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

module Data.Digit where

open import Data.Nat using (; zero; suc; pred; _+_; _*_; _≤?_; _≤′_)
open import Data.Nat.Properties
open SemiringSolver
open import Data.Fin as Fin using (Fin; zero; suc; toℕ)
open import Data.Char using (Char)
open import Data.List.Base
open import Data.Product
open import Data.Vec as Vec using (Vec; _∷_; [])
open import Data.Nat.DivMod
open import Induction.Nat using (<′-rec; <′-Rec)
open import Relation.Nullary using (yes; no)
open import Relation.Nullary.Decidable
open import Relation.Binary using (Decidable)
open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
open import Function

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

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

Digit :   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

Expansion :   Set
Expansion base = List (Fin base)

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

fromDigits :  {base}  Expansion base
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.

toDigits : (base : ) {base≥2 : True (2 ≤? base)} (n : )
λ (ds : Expansion base)  fromDigits ds  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 = λ n   λ ds  fromDigits ds  n

cons :  {m} (r : Fin base)  Pred m  Pred (toℕ r + m * base)
cons r (ds , eq) = (r  ds , P.cong  i  toℕ r + i * base) eq)

open ≤-Reasoning

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  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)

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