CoInductive path x: Prop:=
  | step x': path x' -> P x x' -> path x.CoInductive path x: Prop:=
  | step x': path x' -> P x x' -> path x.CoInductive stutter_path x: nat -> Prop :=
  | step' n1 n2 x':  stutter_path x' n1 -> P x x' -> stutter_path x n2
  | stutter n : stutter_path x n -> stutter_path x (S n).CoInductive stutter_path x: nat -> Prop :=
  | step' n1 n2 x':  stutter_path x' n1 -> P x x' -> stutter_path x n2
  | stutter n : stutter_path x n -> stutter_path x (S n).CoInductive stutter_path x: nat -> Prop :=
  | step' n1 n2 x':  stutter_path x' n1 -> P x x' -> stutter_path x n2
  | stutter n : stutter_path x n -> stutter_path x (S n).Inductive path_gen path : X -> Prop :=
  | _path_gen : forall x x', P x x' -> path x' -> path_gen path x.Inductive path_gen path : X -> Prop :=
  | _path_gen : forall x x', P x x' -> path x' -> path_gen path x.Definition path' := paco1 path_gen.Inductive stutterpath_gen (stutterpath : X -> nat -> Prop) : X -> nat -> Prop:=
  | _stutterpath_step : forall n1 n2 x x', P x x' -> stutterpath x' n1 -> stutterpath_gen stutterpath x n2
  | _stutterpath_stut : forall n x, stutterpath x n -> stutterpath_gen stutterpath x (S n).Inductive stutterpath_gen (stutterpath : X -> nat -> Prop) : X -> nat -> Prop:=
  | _stutterpath_step : forall n1 n2 x x', P x x' -> stutterpath x' n1 -> stutterpath_gen stutterpath x n2
  | _stutterpath_stut : forall n x, stutterpath x n -> stutterpath_gen stutterpath x (S n).Inductive stutterpath_gen (stutterpath : X -> nat -> Prop) : X -> nat -> Prop:=
  | _stutterpath_step : forall n1 n2 x x', P x x' -> stutterpath x' n1 -> stutterpath_gen stutterpath x n2
  | _stutterpath_stut : forall n x, stutterpath x n -> stutterpath_gen stutterpath x (S n).Definition stutterpath' := paco2 stutterpath_gen.

(* Specifications: *)
Theorem theorem : forall x n, stutterpath' bot2 x n -> path' bot1 x.