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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
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(*         *     GNU Lesser General Public License Version 2.1          *)
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(** A constructive proof of a non-standard version of the weak Fan Theorem 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 idea of the proof comes from the proof of the weak König's lemma from separation in second-order arithmetic [[Simpson99]].  [[Simpson99]] Stephen G. Simpson. Subsystems of second order arithmetic, Cambridge University Press, 1999 *)
(** [inductively_barred P l] means that P eventually holds above l *)
Inductive inductively_barred P : list bool -> Prop :=
| now l : P l -> inductively_barred P l
| propagate l :
inductively_barred P (true::l) ->
inductively_barred P (false::l) ->
inductively_barred P l.
(** [approx X l] says that [l] is a boolean representation of a prefix of [X] *)
Fixpoint approx X (l:list bool) :=
match l with
| [] => True
| b::l => approx X l /\ (if b then X (length l) else ~ X (length l))
end.
(** [barred P] means that for any infinite path represented as a predicate, the property [P] holds for some prefix of the path *)
Definition barred P :=
forall (X:nat -> Prop), exists l, approx X l /\ P 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 inductively_barred P (false::l) else ~ inductively_barred P (false::l)
end.
(** [X] is the translation of [Y] as a predicate *)
Definition X P n := exists l, length l = n /\ Y P (true::l).

Theorem WeakFanTheorem : forall P, barred P -> inductively_barred P [].