Fixpoint delete_nth {A} (n: nat) (xs: list A) {struct n} : list A :=
 match n, xs with
 | O, y::ys => ys
 | S n', y::ys =>y :: delete_nth n' ys
 | _ , _ => nil
 end.Inductive find_nth_preds_rec {A: Type} (pred: A -> Prop): nat -> list A -> option (nat * A) -> Prop :=
| find_nth_preds_rec_cons_head: forall n R0 R, pred R0 -> find_nth_preds_rec pred n (R0 :: R) (Some (n, R0))
| find_nth_preds_rec_cons_tail: forall n R0 R R_res, find_nth_preds_rec pred (S n) R R_res -> find_nth_preds_rec pred n (R0 :: R) R_res
| find_nth_preds_rec_nil: forall n, find_nth_preds_rec pred n nil None.Inductive find_nth_preds_rec {A: Type} (pred: A -> Prop): nat -> list A -> option (nat * A) -> Prop :=
| find_nth_preds_rec_cons_head: forall n R0 R, pred R0 -> find_nth_preds_rec pred n (R0 :: R) (Some (n, R0))
| find_nth_preds_rec_cons_tail: forall n R0 R R_res, find_nth_preds_rec pred (S n) R R_res -> find_nth_preds_rec pred n (R0 :: R) R_res
| find_nth_preds_rec_nil: forall n, find_nth_preds_rec pred n nil None.Inductive find_nth_preds_rec {A: Type} (pred: A -> Prop): nat -> list A -> option (nat * A) -> Prop :=
| find_nth_preds_rec_cons_head: forall n R0 R, pred R0 -> find_nth_preds_rec pred n (R0 :: R) (Some (n, R0))
| find_nth_preds_rec_cons_tail: forall n R0 R R_res, find_nth_preds_rec pred (S n) R R_res -> find_nth_preds_rec pred n (R0 :: R) R_res
| find_nth_preds_rec_nil: forall n, find_nth_preds_rec pred n nil None.Inductive find_nth_preds_rec {A: Type} (pred: A -> Prop): nat -> list A -> option (nat * A) -> Prop :=
| find_nth_preds_rec_cons_head: forall n R0 R, pred R0 -> find_nth_preds_rec pred n (R0 :: R) (Some (n, R0))
| find_nth_preds_rec_cons_tail: forall n R0 R R_res, find_nth_preds_rec pred (S n) R R_res -> find_nth_preds_rec pred n (R0 :: R) R_res
| find_nth_preds_rec_nil: forall n, find_nth_preds_rec pred n nil None.(* ensure that we avoid name collision with the above *)
Inductive find_nth_preds {A: Type} (pred: A -> Prop): list A -> option (nat * A) -> Prop :=
| find_nth_preds_constr: forall R R_res, find_nth_preds_rec pred 0 R R_res -> find_nth_preds pred R R_res.(* ensure that we avoid name collision with the above *)
Inductive find_nth_preds {A: Type} (pred: A -> Prop): list A -> option (nat * A) -> Prop :=
| find_nth_preds_constr: forall R R_res, find_nth_preds_rec pred 0 R R_res -> find_nth_preds pred R R_res.

Lemma find_nth_preds_rec_S: forall {A: Type} (pred: A -> Prop) z R n Rn, find_nth_preds_rec pred z R (Some (n, Rn)) -> find_nth_preds_rec pred (S z) R (Some (S n, Rn)).