include "o-basic_pairs.ma".
include "o-saturations.ma".
-lemma xxx : ∀x.carr x → carr1 (setoid1_of_setoid x). intros; assumption; qed.
-coercion xxx.
+definition A : ∀b:BP. unary_morphism1 (form b) (form b).
+intros; constructor 1;
+ [ apply (λx.□_b (Ext⎽b x));
+ | intros; apply (†(†e));]
+qed.
+
+lemma down_p : ∀S:SET1.∀I:SET.∀u:S⇒S.∀c:arrows2 SET1 I S.∀a:I.∀a':I.a=a'→u (c a)=u (c a').
+intros; apply (†(†e));
+qed.
-record concrete_space : Type ≝
+record concrete_space : Type2 ≝
{ bp:> BP;
- downarrow: form bp → oa_P (form bp);
- downarrow_is_sat: is_saturation ? downarrow;
- converges: ∀q1,q2:form bp.
- or_f_minus ?? (⊩) q1 ∧ or_f_minus ?? (⊩) q2 =
- or_f_minus ?? (⊩) ((downarrow q1) ∧ (downarrow q2));
- all_covered: (*⨍^-_bp*) or_f_minus ?? (⊩) (oa_one (form bp)) = oa_one (concr bp);
- il2: ∀I:setoid.∀p:ums I (oa_P (form bp)).
- downarrow (oa_join ? I (mk_unary_morphism ?? (λi:I. downarrow (p i)) ?)) =
- oa_join ? I (mk_unary_morphism ?? (λi:I. downarrow (p i)) ?)
+ (*distr : is_distributive (form bp);*)
+ downarrow: unary_morphism1 (form bp) (form bp);
+ downarrow_is_sat: is_o_saturation ? downarrow;
+ converges: ∀q1,q2.
+ (Ext⎽bp q1 ∧ (Ext⎽bp q2)) = (Ext⎽bp ((downarrow q1) ∧ (downarrow q2)));
+ all_covered: Ext⎽bp (oa_one (form bp)) = oa_one (concr bp);
+ il2: ∀I:SET.∀p:arrows2 SET1 I (form bp).
+ downarrow (∨ { x ∈ I | downarrow (p x) | down_p ???? }) =
+ ∨ { x ∈ I | downarrow (p x) | down_p ???? };
+ il1: ∀q.downarrow (A ? q) = A ? q
}.
-definition bp': concrete_space → basic_pair ≝ λc.bp c.
+interpretation "o-concrete space downarrow" 'downarrow x =
+ (fun11 __ (downarrow _) x).
-coercion bp'.
+definition binary_downarrow :
+ ∀C:concrete_space.binary_morphism1 (form C) (form C) (form C).
+intros; constructor 1;
+[ intros; apply (↓ c ∧ ↓ c1);
+| intros;
+ alias symbol "prop2" = "prop21".
+ alias symbol "prop1" = "prop11".
+ apply ((†e)‡(†e1));]
+qed.
+
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun21 _ _ _ (binary_downarrow _) a b).
-record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
- { rp:> arrows1 ? CS1 CS2;
+record convergent_relation_pair (CS1,CS2: concrete_space) : Type2 ≝
+ { rp:> arrows2 ? CS1 CS2;
respects_converges:
- ∀b,c.
- extS ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
- BPextS CS1 ((extS ?? rp \sub\f b) ↓ (extS ?? rp \sub\f c));
+ ∀b,c. eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (rp\sub\f⎻ b ↓ rp\sub\f⎻ c));
respects_all_covered:
- extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)
+ eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (oa_one (form CS2))))
+ (Ext⎽CS1 (oa_one (form CS1)))
}.
-definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
- λCS1,CS2,c. rp CS1 CS2 c.
-
-coercion rp'.
-
-definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
+definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid2.
intros;
constructor 1;
[ apply (convergent_relation_pair c c1)
| constructor 1;
[ intros;
apply (relation_pair_equality c c1 c2 c3);
- | intros 1; apply refl1;
- | intros 2; apply sym1;
- | intros 3; apply trans1]]
+ | intros 1; apply refl2;
+ | intros 2; apply sym2;
+ | intros 3; apply trans2]]
qed.
-definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1 BP CS1 CS2 ≝
- λCS1,CS2,c.rp ?? c.
-
-coercion rp''.
+definition convergent_relation_space_of_convergent_relation_space_setoid:
+ ∀CS1,CS2.carr2 (convergent_relation_space_setoid CS1 CS2) →
+ convergent_relation_pair CS1 CS2 ≝ λP,Q,c.c.
+coercion convergent_relation_space_of_convergent_relation_space_setoid.
definition convergent_relation_space_composition:
∀o1,o2,o3: concrete_space.
- binary_morphism1
+ binary_morphism2
(convergent_relation_space_setoid o1 o2)
(convergent_relation_space_setoid o2 o3)
(convergent_relation_space_setoid o1 o3).
intros; constructor 1;
- [ intros; whd in c c1 ⊢ %;
+ [ intros; whd in t t1 ⊢ %;
constructor 1;
- [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
+ [ apply (c1 ∘ c);
| intros;
- change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
- change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
- with (c1 \sub \f ∘ c \sub \f);
- change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
- with (c1 \sub \f ∘ c \sub \f);
- apply (.= (extS_com ??????));
- apply (.= (†(respects_converges ?????)));
- apply (.= (respects_converges ?????));
- apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
- apply refl1;
- | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
- apply (.= (extS_com ??????));
- apply (.= (†(respects_all_covered ???)));
- apply (.= respects_all_covered ???);
- apply refl1]
+ change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (b↓c2))));
+ alias symbol "trans" = "trans1".
+ apply (.= († (respects_converges : ?)));
+ apply (respects_converges ?? c (c1\sub\f⎻ b) (c1\sub\f⎻ c2));
+ | change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (oa_one (form o3)))));
+ apply (.= (†(respects_all_covered :?)));
+ apply rule (respects_all_covered ?? c);]
| intros;
- change with (b ∘ a = b' ∘ a');
- change in H with (rp'' ?? a = rp'' ?? a');
- change in H1 with (rp'' ?? b = rp ?? b');
- apply (.= (H‡H1));
- apply refl1]
+ change with (b ∘ a = b' ∘ a');
+ change in e with (rp ?? a = rp ?? a');
+ change in e1 with (rp ?? b = rp ?? b');
+ apply (e‡e1);]
qed.
-definition CSPA: category1.
+definition CSPA: category2.
constructor 1;
[ apply concrete_space
| apply convergent_relation_space_setoid
| intro; constructor 1;
- [ apply id1
- | intros;
- unfold id; simplify;
- apply (.= (equalset_extS_id_X_X ??));
- apply (.= (†((equalset_extS_id_X_X ??)\sup -1‡
- (equalset_extS_id_X_X ??)\sup -1)));
- apply refl1;
- | apply (.= (equalset_extS_id_X_X ??));
- apply refl1]
+ [ apply id2
+ | intros; apply refl1;
+ | apply refl1]
| apply convergent_relation_space_composition
- | intros; simplify;
+ | intros; simplify; whd in a12 a23 a34;
change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
- apply (.= ASSOC1);
- apply refl1
+ apply rule ASSOC;
| intros; simplify;
- change with (a ∘ id1 ? o1 = a);
- apply (.= id_neutral_right1 ????);
- apply refl1
+ change with (a ∘ id2 BP o1 = a);
+ apply (id_neutral_right2 : ?);
| intros; simplify;
- change with (id1 ? o2 ∘ a = a);
- apply (.= id_neutral_left1 ????);
- apply refl1]
+ change with (id2 ? o2 ∘ a = a);
+ apply (id_neutral_left2 : ?);]
qed.
+
+definition concrete_space_of_CSPA : objs2 CSPA → concrete_space ≝ λx.x.
+coercion concrete_space_of_CSPA.
+
+definition convergent_relation_space_setoid_of_arrows2_CSPA :
+ ∀P,Q. arrows2 CSPA P Q → convergent_relation_space_setoid P Q ≝ λP,Q,x.x.
+coercion convergent_relation_space_setoid_of_arrows2_CSPA.
+
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