-(**************************************************************************)
+ (**************************************************************************)
(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
include "o-algebra.ma".
include "o-saturations.ma".
-record basic_topology: Type ≝
+record basic_topology: Type2 ≝
{ carrbt:> OA;
- A: arrows1 SET (oa_P carrbt) (oa_P carrbt);
- J: arrows1 SET (oa_P carrbt) (oa_P carrbt);
- A_is_saturation: is_saturation ? A;
- J_is_reduction: is_reduction ? J;
+ A: carrbt ⇒ carrbt;
+ J: carrbt ⇒ carrbt;
+ A_is_saturation: is_o_saturation ? A;
+ J_is_reduction: is_o_reduction ? J;
compatibility: ∀U,V. (A U >< J V) = (U >< J V)
}.
-record continuous_relation (S,T: basic_topology) : Type ≝
- { cont_rel:> arrows1 ? S T;
+record continuous_relation (S,T: basic_topology) : Type2 ≝
+ { cont_rel:> arrows2 OA 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)
}.
-definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
+definition continuous_relation_setoid: basic_topology → basic_topology → setoid2.
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)));
- | simplify; intros; apply refl1;
- | simplify; intros; apply sym1; apply H
- | simplify; intros; apply trans1; [2: apply H |3: apply H1; |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 → unary_morphism (oa_P (carrbt S)) (oa_P (carrbt T)).
- intros; apply rule cont_rel; apply c;
+ [ alias symbol "eq" = "setoid2 eq".
+ alias symbol "compose" = "category2 composition".
+ apply (λr,s:continuous_relation S T. (r⎻* ) ∘ (A S) = (s⎻* ∘ (A ?)));
+ | simplify; intros; apply refl2;
+ | simplify; intros; apply sym2; apply e
+ | simplify; intros; apply trans2; [2: apply e |3: apply e1; |1: skip]]]
qed.
-coercion cont_rel''.
+definition continuous_relation_of_continuous_relation_setoid:
+ ∀P,Q. continuous_relation_setoid P Q → continuous_relation P Q ≝ λP,Q,c.c.
+coercion continuous_relation_of_continuous_relation_setoid.
+(*
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 o2 o3 →
continuous_relation_setoid o1 o3.
intros (o1 o2 o3 r s); constructor 1;
- [ apply (s ∘ r)
+ [ apply (s ∘ r);
+ | intros;
+ apply sym1;
+ change in match ((s ∘ r) U) with (s (r U));
+ (*<BAD>*) unfold FunClass_1_OF_carr2;
+ 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 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));
+ apply (.= (saturated : ?)\sup -1);
+ [ apply (.= (saturated : ?)); [ assumption | apply refl1 ]
+ | apply refl1]]
qed.
-definition BTop: category1.
+definition BTop: category2.
constructor 1;
[ apply basic_topology
| 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]
+ [ apply id2
+ | intros; apply e;
+ | intros; apply e;]
| 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;
+ change with ((b⎻* ∘ a⎻* ) ∘ A o1 = ((b'⎻* ∘ a'⎻* ) ∘ A o1));
+ change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
+ change in e with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
+ change in e1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
+ apply (.= e‡#);
+ intro x;
+ change with (eq1 ? (b⎻* (a'⎻* (A o1 x))) (b'⎻*(a'⎻* (A o1 x))));
+ apply (.= †(saturated o1 o2 a' (A o1 x) ?)); [
+ apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1);]
+ apply (.= (e1 (a'⎻* (A o1 x))));
+ change with (eq1 ? (b'⎻* (A o2 (a'⎻* (A o1 x)))) (b'⎻*(a'⎻* (A o1 x))));
+ apply (.= †(saturated o1 o2 a' (A o1 x):?)^-1); [
+ apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1);]
+ apply rule #;]
+ | intros; simplify;
+ change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
+ apply rule (#‡ASSOC ^ -1);
+ | intros; simplify;
+ change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_right2 : ?));
+ | intros; simplify;
+ change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_left2 : ?));]
qed.
+definition basic_topology_of_BTop: objs2 BTop → basic_topology ≝ λx.x.
+coercion basic_topology_of_BTop.
+
+definition continuous_relation_setoid_of_arrows2_BTop :
+ ∀P,Q. arrows2 BTop P Q → continuous_relation_setoid P Q ≝ λP,Q,x.x.
+coercion continuous_relation_setoid_of_arrows2_BTop.
+
+(*
(*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.
+*)
projects / helm.git / blobdiff