}.
ndefinition proofs: CProp → setoid.
-#P; napply (mk_setoid …);
+#P; napply mk_setoid;
##[ napply P;
##| #x; #y; napply True;
##|##*: nwhd; nrepeat (#_); napply I;
notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
ndefinition function_space_setoid: setoid → setoid → setoid.
- #A; #B; napply (mk_setoid …);
+ #A; #B; napply mk_setoid;
##[ napply (function_space A B);
##| #f; #f1; napply (∀a:A. proofs (eq … (f a) (f1 a)));
##| nwhd; #x; #a;
- napply (f_ok … x …); (* QUI!! *)
+ napply (f_ok … x); (* QUI!! *)
(* unfold carr; unfold proofs; simplify;
apply (refl A)
| simplify;
ntheorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
#A; #S1; #S2; #S3; #H12; #H23; #x; #H;
- napply (H23 …); napply (H12 …); nassumption;
+ napply H23; napply H12; nassumption;
nqed.
ndefinition overlaps ≝ λA.λU,V:Ω \sup A.∃x:A.x ∈ U ∧ x ∈ V.