(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(** A constructive proof of a version of Weak König's Lemma over a decidable predicate in the formulation of which infinite paths are treated as predicates. The representation of paths as relations avoid the need for classical logic and unique choice. The decidability condition is sufficient to ensure that some required instance of double negation for disjunction of finite paths holds.  The idea of the proof comes from the proof of the weak König's lemma from separation in second-order arithmetic.  Notice that we do not start from a tree but just from an arbitrary predicate. Original Weak Konig's Lemma is the instantiation of the lemma to a tree *)
(** [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 *)
(** [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.
(** 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.
Definition demorgan_or (P:list bool -> Prop) l l' := ~ (P l /\ P l') -> ~ P l \/ ~ P l'.
Definition demorgan_inductively_barred_at P :=
forall n l, demorgan_or (inductively_barred_at P n) (true::l) (false::l).
(** [X] is the translation of [Y] as a predicate *)
Definition X P n := exists l, length l = n /\ Y P (true::l).
(** Main theorem *)

Theorem PreWeakKonigsLemma : forall P,
demorgan_inductively_barred_at P -> infinite_from P [] -> has_infinite_path P.