include "o-basic_pairs.ma".
include "o-saturations.ma".
-notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
-notation > "□_term 90 b" non associative with precedence 90 for @{'box $b}.
-interpretation "Universal image ⊩⎻*" 'box x = (fun12 _ _ (or_f_minus_star _ _) (rel x)).
-
-notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
-notation > "◊_term 90 b" non associative with precedence 90 for @{'diamond $b}.
-interpretation "Existential image ⊩" 'diamond x = (fun12 _ _ (or_f _ _) (rel x)).
-
-notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
-notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
-interpretation "Universal pre-image ⊩*" 'rest x = (fun12 _ _ (or_f_star _ _) (rel x)).
-
-notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
-notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
-interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 _ _ (or_f_minus _ _) (rel x)).
-
definition A : ∀b:BP. unary_morphism1 (form b) (form 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 xxx : ∀x.carr x → carr1 (setoid1_of_setoid x). intros; assumption; qed.
-coercion xxx nocomposites.
-*)
-
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.
{ bp:> BP;
(*distr : is_distributive (form bp);*)
downarrow: unary_morphism1 (form bp) (form bp);
- downarrow_is_sat: is_saturation ? downarrow;
+ 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);
}.
interpretation "o-concrete space downarrow" 'downarrow x =
- (fun_1 __ (downarrow _) x).
-
-definition bp': concrete_space → basic_pair ≝ λc.bp c.
-coercion bp'.
-
-lemma setoid_OF_OA : OA → setoid.
-intros; apply (oa_P o);
-qed.
-
-coercion setoid_OF_OA.
+ (fun11 __ (downarrow _) x).
definition binary_downarrow :
∀C:concrete_space.binary_morphism1 (form C) (form C) (form C).
intros; constructor 1;
[ intros; apply (↓ c ∧ ↓ c1);
-| intros; apply ((†H)‡(†H1));]
+| intros;
+ alias symbol "prop2" = "prop21".
+ alias symbol "prop1" = "prop11".
+ apply ((†e)‡(†e1));]
qed.
-interpretation "concrete_space binary ↓" 'fintersects a b = (fun1 _ _ _ (binary_downarrow _) a b).
+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. eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (rp\sub\f⎻ b ↓ rp\sub\f⎻ c));
respects_all_covered:
(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 (c1 ∘ c);
| intros;
change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (b↓c2))));
- unfold uncurry_arrows;
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)))));
- unfold uncurry_arrows;
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));]
+ 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
+ [ 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 rule ASSOC;
| intros; simplify;
- change with (a ∘ id1 ? o1 = a);
- apply (id_neutral_right1 : ?);
+ change with (a ∘ id2 BP o1 = a);
+ apply (id_neutral_right2 : ?);
| intros; simplify;
- change with (id1 ? o2 ∘ a = a);
- apply (id_neutral_left1 : ?);]
+ 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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