+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)); ##] //;