induction depth; intuition.
inversion H; clear H; subst.
inversion H5; clear H5; subst.
inversion H6; clear H6; subst.
simpl in *.
apply evalDet_steps_bind_done_inv in H9.
destruct H9.
destruct H.
intuition.
inversion H3; clear H3; subst.
inversion H; clear H; subst.
case_eq (P x); intuition;
rewrite H in H6.
inversion H6; clear H6; subst.
simpl in *.
inversion H11; clear H11; subst.
assert (a1 = a0 /\ s'0 = s').
eapply evalDet_steps_done_func; eauto.
intuition.
subst.
econstructor.
rewrite H.
trivial.
assert (x0 = nil).
eapply H2.
rewrite firstn_eq_all_gen; eauto.
subst.
inversion H6; clear H6; subst.
simpl in *.
apply evalDet_steps_bind_done_inv in H11.
destruct H11.
destruct H3.
intuition.
assert (x1 = nil).
eapply evalDet_nil.
eauto.
subst.
assert (evalDet_steps (cs_more c (nil ++ ls)) (cs_done x0 (nil ++ ls))).
eapply evalDet_steps_app_eq.
trivial.
simpl in *.
assert (ls = nil).
eapply H2.
rewrite firstn_eq_all_gen; eauto.
subst.
assert (a = x).
eapply evalDet_steps_nil_eq; eauto.
eapply filter_In; eauto.
subst.
assert (P x = true).
eapply filter_In; eauto.
congruence.
exfalso.
eapply H.
econstructor.
eauto.
inversion H3; clear H3; subst.
inversion H; clear H; subst.
case_eq (P a0); intuition.
exfalso.
eapply H5.
econstructor.
econstructor.
eauto.
simpl.
eapply evalDet_steps_trans.
eapply evalDet_steps_bind_done.
eauto.
rewrite H.
econstructor.
eauto.
simpl.
econstructor.
eapply datRepeat_depth_0.
econstructor.
trivial.
trivial.
econstructor.
intuition.
inversion H3; clear H3; subst.
assert (s'0 = nil).
eapply evalDet_nil; eauto.
subst.
assert (evalDet_steps (cs_more c (nil ++ ls)) (cs_done a1 (nil ++ ls))).
eapply evalDet_steps_app_eq.
trivial.
simpl in *.
assert (ls = nil).
eapply H2.
rewrite firstn_eq_all_gen; eauto.
subst.
assert (a0 = a1 /\ s' = nil).
eapply evalDet_steps_done_func; eauto.
intuition. subst.
assert (a = a1).
eapply evalDet_steps_nil_eq; eauto.
eapply filter_In; eauto.
subst.
assert (P a1 = true).
eapply filter_In; eauto.
congruence.
econstructor.
inversion H; clear H; subst.
inversion H5; clear H5; subst.
inversion H6; clear H6; subst.
simpl in *.
apply evalDet_steps_bind_done_inv in H9.
destruct H9.
destruct H.
intuition.
inversion H3; clear H3; subst.
inversion H; clear H; subst.
assert (x = a1 /\ x0 = s'0).
eapply evalDet_steps_done_func; eauto.
intuition; subst.
assert (s'0 = nil).
eapply H2.
rewrite firstn_eq_all_gen.
eauto.
eauto.
subst.
case_eq (P a1); intuition.
rewrite H in H6.
inversion H6; clear H6; subst.
simpl in *.
inversion H11; clear H11; subst.
econstructor.
trivial.
rewrite H in H6.
inversion H6; clear H6; subst.
simpl in *.
apply evalDet_steps_bind_done_inv in H11.
destruct H11.
destruct H3.
intuition.
assert (x0 = nil).
eapply evalDet_nil.
eauto.
subst.
assert (evalDet_steps (cs_more c (nil ++ ls)) (cs_done x (nil ++ ls))).
eapply evalDet_steps_app_eq.
eauto.
simpl in *.
assert (ls = nil).
eapply H2.
rewrite firstn_eq_all_gen;
eauto.
subst.
assert (a = a1).
eapply evalDet_steps_nil_eq.
eauto.
eapply filter_In; eauto.
subst.
assert (P a1 = true).
eapply filter_In; eauto.
congruence.
exfalso.
eapply H7.
econstructor.
eauto.
edestruct (dat_exists_h depth (true::nil)); eauto.
edestruct (dat_exists_h depth (false::nil)); eauto.
inversion H3; clear H3; subst.
inversion H9; clear H9; subst.
case_eq (P a0); intuition.
exfalso.
eapply H6.
econstructor.
econstructor.
eauto.
simpl.
eapply evalDet_steps_trans.
eapply evalDet_steps_bind_done.
eauto.
rewrite H3.
econstructor.
eauto.
simpl.
econstructor.
destruct repeats.
omega.
eapply datRepeat_depth_leaf_Some_repeat.
trivial.
eapply dat_correct_h_internal; simpl.
intuition.
inversion H9; clear H9; subst.
assert (s'0 = nil).
eapply evalDet_nil.
eauto.
subst.
assert (evalDet_steps (cs_more c (nil ++ ls)) (cs_done a1 (nil ++ ls))).
