include "o-basic_pairs.ma".
include "o-saturations.ma".
-definition A : ∀b:BP. unary_morphism1 (form b) (form b).
+definition A : ∀b:OBP. unary_morphism1 (Oform b) (Oform b).
intros; constructor 1;
[ apply (λx.□_b (Ext⎽b x));
- | do 2 unfold FunClass_1_OF_Type_OF_setoid21; intros; apply (†(†e));]
+ | 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 : Type2 ≝
- { bp:> BP;
+record Oconcrete_space : Type2 ≝
+ { Obp:> OBP;
(*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
+ 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
}.
interpretation "o-concrete space downarrow" 'downarrow x =
- (fun11 __ (downarrow _) x).
+ (fun11 __ (Odownarrow _) x).
-definition bp': concrete_space → basic_pair ≝ λc.bp c.
-coercion bp'.
-
-definition bp'': concrete_space → objs2 BP.
- intro; apply (bp' c);
-qed.
-coercion bp''.
-
-definition binary_downarrow :
- ∀C:concrete_space.binary_morphism1 (form C) (form C) (form C).
+definition Obinary_downarrow :
+ ∀C:Oconcrete_space.binary_morphism1 (Oform C) (Oform C) (Oform C).
intros; constructor 1;
-[ intros; apply (↓ t ∧ ↓ t1);
+[ 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).
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun21 _ _ _ (Obinary_downarrow _) a b).
-record convergent_relation_pair (CS1,CS2: concrete_space) : Type2 ≝
- { rp:> arrows2 ? CS1 CS2;
- respects_converges:
- ∀b,c. eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (rp\sub\f⎻ b ↓ rp\sub\f⎻ c));
- respects_all_covered:
- eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (oa_one (form CS2))))
- (Ext⎽CS1 (oa_one (form CS1)))
+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 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 → setoid2.
- 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 (c2 c3);
+ apply (Orelation_pair_equality c c1 c2 c3);
| intros 1; apply refl2;
| intros 2; apply sym2;
| intros 3; apply trans2]]
qed.
+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.
-definition rp'': ∀CS1,CS2.carr2 (convergent_relation_space_setoid CS1 CS2) → arrows2 BP CS1 CS2 ≝
- λCS1,CS2,c.rp ?? c.
-coercion rp''.
-
-
-definition rp''': ∀CS1,CS2.Type_OF_setoid21 (convergent_relation_space_setoid CS1 CS2) → arrows2 BP CS1 CS2 ≝
- λCS1,CS2,c.rp ?? c.
-coercion rp'''.
-
-definition rp'''': ∀CS1,CS2.Type_OF_setoid21 (convergent_relation_space_setoid CS1 CS2) → carr2 (arrows2 BP CS1 CS2) ≝
- λCS1,CS2,c.rp ?? c.
-coercion rp''''.
-
-definition convergent_relation_space_composition:
- ∀o1,o2,o3: concrete_space.
+definition Oconvergent_relation_space_composition:
+ ∀o1,o2,o3: Oconcrete_space.
binary_morphism2
- (convergent_relation_space_setoid o1 o2)
- (convergent_relation_space_setoid o2 o3)
- (convergent_relation_space_setoid o1 o3).
+ (Oconvergent_relation_space_setoid o1 o2)
+ (Oconvergent_relation_space_setoid o2 o3)
+ (Oconvergent_relation_space_setoid o1 o3).
intros; constructor 1;
[ intros; whd in t t1 ⊢ %;
constructor 1;
- [ apply (t1 ∘ t);
+ [ apply (c1 ∘ c);
| intros;
- change in ⊢ (? ? ? % ?) with (t\sub\c⎻ (t1\sub\c⎻ (Ext⎽o3 (b↓c))));
- unfold FunClass_1_OF_Type_OF_setoid21;
+ change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (b↓c2))));
alias symbol "trans" = "trans1".
- apply (.= († (respects_converges : ?)));
- apply (respects_converges ?? t (t1\sub\f⎻ b) (t1\sub\f⎻ c));
- | change in ⊢ (? ? ? % ?) with (t\sub\c⎻ (t1\sub\c⎻ (Ext⎽o3 (oa_one (form o3)))));
- unfold FunClass_1_OF_Type_OF_setoid21;
- apply (.= (†(respects_all_covered :?)));
- apply rule (respects_all_covered ?? t);]
+ 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 e with (rp'' ?? a = rp'' ?? a');
- change in e1 with (rp'' ?? b = rp ?? b');
+ change in e with (Orp ?? a = Orp ?? a');
+ change in e1 with (Orp ?? b = Orp ?? b');
apply (e‡e1);]
qed.
-definition CSPA: category2.
+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 id2
| intros; apply refl1;
| apply refl1]
- | apply convergent_relation_space_composition
+ | apply Oconvergent_relation_space_composition
| intros; simplify; whd in a12 a23 a34;
change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
apply rule ASSOC;
| intros; simplify;
- change with (a ∘ id2 BP o1 = a);
+ change with (a ∘ id2 OBP o1 = a);
apply (id_neutral_right2 : ?);
| intros; simplify;
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.
+
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