Definition fun_leq' {A B} (f1: Z -> option A) (f2: Z -> option B): Prop :=
  forall p, f1 p -> f2 p.Definition fun_leq {A B} (o1: option (Z -> option A)) (o2: option (Z -> option B)): Prop :=
  match o1, o2 with
  | Some f1, Some f2 => fun_leq' f1 f2
  | None, None => True
  | _, _ => False
  end.Definition option_eq {A B} (a:option A) (b: option B): Prop :=
  match a, b with
  | Some _ , Some _ => True
  | None, None => True
  | _, _ => False
  end.Definition bounded_nat_func_aux {A} (f: nat -> option A) hi: Prop :=
  (forall p, (p >= hi )%nat -> f p = None).Definition bounded_nat_func' {A} (f: nat -> option A) hi: Prop :=
  (forall p, (p > hi )%nat -> f p = None).Definition bounded_func' {A} (f: Z -> option A) hi lo: Prop :=
  (forall p, (p > hi )%Z -> f p = None) /\
  (forall p, (p < lo)%Z -> f p = None).Definition bounded_func_op {A} (f: option (Z -> option A)) hi lo: Prop :=
  match f with
  | Some f' => bounded_func' f' hi lo
  | None => True
  end.Definition bounded_func {A} (f: Z -> option A): Prop :=
  exists hi lo,
  bounded_func' f hi lo.Definition bounded_map {A} (m: PTree.t (Z -> option A)):=
  forall p f, m ! p = Some f -> bounded_func f.Fixpoint strong_tree_leq {A B}
         (t1: PTree.t A) (t2: PTree.t B)
         (leq: option A -> option B -> Prop):=
  match t1, t2 with
  | PTree.Leaf, PTree.Leaf => True
  | PTree.Node l1 o1 r1, PTree.Node l2 o2 r2 =>
    leq o1 o2 /\
    strong_tree_leq l1 l2 leq /\
    strong_tree_leq r1 r2 leq
  | _, _ => False
  end.Definition same_shape {A B} (m1: PTree.t (Z -> option A))(m2: PTree.t (Z -> option B)):=
  strong_tree_leq m1 m2 option_eq.Definition sub_map' {A B} (m1: PTree.t (Z -> option A))(m2: PTree.t (Z -> option B)):=
  forall p f1, m1 ! p = Some f1 ->
          exists f2, m2 ! p = Some f2 /\ fun_leq' f1 f2.Definition sub_map {A B} (m1: PTree.t (Z -> option A))(m2: PTree.t (Z -> option B)):=
  strong_tree_leq m1 m2 fun_leq.

Lemma sub_map_and_shape: forall { A B} m1 m2, @same_shape A B m1 m2 -> sub_map' m1 m2 -> sub_map m1 m2.