Section ForNotations.
Notation In:=(InA eq).
Notation Inf:=(lelistA lt).
Notation Sort:=(sort lt).
Notation NoDup:=(NoDupA eq).
(* ltk ignore the second components *)
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl : core.
Hint Resolve ltk_trans ltk_not_eqk ltk_not_eqke : core.
Hint Immediate eqk_sym eqke_sym : core.
Global Instance eqk_equiv : Equivalence eqk.
Global Instance ltk_compat' : Proper (eqke==>eqke==>iff) ltk.
(* Additional facts *)
Hint Resolve eqk_not_ltk : core.
Hint Immediate ltk_eqk eqk_ltk : core.

Lemma InA_eqke_eqk :
forall x m, InA eqke x m -> InA eqk x m.