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:OBP. unary_morphism1 (Oform b) (Oform 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 ≝
- { bp:> BP;
- downarrow: unary_morphism (oa_P (form bp)) (oa_P (form bp));
- downarrow_is_sat: is_saturation ? downarrow;
- converges: ∀q1,q2.
- 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)) ?)
+record Oconcrete_space : Type2 ≝
+ { Obp:> OBP;
+ (*distr : is_distributive (form bp);*)
+ Odownarrow: unary_morphism1 (Oform Obp) (Oform Obp);
+ Odownarrow_is_sat: is_o_saturation ? Odownarrow;
+ Oconverges: ∀q1,q2.
+ (Ext⎽Obp q1 ∧ (Ext⎽Obp q2)) = (Ext⎽Obp ((Odownarrow q1) ∧ (Odownarrow q2)));
+ Oall_covered: Ext⎽Obp (oa_one (Oform Obp)) = oa_one (Oconcr Obp);
+ Oil2: ∀I:SET.∀p:arrows2 SET1 I (Oform Obp).
+ Odownarrow (∨ { x ∈ I | Odownarrow (p x) | down_p ???? }) =
+ ∨ { x ∈ I | Odownarrow (p x) | down_p ???? };
+ Oil1: ∀q.Odownarrow (A ? q) = A ? q
}.
-definition bp': concrete_space → basic_pair ≝ λc.bp c.
+interpretation "o-concrete space downarrow" 'downarrow x =
+ (fun11 ?? (Odownarrow ?) x).
-coercion bp'.
+definition Obinary_downarrow :
+ ∀C:Oconcrete_space.binary_morphism1 (Oform C) (Oform C) (Oform C).
+intros; constructor 1;
+[ intros; apply (↓ c ∧ ↓ c1);
+| intros;
+ alias symbol "prop2" = "prop21".
+ alias symbol "prop1" = "prop11".
+ apply ((†e)‡(†e1));]
+qed.
-record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
- { rp:> arrows1 ? 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));
- respects_all_covered:
- extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)
- }.
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun21 ? ? ? (Obinary_downarrow ?) a b).
-definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
- λCS1,CS2,c. rp CS1 CS2 c.
-
-coercion rp'.
+record Oconvergent_relation_pair (CS1,CS2: Oconcrete_space) : Type2 ≝
+ { Orp:> arrows2 ? CS1 CS2;
+ Orespects_converges:
+ ∀b,c. eq1 ? (Orp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (Orp\sub\f⎻ b ↓ Orp\sub\f⎻ c));
+ Orespects_all_covered:
+ eq1 ? (Orp\sub\c⎻ (Ext⎽CS2 (oa_one (Oform CS2))))
+ (Ext⎽CS1 (oa_one (Oform CS1)))
+ }.
-definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
- intros;
+definition Oconvergent_relation_space_setoid: Oconcrete_space → Oconcrete_space → setoid2.
+ intros (c c1);
constructor 1;
- [ apply (convergent_relation_pair c c1)
+ [ apply (Oconvergent_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 (c2 c3);
+ apply (Orelation_pair_equality c c1 c2 c3);
+ | 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.
+definition Oconvergent_relation_space_of_Oconvergent_relation_space_setoid:
+ ∀CS1,CS2.carr2 (Oconvergent_relation_space_setoid CS1 CS2) →
+ Oconvergent_relation_pair CS1 CS2 ≝ λP,Q,c.c.
+coercion Oconvergent_relation_space_of_Oconvergent_relation_space_setoid.
-coercion rp''.
-
-definition convergent_relation_space_composition:
- ∀o1,o2,o3: concrete_space.
- binary_morphism1
- (convergent_relation_space_setoid o1 o2)
- (convergent_relation_space_setoid o2 o3)
- (convergent_relation_space_setoid o1 o3).
+definition Oconvergent_relation_space_composition:
+ ∀o1,o2,o3: Oconcrete_space.
+ binary_morphism2
+ (Oconvergent_relation_space_setoid o1 o2)
+ (Oconvergent_relation_space_setoid o2 o3)
+ (Oconvergent_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 (.= († (Orespects_converges : ?)));
+ apply (Orespects_converges ?? c (c1\sub\f⎻ b) (c1\sub\f⎻ c2));
+ | change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (oa_one (Oform o3)))));
+ apply (.= (†(Orespects_all_covered :?)));
+ apply rule (Orespects_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 (Orp ?? a = Orp ?? a');
+ change in e1 with (Orp ?? b = Orp ?? b');
+ apply (e‡e1);]
qed.
-definition CSPA: category1.
+definition OCSPA: category2.
constructor 1;
- [ apply concrete_space
- | apply convergent_relation_space_setoid
+ [ apply Oconcrete_space
+ | apply Oconvergent_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 convergent_relation_space_composition
- | intros; simplify;
+ [ apply id2
+ | intros; apply refl1;
+ | apply refl1]
+ | apply Oconvergent_relation_space_composition
+ | 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 OBP 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 Oconcrete_space_of_OCSPA : objs2 OCSPA → Oconcrete_space ≝ λx.x.
+coercion Oconcrete_space_of_OCSPA.
+
+definition Oconvergent_relation_space_setoid_of_arrows2_OCSPA :
+ ∀P,Q. arrows2 OCSPA P Q → Oconvergent_relation_space_setoid P Q ≝ λP,Q,x.x.
+coercion Oconvergent_relation_space_setoid_of_arrows2_OCSPA.
+