ndefinition union ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∨ x ∈ V }.
interpretation "union" 'union U V = (union ? U V).
-ndefinition substract ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ ¬ x ∈ V }.
-interpretation "substract" 'minus U V = (substract ? U V).
-
-
ndefinition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
-//.nqed.
+ #A; #S; #x; #H; nassumption.
+nqed.
nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
-/3/.nqed.
+ #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
+nqed.
include "properties/relations1.ma".
ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
-#A; @(λS,S'. S ⊆ S' ∧ S' ⊆ S); /2/; ##[ #A B; *; /3/]
-#S T U; *; #H1 H2; *; /4/;
+ #A; @
+ [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
+ | #S; @; napply subseteq_refl
+ | #S; #S'; *; #H1; #H2; @; nassumption
+ | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; @; napply subseteq_trans;
+ ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
nqed.
include "sets/setoids1.ma".
-ndefinition singleton ≝ λA:setoid.λa:A.{ x | a = x }.
-interpretation "singl" 'singl a = (singleton ? a).
-
(* this has to be declared here, so that it is combined with carr *)
ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
ndefinition powerclass_setoid: Type[0] → setoid1.
- #A; @(Ω^A);//.
+ #A; @[ napply (Ω^A)| napply seteq ]
nqed.
+include "hints_declaration.ma".
+
alias symbol "hint_decl" = "hint_decl_Type2".
-unification hint 0 ≔ A;
- R ≟ (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A)))
-(*--------------------------------------------------*)⊢
- carr1 R ≡ Ω^A.
+unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ 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 }.
-notation < "Ω term 90 A \atop ≈" non associative with precedence 90
+notation "Ω term 90 A \atop ≈" non associative with precedence 70
for @{ 'ext_powerclass $A }.
interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
- #A; @ [ napply (λS,S'. S = S') ] /2/.
+ #A; @
+ [ napply (λS,S'. S = S')
+ | #S; napply (refl1 ? (seteq A))
+ | #S; #S'; napply (sym1 ? (seteq A))
+ | #S; #T; #U; napply (trans1 ? (seteq A))]
nqed.
ndefinition ext_powerclass_setoid: setoid → setoid1.
- #A; @ (ext_seteq A).
+ #A; @
+ [ napply (ext_powerclass A)
+ | napply (ext_seteq A) ]
nqed.
unification hint 0 ≔ 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. (setoid1_of_setoid A) ⇒_1 ((𝛀^A) ⇒_1 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/.
+ ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
+ #A; @
+ [ napply (λx,S. x ∈ S)
+ | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
+ ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
+ ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
+ ##]
+ ##]
nqed.
-unification hint 0 ≔ AA : setoid, S : 𝛀^AA, x : carr AA;
- A ≟ carr AA,
+unification hint 0 ≔ A:setoid, x, S;
SS ≟ (ext_carr ? S),
- TT ≟ (mk_unary_morphism1 ??
- (λx:carr1 (setoid1_of_setoid ?).
- mk_unary_morphism1 ??
- (λS:carr1 (ext_powerclass_setoid ?). x ∈ (ext_carr ? S))
- (prop11 ?? (fun11 ?? (mem_ext_powerclass_setoid_is_morph AA) x)))
- (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA))),
- T2 ≟ (ext_powerclass_setoid AA)
-(*---------------------------------------------------------------------------*) ⊢
- fun11 T2 CPROP (fun11 (setoid1_of_setoid AA) (unary_morphism1_setoid1 T2 CPROP) TT x) S ≡ mem A SS x.
-
-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. 𝛀^A ⇒_1 𝛀^A ⇒_1 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;
+ TT ≟ (mk_binary_morphism1 ???
+ (λx:setoid1_of_setoid ?.λS:ext_powerclass_setoid ?. x ∈ S)
+ (prop21 ??? (mem_ext_powerclass_setoid_is_morph A))),
+ XX ≟ (ext_powerclass_setoid A)
+ (*-------------------------------------*) ⊢
+ fun21 (setoid1_of_setoid A) XX CPROP TT x S
+ ≡ mem A SS x.
+
+nlemma subseteq_is_morph: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) CPROP.
+ #A; @
+ [ napply (λS,S'. S ⊆ S')
+ | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
+ [ napply (subseteq_trans … a)
+ [ nassumption | napply (subseteq_trans … b); nassumption ]
+ ##| napply (subseteq_trans … a')
+ [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
nqed.
