Fixpoint assoc (l : list (K * V)) (k : K) : option V :=
    match l with
      | [] => None
      | (k', v) :: l' =>
        if K_eq_dec k k' then
          Some v
        else
          assoc l' k
    end.Definition assoc_default (l : list (K * V)) (k : K) (default : V) : V :=
    match (assoc l k) with
      | Some x => x
      | None => default
    end.Fixpoint assoc_set (l : list (K * V)) (k : K) (v : V) : list (K * V) :=
    match l with
      | [] => [(k, v)]
      | (k', v') :: l' =>
        if K_eq_dec k k' then
          (k, v) :: l'
        else
          (k', v') :: (assoc_set l' k v)
    end.Fixpoint assoc_del (l : list (K * V)) (k : K) : list (K * V) :=
    match l with
      | [] => []
      | (k', v') :: l' =>
        if K_eq_dec k k' then
          assoc_del l' k
        else
          (k', v') :: (assoc_del l' k)
    end.

Lemma assoc_assoc_default_missing: forall (l : list (K * V)) k d, assoc l k = None -> assoc_default l k d = d.