- #A; napply mk_binary_morphism1
- [ napply (λS,S': qpowerclass_setoid ?. S ⊆ S')
- | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; napply mk_iff; #H
- [ napply (subseteq_trans … a' a) (* anche qui, perche' serve a'? *)
- [ nassumption | napply (subseteq_trans … a b); nassumption ]
- ##| napply (subseteq_trans … a a') (* anche qui, perche' serve a'? *)
- [ nassumption | napply (subseteq_trans … a' b'); nassumption ] ##]
+ #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 ] ##]
+nqed.
+
+unification hint 0 ≔ A,a,a'
+ (*-----------------------------------------------------------------*) ⊢
+ eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
+
+nlemma intersect_ok: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
+ #A; #S; #S'; @ (S ∩ S');
+ #a; #a'; #Ha; @; *; #H1; #H2; @
+ [##1,2: napply (. Ha^-1‡#); nassumption;
+##|##3,4: napply (. Ha‡#); nassumption]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 1 ≔
+ A : setoid, B,C : qpowerclass A ⊢
+ pc A (mk_qpowerclass ? (B ∩ C) (mem_ok' ? (intersect_ok ? B C)))
+ ≡ intersect ? (pc ? B) (pc ? C).
+
+nlemma intersect_ok': ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
+ #A; @ (λS,S'. S ∩ S');
+ #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *; #Ka; #Kb; @
+ [ napply Ha1; nassumption
+ | napply Hb1; nassumption
+ | napply Ha2; nassumption
+ | napply Hb2; nassumption]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔
+ A : Type[0], B,C : powerclass A ⊢
+ fun21 …
+ (mk_binary_morphism1 …
+ (λS,S'.S ∩ S')
+ (prop21 … (intersect_ok' A))) B C
+ ≡ intersect ? B C.
+
+ndefinition prop21_mem :
+ ∀A,C.∀f:binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) C.
+ ∀a,a':setoid1_of_setoid A.
+ ∀b,b':qpowerclass_setoid A.a = a' → b = b' → f a b = f a' b'.
+#A; #C; #f; #a; #a'; #b; #b'; #H1; #H2; napply prop21; nassumption;
+nqed.
+
+interpretation "prop21 mem" 'prop2 l r = (prop21_mem ??????? l r).
+
+nlemma intersect_ok'':
+ ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
+ #A; @ (intersect_ok A); nlapply (prop21 … (intersect_ok' A)); #H;
+ #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;