-(* hints for ∩ *)
+unification hint 0 ≔ A,a,a'
+ (*-----------------------------------------------------------------*) ⊢
+ eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
+
nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
-#S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
-##[##1,2: napply (. Exy^-1╪_1#); nassumption;
-##|##3,4: napply (. Exy‡#); nassumption]
+ #A; #S; #S'; @ (S ∩ S');
+ #a; #a'; #Ha; @; *; #H1; #H2; @
+ [##1,2: napply (. Ha^-1‡#); nassumption;
+##|##3,4: napply (. Ha‡#); nassumption]
nqed.
alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ A : setoid, B,C : 𝛀^A;
- AA ≟ carr A,
- BB ≟ ext_carr ? B,
- CC ≟ ext_carr ? C,
- R ≟ (mk_ext_powerclass ?
- (ext_carr ? B ∩ ext_carr ? C)
- (ext_prop ? (intersect_is_ext ? B C)))
+unification hint 0 ≔
+ A : setoid, B,C : ext_powerclass A;
+ R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
+
(* ------------------------------------------*) ⊢
- ext_carr A R ≡ intersect AA BB CC.
-
-nlemma intersect_is_morph: ∀A. Ω^A ⇒_1 Ω^A ⇒_1 Ω^A.
-#A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
-#a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
+ ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
+
+nlemma intersect_is_morph:
+ ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
+ #A; @ (λS,S'. S ∩ S');
+ #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *; #Ka; #Kb; @
+ [ napply Ha1; nassumption
+ | napply Hb1; nassumption
+ | napply Ha2; nassumption
+ | napply Hb2; nassumption]
nqed.
alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ A : Type[0], B,C : Ω^A;
- T ≟ powerclass_setoid A,
- R ≟ mk_unary_morphism1 ??
- (λX. mk_unary_morphism1 ??
- (λY.X ∩ Y) (prop11 ?? (fun11 ?? (intersect_is_morph A) X)))
- (prop11 ?? (intersect_is_morph A))
-(*------------------------------------------------------------------------*) ⊢
- fun11 T T (fun11 T (unary_morphism1_setoid1 T T) 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. 𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
- #A; napply (mk_binary_morphism1 … (intersect_is_ext …));
- #a; #a'; #b; #b'; #Ha; #Hb; napply (prop11 … (intersect_is_morph A)); nassumption.
+unification hint 0 ≔
+ A : Type[0], B,C : Ω^A;
+ R ≟ (mk_binary_morphism1 …
+ (λS,S'.S ∩ S')
+ (prop21 … (intersect_is_morph A)))
+ ⊢
+ fun21 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A) R B C
+ ≡ intersect ? B C.
+
+interpretation "prop21 ext" 'prop2 l r = (prop21 (ext_powerclass_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
+
+nlemma intersect_is_ext_morph:
+ ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
+ #A; @ (intersect_is_ext …); nlapply (prop21 … (intersect_is_morph A));
+#H; #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;
nqed.
unification hint 1 ≔
- AA : setoid, B,C : 𝛀^AA;
- A ≟ carr AA,
- T ≟ ext_powerclass_setoid AA,
- R ≟ (mk_unary_morphism1 ?? (λX:𝛀^AA.
- mk_unary_morphism1 ?? (λY:𝛀^AA.
- mk_ext_powerclass AA
- (ext_carr ? X ∩ ext_carr ? Y)
- (ext_prop AA (intersect_is_ext ? X Y)))
- (prop11 ?? (fun11 ?? (intersect_is_ext_morph AA) X)))
- (prop11 ?? (intersect_is_ext_morph AA))) ,
+ A:setoid, B,C : 𝛀^A;
+ R ≟ (mk_binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A)
+ (λS,S':carr1 (ext_powerclass_setoid A).
+ mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S')))
+ (prop21 … (intersect_is_ext_morph A))) ,
BB ≟ (ext_carr ? B),
CC ≟ (ext_carr ? C)
- (* ---------------------------------------------------------------------------------------*) ⊢
- ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ intersect A BB CC.
-
-
-(* hints for ∪ *)
-nlemma union_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
-#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;
- AA ≟ carr A,
- BB ≟ ext_carr ? B,
- CC ≟ ext_carr ? C,
- R ≟ mk_ext_powerclass ?
- (ext_carr ? B ∪ ext_carr ? C) (ext_prop ? (union_is_ext ? B C))
-(*-------------------------------------------------------------------------*) ⊢
- ext_carr A R ≡ union AA BB CC.
-
-unification hint 0 ≔ S:Type[0], A,B:Ω^S;
- T ≟ powerclass_setoid S,
- MM ≟ mk_unary_morphism1 ??
- (λA.mk_unary_morphism1 ??
- (λB.A ∪ B) (prop11 ?? (fun11 ?? (union_is_morph S) A)))
- (prop11 ?? (union_is_morph S))
-(*--------------------------------------------------------------------------*) ⊢
- fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A ∪ B.
