ndefinition union ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∨ x ∈ V }.
interpretation "union" 'union U V = (union ? U V).
+ndefinition big_union ≝ λA,B.λT:Ω \sup A.λf:A → Ω \sup B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
+
+ndefinition big_intersection ≝ λA,B.λT:Ω \sup A.λf:A → Ω \sup B.{ x | ∀i. i ∈ T → x ∈ f i }.
+
+ndefinition full_set: ∀A. Ω \sup A ≝ λA.{ x | True }.
+(* bug dichiarazione coercion qui *)
+(* ncoercion full_set : ∀A:Type[0]. Ω \sup A ≝ full_set on _A: Type[0] to (Ω \sup ?). *)
+
nlemma subseteq_refl: ∀A.∀S: Ω \sup A. S ⊆ S.
#A; #S; #x; #H; nassumption.
nqed.
#A; napply mk_setoid1
[ napply (Ω \sup A)
| napply seteq ]
-nqed.
+nqed.
+
+unification hint 0 (∀A. (λx,y.True) (carr1 (powerclass_setoid A)) (Ω \sup A)).
(************ SETS OVER SETOIDS ********************)
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')
#A; #B; #f; #x; napply refl;
nqed.
-ndefinition surjective ≝ λA,B.λf:unary_morphism A B. ∀y.∃x. f x = y.
+ndefinition surjective ≝
+ λA,B.λS: qpowerclass A.λT: qpowerclass B.λf:unary_morphism A B.
+ ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
-ndefinition injective ≝ λA,B.λf:unary_morphism A B. ∀x,x'. f x = f x' → x = x'.
+ndefinition injective ≝
+ λA,B.λS: qpowerclass A.λf:unary_morphism A B.
+ ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
nlemma first_omomorphism_theorem_functions2:
- ∀A,B.∀f: unary_morphism A B. surjective ?? (canonical_proj ? (eqrel_of_morphism ?? f)).
- #A; #B; #f; nwhd; #y; napply (ex_intro … y); napply refl.
+ ∀A,B.∀f: unary_morphism A B. surjective ?? (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism ?? f)).
+ #A; #B; #f; nwhd; #y; #Hy; napply (ex_intro … y); napply conj
+ [ napply I | napply refl]
nqed.
nlemma first_omomorphism_theorem_functions3:
- ∀A,B.∀f: unary_morphism A B. injective ?? (quotiented_mor ?? f).
- #A; #B; #f; nwhd; #x; #x'; #H; nassumption.
-nqed.
\ No newline at end of file
+ ∀A,B.∀f: unary_morphism A B. injective ?? (Full_set ?) (quotiented_mor ?? f).
+ #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
+nqed.
+
+nrecord isomorphism (A) (B) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
+ { iso_f:> unary_morphism A B;
+ f_sur: surjective ?? S T iso_f;
+ f_inj: injective ?? S iso_f
+ }.