unfold left_total.
intuition.
edestruct lowDistApprox_repeat_left_total; eauto.
Qed.
Inductive datRepeat (A : Set)(f : DistApproxTree A -> Prop)(P : A -> bool) : nat -> DistApproxTree A -> DistApproxTree A -> Prop :=
| datRepeat_leaf_None : forall n,
datRepeat f P n (dat_leaf None) (dat_leaf None)
| datRepeat_leaf_Some_yes : forall a n,
P a = true ->
datRepeat f P n (dat_leaf (Some a)) (dat_leaf (Some a))
| datRepeat_leaf_Some_no : forall a,
P a = false ->
datRepeat f P O (dat_leaf (Some a)) (dat_leaf None)
| datRepeat_leaf_Some_repeat : forall a t t' n,
f t ->
P a = false ->
datRepeat f P n t t' ->
datRepeat f P (S n) (dat_leaf (Some a)) t'
| datRepeat_internal : forall n t1 t2 t1' t2',
datRepeat f P n t1 t1' ->
datRepeat f P n t2 t2' ->
datRepeat f P n (dat_internal t1 t2) (dat_internal t1' t2').
Definition dat_correct_repeat2(A : Set)(c : Comp A)(P : A -> bool)(n : nat)(t : DistApproxTree A) :=
exists t1 : DistApproxTree A,
dat_correct c n t1 /\
datRepeat (dat_correct c n) P n t1 t. (* was (pred n) *)

Lemma datRepeat_left_total : forall n (A : Set)(eqd : eq_dec A)(t : DistApproxTree A)(f : DistApproxTree A -> Prop) P,
(exists b : DistApproxTree A, f b) ->
exists t', datRepeat f P n t t'.