include "relations.ma".
include "o-algebra.ma".
-definition SUBSETS: objs1 SET → OAlgebra.
+definition POW': objs1 SET → OAlgebra.
intro A; constructor 1;
[ apply (Ω \sup A);
| apply subseteq;
| intros; split; intro;
[ cases f; cases x1; exists [apply w1] exists [apply w] assumption;
| cases e; cases x; exists; [apply w1] [ assumption | exists; [apply w] assumption]]
- | intros; intros 2; cases (f (singleton ? a) ?);
+ | intros; intros 2; cases (f {(a)} ?);
[ exists; [apply a] [assumption | change with (a = a); apply refl1;]
| change in x1 with (a = w); change with (mem A a q); apply (. (x1‡#));
assumption]]
qed.
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (SUBSETS o1) (SUBSETS o2).
+definition powerset_of_POW': ∀A.oa_P (POW' A) → Ω \sup A ≝ λA,x.x.
+coercion powerset_of_POW'.
+
+definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (POW' o1) (POW' o2).
intros;
constructor 1;
[ constructor 1;
qed.
lemma orelation_of_relation_preserves_identity:
- ∀o1:REL. eq2 ? (orelation_of_relation ?? (id1 ? o1)) (id2 OA (SUBSETS o1)).
+ ∀o1:REL. eq2 ? (orelation_of_relation ?? (id1 ? o1)) (id2 OA (POW' o1)).
intros; split; intro; split; whd; intro;
[ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
apply (f a1); change with (a1 = a1); apply refl1;
lemma orelation_of_relation_preserves_composition:
∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
orelation_of_relation ?? (G ∘ F) =
- comp2 OA (SUBSETS o1) (SUBSETS o2) (SUBSETS o3)
+ comp2 OA (POW' o1) (POW' o2) (POW' o3)
?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
[ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
| cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
qed.
-definition SUBSETS': carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
+definition POW: carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
constructor 1;
- [ apply SUBSETS;
+ [ apply POW';
| intros; constructor 1;
[ apply (orelation_of_relation S T);
| intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
| apply orelation_of_relation_preserves_identity;
- | simplify; intros;
- apply (.= (orelation_of_relation_preserves_composition o1 o2 o4 f1 (f3∘f2)));
- apply (#‡(orelation_of_relation_preserves_composition o2 o3 o4 f2 f3)); ]
+ | apply orelation_of_relation_preserves_composition; ]
qed.
-theorem SUBSETS_faithful:
+theorem POW_faithful:
∀S,T.∀f,g:arrows2 (category2_of_category1 REL) S T.
- map_arrows2 ?? SUBSETS' ?? f = map_arrows2 ?? SUBSETS' ?? g → f=g.
- intros; unfold SUBSETS' in e; simplify in e; cases e;
+ map_arrows2 ?? POW ?? f = map_arrows2 ?? POW ?? g → f=g.
+ intros; unfold POW in e; simplify in e; cases e;
unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
- intros 2; lapply (e3 (singleton ? x)); cases Hletin;
+ intros 2; cases (e3 {(x)});
split; intro; [ lapply (s y); | lapply (s1 y); ]
[2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
- |*: cases Hletin1; cases x1; change in f3 with (eq1 ? x w); apply (. f3‡#); assumption;]
+ |*: cases Hletin; cases x1; change in f3 with (x =_1 w); apply (. f3‡#); assumption;]
+qed.
+
+
+lemma currify: ∀A,B,C. binary_morphism1 A B C → A → unary_morphism1 B C.
+intros; constructor 1; [ apply (b c); | intros; apply (#‡e);]
+qed.
+
+(*
+alias symbol "singl" = "singleton".
+alias symbol "eq" = "setoid eq".
+lemma in_singleton_to_eq : ∀A:setoid.∀y,x:A.y ∈ {(x)} → (eq1 A) y x.
+intros; apply sym1; apply f;
+qed.
+
+lemma eq_to_in_singleton : ∀A:setoid.∀y,x:A.eq1 A y x → y ∈ {(x)}.
+intros; apply (e^-1);
+qed.
+*)
+
+interpretation "lifting singl" 'singl x =
+ (fun11 ? (objs2 (POW ?)) (singleton ?) x).
+
+theorem POW_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? POW S T g = f).
