- (* ---------------------------------------------------------------------------------------*) ⊢
- 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.
-