-nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
- #A; @
- [ napply (λS,S'. S ⊆ S')
- | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
- [ napply (subseteq_trans … a)
- [ nassumption | napply (subseteq_trans … b); nassumption ]
- ##| napply (subseteq_trans … a')
- [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
+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‡#); 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)))
+ (* ------------------------------------------*) ⊢
+ ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
+
+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;
+ R ≟ mk_unary_morphism1 ??
+ (λS. mk_unary_morphism1 ?? (λS'.S ∩ S') (prop11 ?? (intersect_is_morph A S)))
+ (prop11 ?? (intersect_is_morph A))
+(*------------------------------------------------------------------------*) ⊢
+ fun11 ?? (fun11 ?? 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,
+ 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)))
+ (prop11 ?? (intersect_is_ext_morph AA))) ,
+ BB ≟ (ext_carr ? B),
+ CC ≟ (ext_carr ? C)
+ (* ------------------------------------------------------*) ⊢
+ ext_carr AA (R B C) ≡ intersect A BB CC.
+
+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)); //;