include "properties/relations1.ma".
ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
- #A; @
- [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
- | /2/
- | #S; #S'; *; /3/
- | #S; #T; #U; *; #H1; #H2; *; /4/]
+#A; @(λS,S'. S ⊆ S' ∧ S' ⊆ S); /2/; ##[ #A B; *; /3/]
+#S T U; *; #H1 H2; *; /4/;
nqed.
include "sets/setoids1.ma".
include "hints_declaration.ma".
alias symbol "hint_decl" = "hint_decl_Type2".
-unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ Ω^A.
+unification hint 0 ≔ A;
+ R ≟ (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A)))
+(*--------------------------------------------------*)⊢
+ carr1 R ≡ Ω^A.
(************ SETS OVER SETOIDS ********************)
include "logic/cprop.ma".
-nrecord ext_powerclass (A: setoid) : Type[1] ≝
- { ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso...
- forse lo si vorrebbe dichiarato con un target più lasco
- ma la sintassi :> non lo supporta *)
+nrecord ext_powerclass (A: setoid) : Type[1] ≝ {
+ ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso...
+ forse lo si vorrebbe dichiarato con un target più lasco
+ ma la sintassi :> non lo supporta *)
ext_prop: ∀x,x':A. x=x' → (x ∈ ext_carr) = (x' ∈ ext_carr)
- }.
+}.
notation > "𝛀 ^ term 90 A" non associative with precedence 70
for @{ 'ext_powerclass $A }.
(* ----------------------------------------------------- *) ⊢
carr1 R ≡ ext_powerclass A.
-(*
-interpretation "prop21 mem" 'prop2 l r = (prop21 (setoid1_of_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
-*)
-
-(*
-ncoercion ext_carr' : ∀A.∀x:ext_powerclass_setoid A. Ω^A ≝ ext_carr
-on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?).
-*)
-
nlemma mem_ext_powerclass_setoid_is_morph:
- ∀A. unary_morphism1 (setoid1_of_setoid A) (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
+ ∀A. (setoid1_of_setoid A) ⇒_1 (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
#A; napply (mk_binary_morphism1 … (λx:setoid1_of_setoid A.λS: 𝛀^A. x ∈ S));
#a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H
[ napply (. (ext_prop … Ha^-1)) | napply (. (ext_prop … Ha)) ] /2/.
unification hint 0 ≔ AA, x, S;
A ≟ carr AA,
SS ≟ (ext_carr ? S),
- TT ≟ (mk_unary_morphism1 …
+ TT ≟ (mk_unary_morphism1 ??
(λx:setoid1_of_setoid ?.
- mk_unary_morphism1 …
+ mk_unary_morphism1 ??
(λS:ext_powerclass_setoid ?. x ∈ S)
- (prop11 … (mem_ext_powerclass_setoid_is_morph AA x)))
- (prop11 … (mem_ext_powerclass_setoid_is_morph AA))),
+ (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA x)))
+ (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA))),
XX ≟ (ext_powerclass_setoid AA)
(*-------------------------------------*) ⊢
fun11 (setoid1_of_setoid AA)
(unary_morphism1_setoid1 XX CPROP) TT x S
≡ mem A SS x.
-nlemma subseteq_is_morph: ∀A. unary_morphism1 (ext_powerclass_setoid A)
- (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
+nlemma set_ext : ∀S.∀A,B:Ω^S.A =_1 B → ∀x:S.(x ∈ A) =_1 (x ∈ B).
+#S A B; *; #H1 H2 x; @; ##[ napply H1 | napply H2] nqed.
+
+nlemma ext_set : ∀S.∀A,B:Ω^S.(∀x:S. (x ∈ A) = (x ∈ B)) → A = B.
+#S A B H; @; #x; ncases (H x); #H1 H2; ##[ napply H1 | napply H2] nqed.
+
+nlemma subseteq_is_morph: ∀A.
+ (ext_powerclass_setoid A) ⇒_1
+ (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
#A; napply (mk_binary_morphism1 … (λS,S':𝛀^A. S ⊆ S'));
#a; #a'; #b; #b'; *; #H1; #H2; *; /5/ by mk_iff, sym1, subseteq_trans;
nqed.
alias symbol "hint_decl" (instance 1) = "hint_decl_Type2".