+ intros; exists;
+ [ constructor 1; constructor 1;
+ [ apply (λx:carr S.λy:carr T. y ∈ f {(x)});
+ | intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
+ [4: apply mem; |6: apply Hletin;|1,2,3,5: skip]
+ lapply (#‡prop11 ?? f ?? (†e)); [6: apply Hletin; |*:skip ]]
+ | whd; split; whd; intro; simplify; unfold map_arrows2; simplify;
+ [ split;
+ [ change with (∀a1.(∀x. a1 ∈ f (singleton S x) → x ∈ a) → a1 ∈ f⎻* a);
+ | change with (∀a1.a1 ∈ f⎻* a → (∀x.a1 ∈ f (singleton S x) → x ∈ a)); ]
+ | split;
+ [ change with (∀a1.(∃y:carr T. y ∈ f (singleton S a1) ∧ y ∈ a) → a1 ∈ f⎻ a);
+ | change with (∀a1.a1 ∈ f⎻ a → (∃y:carr T.y ∈ f (singleton S a1) ∧ y ∈ a)); ]
+ | split;
+ [ change with (∀a1.(∃x:carr S. a1 ∈ f (singleton S x) ∧ x ∈ a) → a1 ∈ f a);
+ | change with (∀a1.a1 ∈ f a → (∃x:carr S. a1 ∈ f (singleton S x) ∧ x ∈ a)); ]
+ | split;
+ [ change with (∀a1.(∀y. y ∈ f (singleton S a1) → y ∈ a) → a1 ∈ f* a);
+ | change with (∀a1.a1 ∈ f* a → (∀y. y ∈ f (singleton S a1) → y ∈ a)); ]]
+ [ intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)^-1) ? a1);
+ [ intros 2; apply (f1 a2); change in f2 with (a2 ∈ f⎻ (singleton ? a1));
+ lapply (. (or_prop3 ?? f (singleton ? a2) (singleton ? a1)));
+ [ cases Hletin; change in x1 with (eq1 ? a1 w);
+ apply (. x1‡#); assumption;
+ | exists; [apply a2] [change with (a2=a2); apply rule #; | assumption]]
+ | change with (a1 = a1); apply rule #; ]
+ | intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)) ? x);
+ [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f⎻* a); apply (. f3^-1‡#);
+ assumption;
+ | lapply (. (or_prop3 ?? f (singleton ? x) (singleton ? a1))^-1);
+ [ cases Hletin; change in x1 with (eq1 ? x w);
+ change with (x ∈ f⎻ (singleton ? a1)); apply (. x1‡#); assumption;
+ | exists; [apply a1] [assumption | change with (a1=a1); apply rule #; ]]]
+ | intros; cases e; cases x; clear e x;
+ lapply (. (or_prop3 ?? f (singleton ? a1) a)^-1);
+ [ cases Hletin; change in x with (eq1 ? a1 w1); apply (. x‡#); assumption;
+ | exists; [apply w] assumption ]
+ | intros; lapply (. (or_prop3 ?? f (singleton ? a1) a));
+ [ cases Hletin; exists; [apply w] split; assumption;
+ | exists; [apply a1] [change with (a1=a1); apply rule #; | assumption ]]
+ | intros; cases e; cases x; clear e x;
+ apply (f_image_monotone ?? f (singleton ? w) a ? a1);
+ [ intros 2; change in f3 with (eq1 ? w a2); change with (a2 ∈ a);
+ apply (. f3^-1‡#); assumption;
+ | assumption; ]
+ | intros; lapply (. (or_prop3 ?? f a (singleton ? a1))^-1);
+ [ cases Hletin; exists; [apply w] split;
+ [ lapply (. (or_prop3 ?? f (singleton ? w) (singleton ? a1)));
+ [ cases Hletin1; change in x3 with (eq1 ? a1 w1); apply (. x3‡#); assumption;
+ | exists; [apply w] [change with (w=w); apply rule #; | assumption ]]
+ | assumption ]
+ | exists; [apply a1] [ assumption; | change with (a1=a1); apply rule #;]]
+ | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)^-1) ? a1);
+ [ apply f1; | change with (a1=a1); apply rule #; ]
+ | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)) ? y);
+ [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f* a);
+ apply (. f3^-1‡#); assumption;
+ | assumption ]]]
qed.
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