Local Notation key := D.t.Definition eqk {elt} : relation (key*elt) := D.eq @@1.Definition eqke {elt} : relation (key*elt) := D.eq * Logic.eq.(** eqk, eqke are equalities *)

#[global]
 Instance eqk_equiv {elt} : Equivalence (@eqk elt) := _.#[global]
 Instance eqke_equiv {elt} : Equivalence (@eqke elt) := _.(** eqke is stricter than eqk *)

#[global]
 Instance eqke_eqk {elt} : subrelation (@eqke elt) (@eqk elt).

Lemma eqke_def {elt} k k' (e e':elt) : eqke (k,e) (k',e') = (D.eq k k' /\ e = e').