Library Coq.Logic.WeakFan

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 K├Ânig's lemma from separation in second-order arithmetic [Simpson99].
[Simpson99] Stephen G. Simpson. Subsystems of second order arithmetic, Cambridge University Press, 1999

Require Import List.
Import ListNotations.

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))

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)

Lemma Y_unique : forall P l1 l2, length l1 = length l2 -> Y P l1 -> Y P l2 -> l1 = l2.

X is the translation of Y as a predicate

Definition X P n := exists l, length l = n /\ Y P (true::l).

Lemma Y_approx : forall P l, approx (X P) l -> Y P l.

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