-unification hint 0 ≔ A,a,a'
- (*-----------------------------------------------------------------*) ⊢
- eq_rel ? (eq0 A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
+unification hint 0 ≔ A,x,y
+(*-----------------------------------------------*) ⊢
+ eq_rel ? (eq0 A) x y ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) x y.
+
+(* XXX capire come mai questa hint non funziona se porto su (setoid1_of_setoid A) *)
nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
- #A; #S; #S'; @ (S ∩ S');
- #a; #a'; #Ha; @; *; #H1; #H2; @
- [##1,2: napply (. Ha^-1‡#); nassumption;
-##|##3,4: napply (. Ha‡#); nassumption]
+#S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
+##[##1,2: napply (. Exy^-1‡#); nassumption;
+##|##3,4: napply (. Exy‡#); nassumption]
nqed.
alias symbol "hint_decl" = "hint_decl_Type1".
unification hint 0 ≔
A : setoid, B,C : ext_powerclass A;
- R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
-
+ R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
(* ------------------------------------------*) ⊢
ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
nlemma intersect_is_morph:
- ∀A. unary_morphism1 (powerclass_setoid A) (unary_morphism1_setoid1 (powerclass_setoid A) (powerclass_setoid A)).
+ ∀A. (powerclass_setoid A) ⇒_1 (unary_morphism1_setoid1 (powerclass_setoid A) (powerclass_setoid A)).
#A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
#a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
nqed.
alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔
- A : Type[0], B,C : Ω^A;
- R ≟ (mk_unary_morphism1 …
- (λS. mk_unary_morphism1 … (λS'.S ∩ S') (prop11 … (intersect_is_morph A S)))
- (prop11 … (intersect_is_morph A)))
- ⊢
- R B C ≡ intersect ? B C.
+unification hint 0 ≔ A : Type[0], B,C : Ω^A;
+ R ≟ mk_unary_morphism1 ??
+ (λS. mk_unary_morphism1 ?? (λS'.S ∩ S') (prop11 ?? (intersect_is_morph A S)))
+ (prop11 ?? (intersect_is_morph A))
+(*------------------------------------------------------------------------*) ⊢
+ fun11 ?? (fun11 ?? R B) C ≡ intersect A B C.
interpretation "prop21 ext" 'prop2 l r =
(prop11 (ext_powerclass_setoid ?)
(unary_morphism1_setoid1 (ext_powerclass_setoid ?) ?) ? ?? l ?? r).
nlemma intersect_is_ext_morph:
- ∀A. unary_morphism1 (ext_powerclass_setoid A)
+ ∀A.
+ (ext_powerclass_setoid A) ⇒_1
(unary_morphism1_setoid1 (ext_powerclass_setoid A) (ext_powerclass_setoid A)).
#A; napply (mk_binary_morphism1 … (intersect_is_ext …));
#a; #a'; #b; #b'; #Ha; #Hb; napply (prop11 … (intersect_is_morph A)); nassumption.
unification hint 1 ≔
AA : setoid, B,C : 𝛀^AA;
A ≟ carr AA,
- R ≟ (mk_unary_morphism1 …
- (λS:ext_powerclass_setoid AA.
+ R ≟ (mk_unary_morphism1 ??
+ (λS:𝛀^AA.
mk_unary_morphism1 ??
- (λS':ext_powerclass_setoid AA.
+ (λS':𝛀^AA.
mk_ext_powerclass AA (S∩S') (ext_prop AA (intersect_is_ext ? S S')))
- (prop11 … (intersect_is_ext_morph AA S)))
- (prop11 … (intersect_is_ext_morph AA))) ,
+ (prop11 ?? (intersect_is_ext_morph AA S)))
+ (prop11 ?? (intersect_is_ext_morph AA))) ,
BB ≟ (ext_carr ? B),
CC ≟ (ext_carr ? C)
(* ------------------------------------------------------*) ⊢
ext_carr AA (R B C) ≡ intersect A BB CC.
+nlemma union_is_morph :
+ ∀A. (powerclass_setoid A) ⇒_1 (unary_morphism1_setoid1 (powerclass_setoid A) (powerclass_setoid A)).
+(*XXX ∀A.Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A). avec non-unif-coerc*)
+#X; napply (mk_binary_morphism1 … (λA,B.A ∪ B));
+#A1 A2 B1 B2 EA EB; napply ext_set; #x;
+nchange in match (x ∈ (A1 ∪ B1)) with (?∨?);
+napply (.= (set_ext ??? EA x)‡#);
+napply (.= #‡(set_ext ??? EB x)); //;
+nqed.
