```------------------------------------------------------------------------
-- The Agda standard library
--
-- Digits and digit expansions
------------------------------------------------------------------------

module Data.Digit where

open import Data.Nat
open import Data.Nat.Properties
open SemiringSolver
open import Data.Fin as Fin using (Fin; zero; suc; toℕ)
open import Relation.Nullary.Decidable
open import Data.Char using (Char)
open import Data.List
open import Data.Vec as Vec using (Vec; _∷_; [])
open import Induction.Nat
open import Data.Nat.DivMod
open ≤-Reasoning
open import Relation.Binary.PropositionalEquality
open import Function

------------------------------------------------------------------------
-- A boring lemma

private

lem : ∀ x k r → 2 + x ≤′ r + (1 + x) * (2 + k)
lem x k r = ≤⇒≤′ \$ begin
2 + x
≤⟨ m≤m+n _ _ ⟩
2 + x + (x + (1 + x) * k + r)
≡⟨ solve 3 (λ x r k → con 2 :+ x :+ (x :+ (con 1 :+ x) :* k :+ r)
:=
r :+ (con 1 :+ x) :* (con 2 :+ k))
refl x r k ⟩
r + (1 + x) * (2 + k)
∎

------------------------------------------------------------------------
-- Digits

-- Digit b is the type of digits in base b.

Digit : ℕ → Set
Digit b = Fin b

-- Some specific digit kinds.

Decimal = Digit 10
Bit     = Digit 2

-- Some named digits.

0b : Bit
0b = zero

1b : Bit
1b = suc zero

------------------------------------------------------------------------
-- Showing digits

-- The characters used to show the first 16 digits.

digitChars : Vec Char 16
digitChars =
'0' ∷ '1' ∷ '2' ∷ '3' ∷ '4' ∷ '5' ∷ '6' ∷ '7' ∷ '8' ∷ '9' ∷
'a' ∷ 'b' ∷ 'c' ∷ 'd' ∷ 'e' ∷ 'f' ∷ []

-- showDigit shows digits in base ≤ 16.

showDigit : ∀ {base} {base≤16 : True (base ≤? 16)} →
Digit base → Char
showDigit {base≤16 = base≤16} d =
Vec.lookup (Fin.inject≤ d (toWitness base≤16)) digitChars

------------------------------------------------------------------------
-- Digit expansions

-- fromDigits takes a digit expansion of a natural number, starting
-- with the _least_ significant digit, and returns the corresponding
-- natural number.

fromDigits : ∀ {base} → List (Fin base) → ℕ
fromDigits        []       = 0
fromDigits {base} (d ∷ ds) = toℕ d + fromDigits ds * base

-- toDigits b n yields the digits of n, in base b, starting with the
-- _least_ significant digit.
--
-- Note that the list of digits is always non-empty.
--
-- This function should be linear in n, if optimised properly (see
-- Data.Nat.DivMod).

data Digits (base : ℕ) : ℕ → Set where
digits : (ds : List (Fin base)) → Digits base (fromDigits ds)

toDigits : (base : ℕ) {base≥2 : True (2 ≤? base)} (n : ℕ) →
Digits base n
toDigits zero       {base≥2 = ()} _
toDigits (suc zero) {base≥2 = ()} _
toDigits (suc (suc k)) n = <-rec Pred helper n
where
base = suc (suc k)
Pred = Digits base

cons : ∀ {n} (r : Fin base) → Pred n → Pred (toℕ r + n * base)
cons r (digits ds) = digits (r ∷ ds)

helper : ∀ n → <-Rec Pred n → Pred n
helper n rec with n divMod base
helper .(toℕ r + 0     * base) rec | result zero    r = digits (r ∷ [])
helper .(toℕ r + suc x * base) rec | result (suc x) r =
cons r (rec (suc x) (lem (pred (suc x)) k (toℕ r)))

theDigits : (base : ℕ) {base≥2 : True (2 ≤? base)} (n : ℕ) →
List (Fin base)
theDigits base {base≥2} n       with toDigits base {base≥2} n
theDigits base .(fromDigits ds) | digits ds = ds
```