Section Elt.
Variable elt : Type.
Notation key:=t.
Definition eqk (p p':key*elt) := eq (fst p) (fst p').
Definition eqke (p p':key*elt) :=
eq (fst p) (fst p') /\ (snd p) = (snd p').
Hint Unfold eqk eqke : core.
Hint Extern 2 (eqke ?a ?b) => split : core.
(* eqke is stricter than eqk *)
(* eqk, eqke are equalities *)
Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl : core.
Hint Immediate eqk_sym eqke_sym : core.
Global Instance eqk_equiv : Equivalence eqk.
Hint Resolve InA_eqke_eqk : core.
Definition MapsTo (k:key)(e:elt):= InA eqke (k,e).
Definition In k m := exists e:elt, MapsTo k e m.
Hint Unfold MapsTo In : core.
(* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *)

Lemma In_inv_3 : forall x x' l,
InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.