Record dict (K V : Type) := Dict { dict_elems : list (K * V) }.Record dict (K V : Type) := Dict { dict_elems : list (K * V) }.Record dict (K V : Type) := Dict { dict_elems : list (K * V) }.Fixpoint lookup_falist {K V : Type} `{Ord K} (v : V) (k : K) (kvs : list (K * V)) :=
  match kvs with
  | [] => v
  | (k' , v') :: kvs' =>
    match ord_dec k k' with
    | Lt => v
    | Eq => v'
    | Gt => lookup_falist v k kvs'
    end
  end.Fixpoint update_falist {K V : Type} `{Ord K} (k : K) (v : V) (kvs : list (K * V)) : list (K * V) :=
  match kvs with
  | [] => [ (k , v) ]
  | (k' , v') :: kvs' =>
    match ord_dec k k' with
    | Lt => (k , v) :: (k' , v') :: kvs'
    | Eq => (k , v) :: kvs'
    | Gt => (k' , v') :: update_falist k v kvs'
    end
  end.Definition lookup_dict {K V : Type} `{Ord K} (v : V) (k : K) (kvs : dict K V) := 
  lookup_falist v k (dict_elems kvs).Definition update_dict {K V : Type} `{Ord K} (k : K) (v : V) (kvs : dict K V) : dict K V :=
  Dict (update_falist k v (dict_elems kvs)).

(* Specifications: *)
Lemma lookup_update_diffid {K V} `{Ord K} : forall (k k' : K) (v v_def : V) m, k <> k' -> lookup_dict v_def k (update_dict k' v m) = lookup_dict v_def k m.
Lemma lookup_update_sameid {K V} `{Ord K} : forall (k : K) (v v_def : V) m, lookup_dict v_def k (update_dict k v m) = v.
Lemma lookup_update_diffid {K V} `{Ord K} : forall (k k' : K) (v v_def : V) m, k <> k' -> lookup_dict v_def k (update_dict k' v m) = lookup_dict v_def k m.
Lemma lookup_update_sameid {K V} `{Ord K} : forall (k : K) (v v_def : V) m, lookup_dict v_def k (update_dict k v m) = v.