open import 1Lab.Prelude

open import Data.Nat.Properties
open import Data.Nat.Divisible
open import Data.Nat.Solver
open import Data.Nat.Order
open import Data.Dec.Base
open import Data.Nat.Base
open import Data.Sum.Base

module Data.Nat.DivMod where


# Natural divisionπ

This module implements the basics of the theory of division (not divisibility, see there) for the natural numbers. In particular, we define what it means to divide in the naturals (the type DivMod), and implement a division procedure that works for positive denominators.

The result of division isnβt a single number, but rather a pair of numbers such that The number is the quotient and the number is the remainder

record DivMod (a b : Nat) : Type where
constructor divmod

field
quot : Nat
rem  : Nat
.quotient : a β‘ quot * b + rem
.smaller  : rem < b


The easy case is to divide zero by anything, in which case the result is zero with remainder zero. The more interesting case comes when we have some successor, and we want to divide it.

divide-pos : β a b β .β¦ _ : Positive b β¦ β DivMod a b
divide-pos zero (suc b) = divmod 0 0 refl (sβ€s 0β€x)


It suffices to assume β since is smaller than β that we have already computed numbers with Since the ordering on is trichotomous, we can proceed by cases on whether

divide-pos (suc a) b with divide-pos a b
divide-pos (suc a) b | divmod q' r' p s with β€-split (suc r') b


First, suppose that i.e., can serve as a remainder for the division of In that case, we have our divisor! It remains to show, by a slightly annoying computation, that

divide-pos (suc a) b | divmod q' r' p s | inl r'+1<b =
divmod q' (suc r') (ap suc p β sym (+-sucr (q' * b) r')) r'+1<b


The other case β that in which β is more interesting. Then, rather than incrementing the remainder, our remainder has βoverflownβ, and we have to increment the quotient instead. We take, in this case, and which works out because ( and) of some arithmetic. See for yourself:

divide-pos (suc a) (suc b') | divmod q' r' p s | inr (inr r'+1=b) =
divmod (suc q') 0
( suc a                           β‘β¨ ap suc p β©β‘
suc (q' * (suc b') + r')        β‘Λβ¨ ap (Ξ» e β suc (q' * e + r')) r'+1=b β©β‘Λ
suc (q' * (suc r') + r')        β‘β¨ nat! β©β‘
suc (r' + q' * (suc r') + zero) β‘β¨ ap (Ξ» e β e + q' * e + 0) r'+1=b β©β‘
(suc b') + q' * (suc b') + 0    β )
(sβ€s 0β€x)


Note that we actually have one more case to consider β or rather, discard β that in which Itβs impossible because, by the definition of division, we have meaning

divide-pos (suc a) (suc b') | divmod q' r' p s | inr (inl b<r'+1) =
absurd (<-not-equal b<r'+1
(β€-antisym (β€-sucr (β€-peel b<r'+1)) (recover s)))


As a finishing touch, we define short operators to produce the result of applying divide-pos to a pair of numbers. Note that the procedure above only works when the denominator is nonzero, so we have a degree of freedom when defining and The canonical choice is to yield in both cases.

_%_ : Nat β Nat β Nat
a % zero = zero
a % suc b = divide-pos a (suc b) .DivMod.rem

_/β_ : Nat β Nat β Nat
a /β zero = zero
a /β suc b = divide-pos a (suc b) .DivMod.quot

is-divmod : β x y β .β¦ _ : Positive y β¦ β x β‘ (x /β y) * y + x % y
is-divmod x (suc y) with divide-pos x (suc y)
... | divmod q r Ξ± Ξ² = recover Ξ±

x%y<y : β x y β .β¦ _ : Positive y β¦ β (x % y) < y
x%y<y x (suc y) with divide-pos x (suc y)
... | divmod q r Ξ± Ξ² = recover Ξ²


With this, we can decide whether two numbers divide each other by checking whether the remainder is zero!

instance
Dec-β£ : β {n m} β Dec (n β£ m)
Dec-β£ {zero} {m} = m β‘? 0
Dec-β£ n@{suc _} {m} with divide-pos m n
... | divmod q 0 Ξ± Ξ² = yes (q , sym (+-zeror _) β sym (recover Ξ±))
... | divmod q r@(suc _) Ξ± Ξ² = no Ξ» (q' , p) β
let
nβ£r : (q' - q) * n β‘ r
nβ£r = monus-distribr q' q n β sym (monus-swapl _ _ _ (sym (p β recover Ξ±)))
in <-β€-asym (recover Ξ²) (mβ£snβmβ€sn (q' - q , nβ£r))