Section MapFoldDef.
Variable M : Type.
Variable neutral : M.
Variable op : M -> M -> M.
Fixpoint MapFold1 (f:ad -> A -> M) (pf:ad -> ad)
(m:Map A) {struct m} : M :=
match m with
| M0 => neutral
| M1 a y => f (pf a) y
| M2 m1 m2 =>
op (MapFold1 f (fun a0:ad => pf (N.double a0)) m1)
(MapFold1 f (fun a0:ad => pf (Ndouble_plus_one a0)) m2)
end.
Definition MapFold (f:ad -> A -> M) (m:Map A) :=
MapFold1 f (fun a:ad => a) m.
Variable State : Type.
Variable f : State -> ad -> A -> State * M.
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 (N.double 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).
End MapFoldDef.
Definition alist := list (ad * A).
Definition anil := nil (A:=(ad * A)).
Definition acons := cons (A:=(ad * A)).
Definition aapp := app (A:=(ad * A)).
Definition alist_of_Map :=
MapFold alist anil aapp (fun (a:ad) (y:A) => acons (a, y) anil).
Fixpoint alist_semantics (l:alist) : ad -> option A :=
match l with
| nil => fun _:ad => None
| (a, y) :: l' =>
fun a0:ad => if N.eqb a a0 then Some y else alist_semantics l' a0
end.

Lemma alist_of_Map_semantics_1_1 :
forall (m:Map A) (pf:ad -> ad) (a:ad) (y:A),
alist_semantics
(MapFold1 alist anil aapp (fun (a0:ad) (y:A) => acons (a0, y) anil) pf
m) a = Some y -> {a' : ad | a = pf a'}.