Let ~p~ be a function from the natural numbers to the booleans. Suppose that for all ~x~, if ~p(x)~ is true, then ~p(x+1)~ is true. Prove that either:

1. ~p(x)~ is false for all ~x~.
2. There exists ~n~ such that ~p(x) = \begin{cases}\text{false} & \text{if }x < n \\ \text{true} & \text{if }x \ge n\end{cases}~.

This proof has applications, too. For example, let ~q(x)~ denote the truth value of "~\pi~ does not contain ~x~ consecutive ~9~'s in its decimal representation." Either ~q(x) = \text{false}~, or there exists ~n~ such that ~q(x) = (x \ge n)~. Therefore, ~q~ has a constant time algorithm, but the exact algorithm remains unknown.

Definitions

-- header.lean
def Simple (f : Nat → Bool) : Prop :=
  f = (fun _ => false) ∨
  ∃ n, f = (fun x => if x < n then false else true)

def NonconstructiveConstantTime : Prop :=
  ∀ p : Nat → Bool,
  (∀ n, p n → p (n + 1)) →
  Simple p

macro "check0123456789abcdef" t:ident : command => `(
  example : NonconstructiveConstantTime := $t
  #print axioms $t
)

Note: The macro's name is randomly generated on each submission, and will follow the format check[0-9a-f]{16}.

Proof Format

Your goal is to define a term proof with the type NonconstructiveConstantTime. You may use this template for your submission:

-- submission.lean
open Classical

def proof : NonconstructiveConstantTime := by
  admit

The judge will automatically prepend import header to your submission.

You may use the following axioms: Classical.choice, Quot.sound, propext

Checker

-- entry.lean
import header
import submission
check0123456789abcdef proof  -- 'proof' depends on axioms: [Classical.choice, Quot.sound, propext]

If you find a way to fool the checker, please open a ticket by clicking the "Report an issue" button at the bottom of the page, then put your submission in the ticket.