Section Relations.
Context {R1 R2 : Type}.
Variable RR : R1 -> R2 -> Prop.
Definition map_default_eq d {E}
: (stateT map (itree E) R1) -> (stateT map (itree E) R2) -> Prop :=
fun t1 t2 => forall s1 s2, (@eq_map _ _ _ _ d) s1 s2 -> eutt (prod_rel (@eq_map _ _ _ _ d) RR) (t1 s1) (t2 s2).
End Relations.

Lemma eq_map_remove:
forall (d : V) (s1 s2 : map) (k : K), (@eq_map _ _ _ _ d) s1 s2 -> (@eq_map _ _ _ _ d) (remove k s1) (remove k s2).