eapply evalDet_steps_app_eq.
trivial.
simpl in *.
assert (ls = nil).
eapply H2.
rewrite firstn_eq_all_gen;
eauto.
subst.
assert (a = a0).
eapply evalDet_steps_nil_eq.
eauto.
eapply filter_In; eauto.
subst.
assert (P a0 = true).
eapply filter_In; eauto.
congruence.
eauto.
eauto.
assert (s' = nil).
eapply H2.
rewrite firstn_eq_all_gen; eauto.
subst.
eapply datRepeat_depth_internal.
eapply IHdepth.
eapply dat_correct_h_repeat_app.
eauto.
trivial.
eauto.
trivial.
eauto.
intuition.
destruct n'; simpl in *.
eapply evalDet_nil; eauto.
edestruct (@evalDet_dec _ c nil); eauto.
destruct H11.
inversion H11; clear H11; subst.
assert (s' = nil).
eapply evalDet_nil.
eauto.
subst.
assert (evalDet_steps (cs_more c (nil ++ ls)) (cs_done x1 (nil ++ ls))).
eapply evalDet_steps_app_eq.
trivial.
simpl in *.
assert (ls = nil).
eapply H2.
rewrite firstn_eq_all_gen; eauto.
subst.
assert (a = a0).
eapply evalDet_steps_nil_eq.
eauto.
eapply filter_In; eauto.
subst.
assert (P a0 = true).
eapply filter_In; eauto.
congruence.
rewrite firstn_nil in H9.
eapply evalDet_app_nil;
eauto.
trivial.
omega.
eapply IHdepth.
eapply dat_correct_h_repeat_app.
eauto.
trivial.
eauto.
trivial.
eauto.
intuition.
destruct n'; simpl in *.
eapply evalDet_nil; eauto.
edestruct (@evalDet_dec _ c nil); eauto.
destruct H11.
inversion H11; clear H11; subst.
assert (s' = nil).
eapply evalDet_nil.
eauto.
subst.
assert (evalDet_steps (cs_more c (nil ++ ls)) (cs_done x1 (nil ++ ls))).
eapply evalDet_steps_app_eq.
trivial.
simpl in *.
assert (ls = nil).
eapply H2.
rewrite firstn_eq_all_gen; eauto.
subst.
assert (a = a0).
eapply evalDet_steps_nil_eq.
eauto.
eapply filter_In; eauto.
subst.
assert (P a0 = true).
eapply filter_In; eauto.
congruence.
rewrite firstn_nil in H9.
eapply evalDet_app_nil;
eauto.
trivial.
omega.
eapply datRepeat_depth_internal.
eapply IHdepth;
eauto.
intuition.
edestruct evalDet_steps_dec; eauto.
destruct H9.
destruct H9.
exfalso.
eapply H10.
econstructor.
eauto.
eapply evalDet_app_nil.
eauto.
econstructor.
eauto.
destruct (ge_dec n' (length (ls ++ true :: nil))).
rewrite firstn_ge_all in H3.
eauto.
trivial.
apply not_ge in n.
exfalso.
assert (n' <= length ls)%nat.
rewrite app_length in n.
simpl in *.
omega.
rewrite firstn_app in H3.
assert (ls = firstn n' ls ++ skipn n' ls).
symmetry.
eapply firstn_skipn.
assert (evalDet_steps (cs_more c ((firstn n' ls) ++ (skipn n' ls))) (cs_done a' (ls' ++ (skipn n' ls)))).
eapply evalDet_steps_app_eq.
trivial.
specialize (H10 a').
eapply H10.
eapply (@evalDet_done _ _ _ a' (ls' ++ skipn n' ls)).
rewrite H14 at 1.
eapply H15.
trivial.
omega.
eapply IHdepth;
eauto.
intuition.
edestruct evalDet_steps_dec; eauto.
destruct H9.
destruct H9.
exfalso.
eapply H10.
econstructor.
eauto.
eapply evalDet_app_nil.
eauto.
econstructor.
eapply H9.
destruct (ge_dec n' (length (ls ++ false :: nil))).
rewrite firstn_ge_all in H3.
eauto.
trivial.
apply not_ge in n.
exfalso.
assert (n' <= length ls)%nat.
rewrite app_length in n.
simpl in *.
omega.
rewrite firstn_app in H3.
assert (ls = firstn n' ls ++ skipn n' ls).
symmetry.
eapply firstn_skipn.
assert (evalDet_steps (cs_more c ((firstn n' ls) ++ (skipn n' ls))) (cs_done a' (ls' ++ (skipn n' ls)))).
eapply evalDet_steps_app_eq.
trivial.
specialize (H10 a').
eapply H10.
eapply (@evalDet_done _ _ _ a' (ls' ++ skipn n' ls)).
rewrite H14 at 1.
eapply H15.
trivial.
omega.
Qed.

Lemma dat_correct_repeat_same:
forall n (A : Set) (t : DistApproxTree A) (c : Comp A) (P : A -> bool),
well_formed_comp c ->
(exists a, In a (filter P (getSupport c))) ->
dat_correct (Repeat c P) n t ->
dat_correct_repeat c P n t.