Fixpoint MapFold1_state (state:State) (pf:ad -> ad) 
     (m:Map A) {struct m} : State * M :=
      match m with
      | M0 => (state, neutral)
      | M1 a y => f state (pf a) y
      | M2 m1 m2 =>
          match MapFold1_state state (fun a0:ad => pf (Ndouble a0)) m1 with
          | (state1, x1) =>
              match
                MapFold1_state state1
                  (fun a0:ad => pf (Ndouble_plus_one a0)) m2
              with
              | (state2, x2) => (state2, op x1 x2)
              end
          end
      end.Definition MapFold_state (state:State) :=
      MapFold1_state state (fun a:ad => a).

(* Specifications: *)
Lemma pair_sp : forall (B C:Type) (x:B * C), x = (fst x, snd x).
Lemma MapFold_state_stateless_1 : forall (m:Map A) (g:ad -> A -> M) (pf:ad -> ad), (forall (state:State) (a:ad) (y:A), snd (f state a y) = g a y) -> forall state:State, snd (MapFold1_state state pf m) = MapFold1 g pf m.
Lemma MapFold_state_stateless : forall g:ad -> A -> M, (forall (state:State) (a:ad) (y:A), snd (f state a y) = g a y) -> forall (state:State) (m:Map A), snd (MapFold_state state m) = MapFold g m.
Lemma MapCollect_as_Fold : forall (f:ad -> A -> Map B) (m:Map A), MapCollect f m = MapFold (Map B) (M0 B) (MapMerge B) f m.