+
+nlemma union_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
+ #S A B; @ (A ∪ B); #x y Exy; @; *; #H1;
+##[##1,3: @; ##|##*: @2 ]
+##[##1,3: napply (. (Exy^-1)╪_1#)
+##|##2,4: napply (. Exy╪_1#)]
+nassumption;
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔
+ A : setoid, B,C : 𝛀^A;
+ R ≟ (mk_ext_powerclass ? (B ∪ C) (ext_prop ? (union_is_ext ? B C)))
+(*-------------------------------------------------------------------------*) ⊢
+ ext_carr A R ≡ union ? (ext_carr ? B) (ext_carr ? C).
+
+unification hint 0 ≔ S:Type[0], A,B:Ω^S;
+ MM ≟ mk_unary_morphism1 ??
+ (λA.mk_unary_morphism1 ?? (λB.A ∪ B) (prop11 ?? (union_is_morph S A)))
+ (prop11 ?? (union_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+ fun11 ?? (fun11 ?? MM A) B ≡ A ∪ B.
+
+nlemma union_is_ext_morph:∀A.
+ (ext_powerclass_setoid A) ⇒_1
+ (unary_morphism1_setoid1 (ext_powerclass_setoid A) (ext_powerclass_setoid A)).
+(*XXX ∀A:setoid.𝛀^A ⇒_1 (𝛀^A ⇒_1 𝛀^A). with coercion non uniformi *)
+#A; napply (mk_binary_morphism1 … (union_is_ext …));
+#x1 x2 y1 y2 Ex Ey; napply (prop11 … (union_is_morph A)); nassumption.
+nqed.
+
+unification hint 1 ≔
+ AA : setoid, B,C : 𝛀^AA;
+ A ≟ carr AA,
+ R ≟ (mk_unary_morphism1 ??
+ (λS:𝛀^AA.
+ mk_unary_morphism1 ??
+ (λS':𝛀^AA.
+ mk_ext_powerclass AA (S ∪ S') (ext_prop AA (union_is_ext ? S S')))
+ (prop11 ?? (union_is_ext_morph AA S)))
+ (prop11 ?? (union_is_ext_morph AA))) ,
+ BB ≟ (ext_carr ? B),
+ CC ≟ (ext_carr ? C)
+(*------------------------------------------------------*) ⊢
+ ext_carr AA (R B C) ≡ union A BB CC.
+
(*
alias symbol "hint_decl" = "hint_decl_Type2".
unification hint 0 ≔
(******************* first omomorphism theorem for sets **********************)
ndefinition eqrel_of_morphism:
- ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
+ ∀A,B. A ⇒_0 B → compatible_equivalence_relation A.
#A; #B; #f; @
[ @ [ napply (λx,y. f x = f y) ] /2/;
##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
napply (.= (†H)); // ]
nqed.
-ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
+ndefinition canonical_proj: ∀A,R. A ⇒_0 (quotient A R).
#A; #R; @
[ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
nqed.
ndefinition quotiented_mor:
- ∀A,B.∀f:unary_morphism A B.
- unary_morphism (quotient … (eqrel_of_morphism … f)) B.
+ ∀A,B.∀f:A ⇒_0 B.(quotient … (eqrel_of_morphism … f)) ⇒_0 B.
#A; #B; #f; @ [ napply f ] //.
nqed.
alias symbol "eq" = "setoid eq".
ndefinition surjective ≝
- λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
+ λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:A ⇒_0 B.
∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
ndefinition injective ≝
- λA,B.λS: ext_powerclass A.λf:unary_morphism A B.
+ λA,B.λS: ext_powerclass A.λf:A ⇒_0 B.
∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
nlemma first_omomorphism_theorem_functions2:
- ∀A,B.∀f: unary_morphism A B.
+ ∀A,B.∀f:A ⇒_0 B.
surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
/3/. nqed.
nlemma first_omomorphism_theorem_functions3:
- ∀A,B.∀f: unary_morphism A B.
+ ∀A,B.∀f:A ⇒_0 B.
injective … (Full_set ?) (quotiented_mor … f).
#A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
nqed.
nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
- { iso_f:> unary_morphism A B;
+ { iso_f:> A ⇒_0 B;
f_closed: ∀x. x ∈ S → iso_f x ∈ T;
f_sur: surjective … S T iso_f;
f_inj: injective … S iso_f
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