[ 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.
(* 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).
+ 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'));
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 ?)
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.
-(* hints for â\88© *)
+(* hints for â\88ª *)
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;
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.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
#A; napply (mk_binary_morphism1 … (union_is_ext …));
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 (R B C) ≡ union A BB CC.
+ ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ union A BB CC.
(* hints for - *)
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 ? (substract_is_ext ? B C)))
-(*-------------------------------------------------------------------------*) ⊢
- ext_carr A R ≡ substract ? (ext_carr ? B) (ext_carr ? 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 ? (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 ?? (substract_is_morph S A)))
+ (λA.mk_unary_morphism1 ??
+ (λB.A - B) (prop11 ?? (fun11 ?? (substract_is_morph S) A)))
(prop11 ?? (substract_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 substract_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
#A; napply (mk_binary_morphism1 … (substract_is_ext …));
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 (substract_is_ext ? S S')))
- (prop11 ?? (substract_is_ext_morph AA S)))
- (prop11 ?? (substract_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 (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) ≡ substract 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 a; @; ##[ napply ({(a)}); ##] #x y E; @; #H; /3/; nqed.
alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ A : setoid, a:A;
+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:A;
+unification hint 0 ≔ A:setoid, a : carr A;
+ T ≟ setoid1_of_setoid A,
+ AA ≟ carr A,
MM ≟ mk_unary_morphism1 ??
- (λa:setoid1_of_setoid A.{(a)}) (prop11 ?? (single_is_morph A))
+ (λa:carr1 (setoid1_of_setoid A).{(a)}) (prop11 ?? (single_is_morph A))
(*--------------------------------------------------------------------------*) ⊢
- fun11 ?? MM a ≡ {(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: AA;
+unification hint 1 ≔ AA : setoid, a: carr AA;
+ T ≟ ext_powerclass_setoid AA,
R ≟ mk_unary_morphism1 ??
- (λa:setoid1_of_setoid AA.
+ (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 (R a) ≡ singleton AA a.
-
-
-
-
+ ext_carr AA (fun11 (setoid1_of_setoid AA) T R a) ≡ singleton AA a.
(*