--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(******************* SETS OVER TYPES *****************)
+
+include "logic/connectives.ma".
+
+nrecord powerclass (A: Type[0]) : Type[1] ≝ { mem: A → CProp[0] }.
+
+interpretation "mem" 'mem a S = (mem ? S a).
+interpretation "powerclass" 'powerset A = (powerclass A).
+interpretation "subset construction" 'subset \eta.x = (mk_powerclass ? x).
+
+ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
+interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
+
+ndefinition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
+interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
+
+ndefinition intersect ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ x ∈ V }.
+interpretation "intersect" 'intersects U V = (intersect ? U V).
+
+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.
+
+nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
+/3/.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/;
+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);//.
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type2".
+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 *)
+ 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
+for @{ 'ext_powerclass $A }.
+
+interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
+
+ndefinition Full_set: ∀A. 𝛀^A.
+ #A; @[ napply A | #x; #x'; #H; napply refl1]
+nqed.
+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/.
+nqed.
+
+ndefinition ext_powerclass_setoid: setoid → setoid1.
+ #A; @ (ext_seteq A).
+nqed.
+
+unification hint 0 ≔ A;
+ R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
+ (* ----------------------------------------------------- *) ⊢
+ carr1 R ≡ ext_powerclass A.
+
+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/.
+nqed.
+
+unification hint 0 ≔ AA : setoid, S : 𝛀^AA, x : carr AA;
+ A ≟ carr AA,
+ 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;
+nqed.
+
+(* hints for ∩ *)
+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]
+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)))
+ (* ------------------------------------------*) ⊢
+ 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/.
+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.
+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))) ,
+ 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.
+
+
+(*
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔
+ A : setoid, B,C : 𝛀^A ;
+ CC ≟ (ext_carr ? C),
+ BB ≟ (ext_carr ? B),
+ C1 ≟ (carr1 (powerclass_setoid (carr A))),
+ C2 ≟ (carr1 (ext_powerclass_setoid A))
+ ⊢
+ eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡
+ eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
+
+unification hint 0 ≔
+ A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.
+
+nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
+ ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
+ #U; #A; #B; #H; #x; #y; #K; #K2;
+ alias symbol "prop2" = "prop21 mem".
+ alias symbol "invert" = "setoid1 symmetry".
+ napply (. K^-1‡H);
+ nassumption;
+nqed.
+
+
+nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
+ #A; @
+ [ #S; #S'; @
+ [ napply (S ∩ S')
+ | #a; #a'; #Ha;
+ nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
+ [##1,2: napply (. Ha^-1‡#); nassumption;
+ ##|##3,4: napply (. Ha‡#); nassumption]##]
+ ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
+ [ alias symbol "invert" = "setoid1 symmetry".
+ alias symbol "refl" = "refl".
+alias symbol "prop2" = "prop21".
+napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+ | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
+nqed.
+
+(* unfold if intersect, exposing fun21 *)
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔
+ A : setoid, B,C : ext_powerclass A ⊢
+ pc A (fun21 …
+ (mk_binary_morphism1 …
+ (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
+ (prop21 … (intersect_ok A)))
+ B
+ C)
+ ≡ intersect ? (pc ? B) (pc ? C).
+
+nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
+ #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
+nqed.
+*)
+
+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}.
+
+ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
+ λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
+
+(******************* compatible equivalence relations **********************)
+
+nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
+ { rel:> equivalence_relation A;
+ compatibility: ∀x,x':A. x=x' → rel x x'
+ }.
+
+ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
+ #A; #R; @ A R;
+nqed.
+
+(******************* first omomorphism theorem for sets **********************)
+
+ndefinition eqrel_of_morphism:
+ ∀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. A ⇒_0 (quotient A R).
+ #A; #R; @
+ [ 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 ] //.
+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.
+
+alias symbol "eq" = "setoid eq".
+ndefinition surjective ≝
+ 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:A ⇒_0 B.
+ ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
+
+nlemma first_omomorphism_theorem_functions2:
+ ∀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: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:> 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
+ }.
+
+
+(*
+nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
+ { iso_f:> unary_morphism A B;
+ f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
+
+
+ncheck (λA:?.
+ λB:?.
+ λS:?.
+ λT:?.
+ λxxx:isomorphism A B S T.
+ match xxx
+ return λxxx:isomorphism A B S T.
+ ∀x: carr A.
+ ∀x_72: mem (carr A) (pc A S) x.
+ mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
+ with [ mk_isomorphism _ yyy ⇒ yyy ] ).
+
+ ;
+ }.
+*)
+
+(* 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.
+
projects / helm.git / blobdiff