+nlemma decidable_option :
+∀T.(Πx,y:T.decidable (x = y)) →
+ (∀x,y:option T.decidable (x = y)).
+ #T; #H; #x; nelim x;
+ ##[ ##1: #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+ ##| ##2: #yy; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H1;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_none_some T … H1)
+ ##]
+ ##| ##2: #xx; #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_some_none T … H2)
+ ##| ##2: #yy; nnormalize; napply (or2_elim (xx = yy) (xx ≠ yy) ? (H …));
+ ##[ ##2: #H1; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H2;
+ napply (H1 (option_destruct_some_some T … H2))
+ ##| ##1: #H1; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H1; napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+