`Nat.toDigits` with base `1`

[marcusrossel]

Prerequisites

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Description

When applying Nat.toDigits with (base := 1), the result might be considered unexpected, as it does not represent a unary encoding of the input value:

#eval Nat.toDigits (base := 1) 0 -- ['0']
#eval Nat.toDigits (base := 1) 1 -- ['0', '0']
#eval Nat.toDigits (base := 1) 2 -- ['0', '0', '0']

Context

In prior discussion on Zulip, @TwoFX states "I would say this is a defect in the documentation at the very least."

Steps to Reproduce

#eval Nat.toDigits (base := 1) 0 -- ['0']
#eval Nat.toDigits (base := 1) 1 -- ['0', '0']
#eval Nat.toDigits (base := 1) 2 -- ['0', '0', '0']

Expected behavior: Either the documentation for Nat.toDigits should state that the results are only meaningful for base > 1, or Nat.toDigits (base := 1) n should return a list of n '0's.

Actual behavior: Nat.toDigits (base := 1) n returns a list of n + 1 '0's.

Versions

Lean 4.21.0-nightly-2025-06-08
Target: x86_64-unknown-linux-gnu

Impact

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[Rob23oba]

I feel like the current behavior for base 1 is the best you can do if you want to make sure that:

1. toDigits is injective (exclude base 0 and base 17 onwards)
2. All digits in the output are valid (Nat.digitChar n with n < base)
3. toDigits doesn't return an empty list

[eric-wieser]

FWIW, mathlib provides:

import Mathlib

#eval Nat.digits 1 0 -- []
#eval Nat.digits 1 1 -- [1]
#eval Nat.digits 1 2 -- [1, 1]