 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; apply H;
+ | intros; apply H;]
| 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;
- 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));
- [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;
- apply (.= †(ASSOC1‡#));
- 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]
+ | intros; simplify; (*intro x; simplify;*)
+ change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
+ change in H with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
+ change in H1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
+ apply (.= H‡#);
+ intro x;
+
+ change with (eq1 (oa_P (carrbt o3)) (b⎻* (a'⎻* (A o1 x))) (b'⎻*(a'⎻* (A o1 x))));
+ lapply (saturated o1 o2 a' (A o1 x):?) as X;
+ [ apply ((saturation_idempotent ?? (A_is_saturation o1) x)^-1) ]
+ change in X with (eq1 (oa_P (carrbt o2)) (a'⎻* (A o1 x)) (A o2 (a'⎻* (A o1 x))));
+ unfold uncurry_arrows;
+ apply (.= †X); whd in H1;
+ lapply (H1 (a'⎻* (A o1 x))) as X1;
+ change in X1 with (eq1 (oa_P (carrbt o3)) (b⎻* (A o2 (a'⎻* (A o1 x)))) (b'⎻* (A o2 (a' \sup ⎻* (A o1 x)))));
+ apply (.= X1);
+ unfold uncurry_arrows;
+ apply (†(X\sup -1));]
+ | intros; simplify;
+ change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
+ apply rule (#‡ASSOC1\sup -1);
+ | intros; simplify;
+ change with ((a⎻* ∘ (id1 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_right1 : ?));
+ | intros; simplify;
+ change with (((id1 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_left1 : ?));]
qed.
+(*
(*CSC: unused! *)
(* this proof is more logic-oriented than set/lattice oriented *)
theorem continuous_relation_eqS:
[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