+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔ A ⊢ carr1 (powerclass_setoid A) ≡ Ω^A.
+
+(************ SETS OVER SETOIDS ********************)
+
+include "logic/cprop.ma".
+
+nrecord qpowerclass (A: setoid) : Type[1] ≝
+ { pc:> Ω^A; (* qui pc viene dichiarato con un target preciso...
+ forse lo si vorrebbe dichiarato con un target più lasco
+ ma la sintassi :> non lo supporta *)
+ mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc)
+ }.
+
+ndefinition Full_set: ∀A. qpowerclass A.
+ #A; @[ napply A | #x; #x'; #H; napply refl1]
+nqed.
+ncoercion Full_set: ∀A. qpowerclass A ≝ Full_set on A: setoid to qpowerclass ?.
+
+ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
+ #A; @
+ [ napply (λS,S'. 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 (qpowerclass A)
+ | napply (qseteq A) ]
+nqed.
+
+unification hint 0 ≔ A ⊢
+ carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
+
+nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
+ #A; @
+ [ napply (λx,S. x ∈ S)
+ | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
+ ##[ napply Hb1; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha^-1;##]
+ ##| napply Hb2; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha;##]
+ ##]
+ ##]
+nqed.
+
+unification hint 0 ≔ A:setoid, x, S;
+ SS ≟ (pc ? S)
+ (*-------------------------------------*) ⊢
+ fun21 ??? (mem_ok A) x S ≡ mem A SS x.
+
+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 ] ##]
+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. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
+ #A; @
+ [ #S; #S'; @
+ [ napply (S ∩ S')
+ | #a; #a'; #Ha;
+ nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
+ [##1,2: napply (. Ha^-1‡#); nassumption;
+ ##|##3,4: napply (. Ha‡#); nassumption]##]
+ ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
+ [ alias symbol "invert" = "setoid1 symmetry".
+ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+ | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
+nqed.
+
+(* unfold if intersect, exposing fun21 *)
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔
+ A : setoid, B,C : qpowerclass A ⊢
+ pc A (intersect_ok A B C) ≡ intersect ? (pc ? B) (pc ? C).
+
+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; napply (. (H^-1‡#)); nassumption.
+nqed.
+
+ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
+ λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
+ {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
+
+ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
+ λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
+
+(******************* compatible equivalence relations **********************)
+
+nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
+ { rel:> equivalence_relation A;
+ compatibility: ∀x,x':A. x=x' → rel x x'
+ }.
+
+ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
+ #A; #R; @ A R;
+nqed.
+
+(******************* first omomorphism theorem for sets **********************)