+(************ SETS OVER SETOIDS ********************)
+
+include "logic/cprop.ma".
+
+nrecord qpowerclass (A: setoid) : Type[1] ≝
+ { pc:> Ω \sup A;
+ mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc)
+ }.
+
+ndefinition Full_set: ∀A. qpowerclass A.
+ #A; napply mk_qpowerclass
+ [ napply (full_set A)
+ | #x; #x'; #H; nnormalize in ⊢ (?%?%%); (* bug universi qui napply refl1;*)
+ napply mk_iff; #K; nassumption ]
+nqed.
+
+ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
+ #A; napply mk_equivalence_relation1
+ [ napply (λS,S':qpowerclass A. eq_rel1 ? (eq1 (powerclass_setoid A)) S S')
+ | #S; napply (refl1 ? (seteq A))
+ | #S; #S'; napply (sym1 ? (seteq A))
+ | #S; #T; #U; napply (trans1 ? (seteq A))]
+nqed.
+
+ndefinition qpowerclass_setoid: setoid → setoid1.
+ #A; napply mk_setoid1
+ [ napply (qpowerclass A)
+ | napply (qseteq A) ]
+nqed.
+
+unification hint 0 (∀A. (λx,y.True) (carr1 (qpowerclass_setoid A)) (qpowerclass A)).
+ncoercion qpowerclass_hint: ∀A: setoid. ∀S: qpowerclass_setoid A. Ω \sup A ≝ λA.λS.S
+ on _S: (carr1 (qpowerclass_setoid ?)) to (Ω \sup ?).
+
+nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
+ #A; napply mk_binary_morphism1
+ [ napply (λx.λS: qpowerclass_setoid A. x ∈ S) (* CSC: ??? *)
+ | #a; #a'; #b; #b'; #Ha; #Hb; (* CSC: qui *; non funziona *)
+ nwhd; nwhd in ⊢ (? (? % ??) (? % ??)); napply mk_iff; #H
+ [ ncases Hb; #Hb1; #_; napply Hb1; napply (. (mem_ok' …))
+ [ nassumption | napply Ha^-1 | ##skip ]
+ ##| ncases Hb; #_; #Hb2; napply Hb2; napply (. (mem_ok' …))
+ [ nassumption | napply Ha | ##skip ]##]
+nqed.
+
+unification hint 0 (∀A,x,S. (λx,y.True) (fun21 ??? (mem_ok A) x S) (mem A S x)).
+
+nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
+ #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 ] ##]
+nqed.
+
+nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
+ #A; napply mk_binary_morphism1
+ [ #S; #S'; napply mk_qpowerclass
+ [ napply (S ∩ S')
+ | #a; #a'; #Ha; nwhd in ⊢ (? ? ? % %); napply mk_iff; *; #H1; #H2; napply conj
+ [##1,2: napply (. (mem_ok' …)^-1) [##3,6: nassumption |##1,4: nassumption |##*: ##skip]
+ ##|##3,4: napply (. (mem_ok' …)) [##2,5: nassumption |##1,4: nassumption |##*: ##skip]##]##]
+ ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; napply conj; #x; nwhd in ⊢ (% → %); #H
+ [ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+ | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
+nqed.
+
+unification hint 0 (∀A.∀U,V.(λx,y.True) (fun21 ??? (intersect_ok A) U V) (intersect A U V)).
+
+nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
+ #A; #U; #V; #x; #x'; #H; #p;
+ (* CSC: senza la change non funziona! *)
+ nchange with (x' ∈ (fun21 ??? (intersect_ok A) U V));
+ napply (. (H^-1‡#)); nassumption.
+nqed.