Definition Map (K V : Type) : Type :=
  list (K * V).Definition well_formed {K V} (m : Map K V) : Prop :=
  NoDup (map fst m).(* no repeat keys *)

Class DecEq (X : Type) := {
  dec_eq : forall x x' : X, { x = x' } + { x <> x' }
  }.(* no repeat keys *)

Class DecEq (X : Type) := {
  dec_eq : forall x x' : X, { x = x' } + { x <> x' }
  }.

Lemma In_insert_invert {K V} `{DecEq K} {p : K * V} {k : K} {v : V} {m : Map K V} : In p (insert k v m) -> p = (k, v) \/ In p m.