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 }.
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^?).
#A; @(Ω^A);//.
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 ≡ ext_powerclass A.
nlemma mem_ext_powerclass_setoid_is_morph:
- ∀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/.
+ ∀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, x, S;
+unification hint 0 ≔ AA : setoid, S : 𝛀^AA, x : carr AA;
A ≟ carr AA,
SS ≟ (ext_carr ? S),
TT ≟ (mk_unary_morphism1 ??
- (λx:setoid1_of_setoid ?.
+ (λx:carr1 (setoid1_of_setoid ?).
mk_unary_morphism1 ??
- (λS:ext_powerclass_setoid ?. x ∈ S)
- (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA x)))
+ (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))),
- XX ≟ (ext_powerclass_setoid AA)
- (*-------------------------------------*) ⊢
- fun11 (setoid1_of_setoid AA)
- (unary_morphism1_setoid1 XX CPROP) TT x S
- ≡ mem A SS x.
+ 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.
- (ext_powerclass_setoid A) ⇒_1
- (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
+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.
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type2".
-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) *)
-
+(* 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^-1â\80¡#); nassumption;
+##[##1,2: napply (. Exy^-1â\95ª_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)))
+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 ? (ext_carr ? B) (ext_carr ? C).
-
-nlemma intersect_is_morph:
- ∀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/.
+ 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 ??
- (λS. mk_unary_morphism1 ?? (λS'.S ∩ S') (prop11 ?? (intersect_is_morph A S)))
+ (λX. mk_unary_morphism1 ??
+ (λY.X ∩ Y) (prop11 ?? (fun11 ?? (intersect_is_morph A) X)))
(prop11 ?? (intersect_is_morph A))
(*------------------------------------------------------------------------*) ⊢
- fun11 ?? (fun11 ?? R B) C ≡ intersect A B C.
+ 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.
- (ext_powerclass_setoid A) ⇒_1
- (unary_morphism1_setoid1 (ext_powerclass_setoid A) (ext_powerclass_setoid A)).
+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,
- R ≟ (mk_unary_morphism1 ??
- (λS:𝛀^AA.
- mk_unary_morphism1 ??
- (λS':𝛀^AA.
- mk_ext_powerclass AA (S∩S') (ext_prop AA (intersect_is_ext ? S S')))
- (prop11 ?? (intersect_is_ext_morph AA S)))
+ 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 (R B C) ≡ intersect A BB CC.
+ (* ---------------------------------------------------------------------------------------*) ⊢
+ ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) 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*)
+
+(* 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 (?∨?);
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)))
+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 ? (ext_carr ? B) (ext_carr ? 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 ?? (union_is_morph S A)))
+ (λA.mk_unary_morphism1 ??
+ (λB.A ∪ B) (prop11 ?? (fun11 ?? (union_is_morph S) A)))
(prop11 ?? (union_is_morph S))
(*--------------------------------------------------------------------------*) ⊢
- fun11 ?? (fun11 ?? MM A) B ≡ A ∪ B.
+ fun11 T T (fun11 T (unary_morphism1_setoid1 T T) 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 *)
+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,
- 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))) ,
+ 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 (R B C) ≡ union A BB CC.
+ 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".
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; *; /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.
(*
nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
;
}.
-*)
\ No newline at end of file
+*)
+
+(* 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