-
-nlemma union_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
-#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,
- T ≟ ext_powerclass_setoid AA,
- R ≟ mk_unary_morphism1 ?? (λX:𝛀^AA.
- mk_unary_morphism1 ?? (λY:𝛀^AA.
- mk_ext_powerclass AA
- (ext_carr ? X ∪ ext_carr ? Y) (ext_prop AA (union_is_ext ? X Y)))
- (prop11 ?? (fun11 ?? (union_is_ext_morph AA) X)))
- (prop11 ?? (union_is_ext_morph AA)),
- BB ≟ (ext_carr ? B),
- CC ≟ (ext_carr ? C)
-(*------------------------------------------------------*) ⊢
- ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ union A BB CC.
-
-
-(* hints for - *)
-nlemma substract_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
-#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)‡#); @; *; #H H1; @; //; #H2; napply H1;
-##[ napply (. (set_ext ??? EB x)); ##| napply (. (set_ext ??? EB^-1 x)); ##] //;
-nqed.
-
-nlemma substract_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
- #S A B; @ (A - B); #x y Exy; @; *; #H1 H2; @; ##[##2,4: #H3; napply H2]
-##[##1,4: napply (. Exy╪_1#); // ##|##2,3: napply (. Exy^-1╪_1#); //]
-nqed.
-
-alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ A : setoid, B,C : 𝛀^A;
- AA ≟ carr A,
- BB ≟ ext_carr ? B,
- CC ≟ ext_carr ? C,
- R ≟ mk_ext_powerclass ?
- (ext_carr ? B - ext_carr ? C)
- (ext_prop ? (substract_is_ext ? B C))
-(*---------------------------------------------------*) ⊢
- ext_carr A R ≡ substract AA BB CC.
-
-unification hint 0 ≔ S:Type[0], A,B:Ω^S;
- T ≟ powerclass_setoid S,
- MM ≟ mk_unary_morphism1 ??
- (λA.mk_unary_morphism1 ??
- (λB.A - B) (prop11 ?? (fun11 ?? (substract_is_morph S) A)))
- (prop11 ?? (substract_is_morph S))
-(*--------------------------------------------------------------------------*) ⊢
- fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A - B.
-
-nlemma substract_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
-#A; napply (mk_binary_morphism1 … (substract_is_ext …));
-#x1 x2 y1 y2 Ex Ey; napply (prop11 … (substract_is_morph A)); nassumption.
-nqed.
-
-unification hint 1 ≔
- AA : setoid, B,C : 𝛀^AA;
- A ≟ carr AA,
- T ≟ ext_powerclass_setoid AA,
- R ≟ mk_unary_morphism1 ?? (λX:𝛀^AA.
- mk_unary_morphism1 ?? (λY:𝛀^AA.
- mk_ext_powerclass AA
- (ext_carr ? X - ext_carr ? Y)
- (ext_prop AA (substract_is_ext ? X Y)))
- (prop11 ?? (fun11 ?? (substract_is_ext_morph AA) X)))
- (prop11 ?? (substract_is_ext_morph AA)),
- BB ≟ (ext_carr ? B),
- CC ≟ (ext_carr ? C)
-(*------------------------------------------------------*) ⊢
- ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ substract A BB CC.
-
-(* hints for {x} *)
-nlemma single_is_morph : ∀A:setoid. (setoid1_of_setoid A) ⇒_1 Ω^A.
-#X; @; ##[ napply (λx.{(x)}); ##]
-#a b E; napply ext_set; #x; @; #H; /3/; nqed.
-
-nlemma single_is_ext: ∀A:setoid. A → 𝛀^A.
-#X a; @; ##[ napply ({(a)}); ##] #x y E; @; #H; /3/; nqed.
-
-alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ A : setoid, a : carr A;
- R ≟ (mk_ext_powerclass ? {(a)} (ext_prop ? (single_is_ext ? a)))
-(*-------------------------------------------------------------------------*) ⊢
- ext_carr A R ≡ singleton A a.
-
-unification hint 0 ≔ A:setoid, a : carr A;
- T ≟ setoid1_of_setoid A,
- AA ≟ carr A,
- MM ≟ mk_unary_morphism1 ??
- (λa:carr1 (setoid1_of_setoid A).{(a)}) (prop11 ?? (single_is_morph A))
-(*--------------------------------------------------------------------------*) ⊢
- fun11 T (powerclass_setoid AA) MM a ≡ {(a)}.
-
-nlemma single_is_ext_morph:∀A:setoid.(setoid1_of_setoid A) ⇒_1 𝛀^A.
-#A; @; ##[ #a; napply (single_is_ext ? a); ##] #a b E; @; #x; /3/; nqed.
