Definition MapSweep2 (a0:ad) (y:A) :=
    if f a0 y then Some (a0, y) else None.Fixpoint MapSweep1 (pf:ad -> ad) (m:Map A) {struct m} : 
   option (ad * A) :=
    match m with
    | M0 => None
    | M1 a y => MapSweep2 (pf a) y
    | M2 m m' =>
        match MapSweep1 (fun a:ad => pf (Ndouble a)) m with
        | Some r => Some r
        | None => MapSweep1 (fun a:ad => pf (Ndouble_plus_one a)) m'
        end
    end.Definition MapSweep (m:Map A) := MapSweep1 (fun a:ad => a) m.

(* Specifications: *)
Lemma MapSweep_semantics_1_1 : forall (m:Map A) (pf:ad -> ad) (a:ad) (y:A), MapSweep1 pf m = Some (a, y) -> f a y = true.
Lemma MapSweep_semantics_1 : forall (m:Map A) (a:ad) (y:A), MapSweep m = Some (a, y) -> f a y = true.
Lemma MapSweep_semantics_2_1 : forall (m:Map A) (pf:ad -> ad) (a:ad) (y:A), MapSweep1 pf m = Some (a, y) -> {a' : ad | a = pf a'}.
Lemma MapSweep_semantics_2_2 : forall (m:Map A) (pf fp:ad -> ad), (forall a0:ad, fp (pf a0) = a0) -> forall (a:ad) (y:A), MapSweep1 pf m = Some (a, y) -> MapGet A m (fp a) = Some y.
Lemma MapSweep_semantics_2 : forall (m:Map A) (a:ad) (y:A), MapSweep m = Some (a, y) -> MapGet A m a = Some y.
Lemma MapSweep_semantics_3_1 : forall (m:Map A) (pf:ad -> ad), MapSweep1 pf m = None -> forall (a:ad) (y:A), MapGet A m a = Some y -> f (pf a) y = false.
Lemma MapSweep_semantics_3 : forall m:Map A, MapSweep m = None -> forall (a:ad) (y:A), MapGet A m a = Some y -> f a y = false.
Lemma MapSweep_semantics_4_1 : forall (m:Map A) (pf:ad -> ad) (a:ad) (y:A), MapGet A m a = Some y -> f (pf a) y = true -> {a' : ad & {y' : A | MapSweep1 pf m = Some (a', y')}}.
Lemma MapSweep_semantics_4 : forall (m:Map A) (a:ad) (y:A), MapGet A m a = Some y -> f a y = true -> {a' : ad & {y' : A | MapSweep m = Some (a', y')}}.