Fixpoint lookup {K V} `{DecEq K} (k : K)
  (m : Map K V) : option V :=
  match m with
  | [] => None
  | (k',v) :: m' =>
    match dec_eq k k' with
    | left _ => Some v
    | right _ => lookup k m'
    end
  end.

(* Specifications: *)
Lemma lookup_insert {K V} `{DecEq K} (k : K) (v : V) (m : Map K V) : well_formed m -> lookup k (insert k v m) = Some v.
Lemma lookup_insert {K V} `{DecEq K} (k : K) (v : V) (m : Map K V) : well_formed m -> lookup k (insert k v m) = Some v.