(** [is_path_from P n l] means that there exists a path of length [n]
    from [l] on which [P] does not hold *)

Inductive is_path_from (P:list bool -> Prop) : nat -> list bool -> Prop :=
| here l : ~ P l -> is_path_from P 0 l
| next_left l n : ~ P l -> is_path_from P n (true::l) -> is_path_from P (S n) l
| next_right l n : ~ P l -> is_path_from P n (false::l) -> is_path_from P (S n) l.(** [is_path_from P n l] means that there exists a path of length [n]
    from [l] on which [P] does not hold *)

Inductive is_path_from (P:list bool -> Prop) : nat -> list bool -> Prop :=
| here l : ~ P l -> is_path_from P 0 l
| next_left l n : ~ P l -> is_path_from P n (true::l) -> is_path_from P (S n) l
| next_right l n : ~ P l -> is_path_from P n (false::l) -> is_path_from P (S n) l.(** [is_path_from P n l] means that there exists a path of length [n]
    from [l] on which [P] does not hold *)

Inductive is_path_from (P:list bool -> Prop) : nat -> list bool -> Prop :=
| here l : ~ P l -> is_path_from P 0 l
| next_left l n : ~ P l -> is_path_from P n (true::l) -> is_path_from P (S n) l
| next_right l n : ~ P l -> is_path_from P n (false::l) -> is_path_from P (S n) l.(** [is_path_from P n l] means that there exists a path of length [n]
    from [l] on which [P] does not hold *)

Inductive is_path_from (P:list bool -> Prop) : nat -> list bool -> Prop :=
| here l : ~ P l -> is_path_from P 0 l
| next_left l n : ~ P l -> is_path_from P n (true::l) -> is_path_from P (S n) l
| next_right l n : ~ P l -> is_path_from P n (false::l) -> is_path_from P (S n) l.(** We give the characterization of is_path_from in terms of a more common arithmetical formula *)

Proposition is_path_from_characterization P n l :
  is_path_from P n l <-> exists l', length l' = n /\ forall n', n'<=n -> ~ P (rev (firstn n' l') ++ l).(** [infinite_from P l] means that we can find arbitrary long paths
    along which [P] does not hold above [l] *)

Definition infinite_from (P:list bool -> Prop) l := forall n, is_path_from P n l.(** [has_infinite_path P] means that there is an infinite path
    (represented as a predicate) along which [P] does not hold at all *)

Definition has_infinite_path (P:list bool -> Prop) :=
  exists (X:nat -> Prop), forall l, approx X l -> ~ P l.(** [inductively_barred_at P n l] means that [P] eventually holds above
    [l] after at most [n] steps upwards *)

Inductive inductively_barred_at (P:list bool -> Prop) : nat -> list bool -> Prop :=
| now_at l n : P l -> inductively_barred_at P n l
| propagate_at l n :
    inductively_barred_at P n (true::l) ->
    inductively_barred_at P n (false::l) ->
    inductively_barred_at P (S n) l.(** [inductively_barred_at P n l] means that [P] eventually holds above
    [l] after at most [n] steps upwards *)

Inductive inductively_barred_at (P:list bool -> Prop) : nat -> list bool -> Prop :=
| now_at l n : P l -> inductively_barred_at P n l
| propagate_at l n :
    inductively_barred_at P n (true::l) ->
    inductively_barred_at P n (false::l) ->
    inductively_barred_at P (S n) l.(** [inductively_barred_at P n l] means that [P] eventually holds above
    [l] after at most [n] steps upwards *)

Inductive inductively_barred_at (P:list bool -> Prop) : nat -> list bool -> Prop :=
| now_at l n : P l -> inductively_barred_at P n l
| propagate_at l n :
    inductively_barred_at P n (true::l) ->
    inductively_barred_at P n (false::l) ->
    inductively_barred_at P (S n) l.(** The proof proceeds by building a set [Y] of finite paths
   approximating either the smallest unbarred infinite path in [P], if
   there is one (taking [true]>[false]), or the path
   true::true::... if [P] happens to be inductively_barred *)

Fixpoint Y P (l:list bool) :=
  match l with
  | [] => True
  | b::l =>
      Y P l /\
      if b then exists n, inductively_barred_at P n (false::l) else infinite_from P (false::l)
  end.

(* Specifications: *)
Lemma is_path_from_restrict : forall P n n' l, n <= n' -> is_path_from P n' l -> is_path_from P n l.
Lemma inductively_barred_at_monotone : forall P l n n', n' <= n -> inductively_barred_at P n' l -> inductively_barred_at P n l.