-
-unification hint 1 ≔ AA : setoid, a: carr AA;
- T ≟ ext_powerclass_setoid AA,
- R ≟ mk_unary_morphism1 ??
- (λa:carr1 (setoid1_of_setoid AA).
- mk_ext_powerclass AA {(a)} (ext_prop ? (single_is_ext AA a)))
- (prop11 ?? (single_is_ext_morph AA))
-(*------------------------------------------------------*) ⊢
- ext_carr AA (fun11 (setoid1_of_setoid AA) T R a) ≡ singleton AA a.
-
+ (* ------------------------------------------------------*) ⊢
+ ext_carr A
+ (fun21
+ (ext_powerclass_setoid A)
+ (ext_powerclass_setoid A)
+ (ext_powerclass_setoid A) R B C) ≡
+ intersect (carr A) BB CC.
(*
alias symbol "hint_decl" = "hint_decl_Type2".
ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
- {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq0 B) (f x) y}.
+ {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
(******************* first omomorphism theorem for sets **********************)
ndefinition eqrel_of_morphism:
- ∀A,B. A ⇒_0 B → compatible_equivalence_relation A.
+ ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
#A; #B; #f; @
- [ @ [ napply (λx,y. f x = f y) ] /2/;
+ [ @
+ [ napply (λx,y. f x = f y)
+ | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
-napply (.= (†H)); // ]
+napply (.= (†H)); napply refl ]
nqed.
-ndefinition canonical_proj: ∀A,R. A ⇒_0 (quotient A R).
+ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
#A; #R; @
- [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
+ [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
nqed.
ndefinition quotiented_mor:
- ∀A,B.∀f:A ⇒_0 B.(quotient … (eqrel_of_morphism … f)) ⇒_0 B.
- #A; #B; #f; @ [ napply f ] //.
+ ∀A,B.∀f:unary_morphism A B.
+ unary_morphism (quotient … (eqrel_of_morphism … f)) B.
+ #A; #B; #f; @
+ [ napply f | #a; #a'; #H; nassumption]
nqed.
nlemma first_omomorphism_theorem_functions1:
∀A,B.∀f: unary_morphism A B.
∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
-//. nqed.
+ #A; #B; #f; #x; napply refl;
+nqed.
alias symbol "eq" = "setoid eq".
ndefinition surjective ≝
- λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:A ⇒_0 B.
+ λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
ndefinition injective ≝
- λA,B.λS: ext_powerclass A.λf:A ⇒_0 B.
+ λA,B.λS: ext_powerclass A.λf:unary_morphism A B.
∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
nlemma first_omomorphism_theorem_functions2:
- ∀A,B.∀f:A ⇒_0 B.
+ ∀A,B.∀f: unary_morphism A B.
surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
-/3/. nqed.
+ #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl;
+ (* bug, prova @ I refl *)
+nqed.
nlemma first_omomorphism_theorem_functions3:
- ∀A,B.∀f:A ⇒_0 B.
+ ∀A,B.∀f: unary_morphism A 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:> A ⇒_0 B;
+ { iso_f:> unary_morphism A B;
f_closed: ∀x. x ∈ S → iso_f x ∈ T;
f_sur: surjective … S T iso_f;
f_inj: injective … S iso_f
}.
+nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
+#A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
+nqed.
+
+nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
+#A; #U; #V; #W; #H; #H1; #x; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
+nqed.
+
+nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
+#A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
+nqed.
(*
nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
;
}.
*)
-
-(* Set theory *)
-
-nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
-#A; #U; #V; #W; *; #H; #x; *; /2/.
-nqed.
-
-nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
-#A; #U; #V; #W; #H; #H1; #x; *; /2/.
-nqed.
-
-nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
-/3/. nqed.
-
-nlemma cupC : ∀S. ∀a,b:Ω^S.a ∪ b = b ∪ a.
-#S a b; @; #w; *; nnormalize; /2/; nqed.
-
-nlemma cupID : ∀S. ∀a:Ω^S.a ∪ a = a.
-#S a; @; #w; ##[*; //] /2/; nqed.
-
-(* XXX Bug notazione \cup, niente parentesi *)
-nlemma cupA : ∀S.∀a,b,c:Ω^S.a ∪ b ∪ c = a ∪ (b ∪ c).
-#S a b c; @; #w; *; /3/; *; /3/; nqed.
-
-ndefinition Empty_set : ∀A.Ω^A ≝ λA.{ x | False }.
-
-notation "∅" non associative with precedence 90 for @{ 'empty }.
-interpretation "empty set" 'empty = (Empty_set ?).
-
-nlemma cup0 :∀S.∀A:Ω^S.A ∪ ∅ = A.
-#S p; @; #w; ##[*; //| #; @1; //] *; nqed.
-
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