------------------------------------------------------------------------
-- The Agda standard library
--
-- Natural number division
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Data.Nat.DivMod where

open import Agda.Builtin.Nat using (div-helper; mod-helper)

open import Data.Fin.Base using (Fin; toℕ; fromℕ<)
open import Data.Fin.Properties using (toℕ-fromℕ<)
open import Data.Nat.Base as Nat
open import Data.Nat.DivMod.Core
open import Data.Nat.Divisibility.Core
open import Data.Nat.Induction
open import Data.Nat.Properties
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary using (yes; no)
open import Relation.Nullary.Decidable using (False; toWitnessFalse)

import Algebra.Properties.CommutativeSemigroup *-commutativeSemigroup as *-CS

open ≤-Reasoning

------------------------------------------------------------------------
-- Definitions

-- The division and modulus operations are only defined when the divisor
-- is non-zero. The proof of non-zero-ness is provided as an irrelevant
-- instance argument which is defined in terms of `⊤` and `⊥`. This
-- allows it to be automatically inferred when the divisor is of the
-- form `suc n`, and hence minimises the number of these proofs that
-- need be passed around. You can therefore write `m / suc n` without
-- further elaboration.

infixl 7 _/_ _%_

-- Natural division

_/_ : (dividend divisor : ) .{{_ : NonZero divisor}}  
m / (suc n) = div-helper 0 n m n

-- Natural remainder/modulus

_%_ : (dividend divisor : ) .{{_ : NonZero divisor}}  
m % (suc n) = mod-helper 0 n m n

------------------------------------------------------------------------
-- Relationship between _%_ and _div_

m≡m%n+[m/n]*n :  m n .{{_ : NonZero n}}  m  m % n + (m / n) * n
m≡m%n+[m/n]*n m (suc n) = div-mod-lemma 0 0 m n

m%n≡m∸m/n*n :  m n .{{_ : NonZero n}}  m % n  m  (m / n) * n
m%n≡m∸m/n*n m n = begin-equality
  m % n                  ≡˘⟨ m+n∸n≡m (m % n) m/n*n 
  m % n + m/n*n  m/n*n  ≡˘⟨ cong (_∸ m/n*n) (m≡m%n+[m/n]*n m n) 
  m  m/n*n              
  where m/n*n = (m / n) * n

------------------------------------------------------------------------
-- Properties of _%_

n%1≡0 :  n  n % 1  0
n%1≡0 = a[modₕ]1≡0

n%n≡0 :  n .{{_ : NonZero n}}  n % n  0
n%n≡0 (suc n-1) = n[modₕ]n≡0 0 n-1

m%n%n≡m%n :  m n .{{_ : NonZero n}}  m % n % n  m % n
m%n%n≡m%n m (suc n-1) = modₕ-idem 0 m n-1

[m+n]%n≡m%n :  m n .{{_ : NonZero n}}  (m + n) % n  m % n
[m+n]%n≡m%n m (suc n-1) = a+n[modₕ]n≡a[modₕ]n 0 m n-1

[m+kn]%n≡m%n :  m k n .{{_ : NonZero n}}  (m + k * n) % n  m % n
[m+kn]%n≡m%n m zero    n = cong (_% n) (+-identityʳ m)
[m+kn]%n≡m%n m (suc k) n = begin-equality
  (m + (n + k * n)) % n ≡⟨ cong (_% n) (sym (+-assoc m n (k * n))) 
  (m + n + k * n)   % n ≡⟨ [m+kn]%n≡m%n (m + n) k n 
  (m + n)           % n ≡⟨ [m+n]%n≡m%n m n 
  m                 % n 

m*n%n≡0 :  m n .{{_ : NonZero n}}  (m * n) % n  0
m*n%n≡0 m (suc n-1) = [m+kn]%n≡m%n 0 m (suc n-1)

m%n<n :  m n .{{_ : NonZero n}}  m % n < n
m%n<n m (suc n-1) = s≤s (a[modₕ]n<n 0 m n-1)

m%n≤n :  m n .{{_ : NonZero n}}  m % n  n
m%n≤n m n = <⇒≤ (m%n<n m n)

m%n≤m :  m n .{{_ : NonZero n}}  m % n  m
m%n≤m m (suc n-1) = a[modₕ]n≤a 0 m n-1

m≤n⇒m%n≡m :  {m n}  m  n  m % suc n  m
m≤n⇒m%n≡m {m} {n} m≤n with ≤⇒≤″ m≤n
... | less-than-or-equal {k} refl = a≤n⇒a[modₕ]n≡a 0 (m + k) m k

%-pred-≡0 :  {m n} .{{_ : NonZero n}}  (suc m % n)  0  (m % n)  n  1
%-pred-≡0 {m} {suc n-1} eq = a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 n-1 m eq

m<[1+n%d]⇒m≤[n%d] :  {m} n d .{{_ : NonZero d}}  m < suc n % d  m  n % d
m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-1

[1+m%d]≤1+n⇒[m%d]≤n :  m n d .{{_ : NonZero d}}  0 < suc m % d  suc m % d  suc n  m % d  n
[1+m%d]≤1+n⇒[m%d]≤n m n (suc d-1) leq = 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 n m d-1 leq

%-distribˡ-+ :  m n d .{{_ : NonZero d}}  (m + n) % d  ((m % d) + (n % d)) % d
%-distribˡ-+ m n d@(suc d-1) = begin-equality
  (m + n)                         % d ≡⟨ cong  v  (v + n) % d) (m≡m%n+[m/n]*n m d) 
  (m % d +  m / d * d + n)        % d ≡⟨ cong (_% d) (+-assoc (m % d) _ n) 
  (m % d + (m / d * d + n))       % d ≡⟨ cong  v  (m % d + v) % d) (+-comm _ n) 
  (m % d + (n + m / d * d))       % d ≡⟨ cong (_% d) (sym (+-assoc (m % d) n _)) 
  (m % d +  n + m / d * d)        % d ≡⟨ [m+kn]%n≡m%n (m % d + n) (m / d) d 
  (m % d +  n)                    % d ≡⟨ cong  v  (m % d + v) % d) (m≡m%n+[m/n]*n n d) 
  (m % d + (n % d + (n / d) * d)) % d ≡⟨ sym (cong (_% d) (+-assoc (m % d) (n % d) _)) 
  (m % d +  n % d + (n / d) * d)  % d ≡⟨ [m+kn]%n≡m%n (m % d + n % d) (n / d) d 
  (m % d +  n % d)                % d 

%-distribˡ-* :  m n d .{{_ : NonZero d}}  (m * n) % d  ((m % d) * (n % d)) % d
%-distribˡ-* m n d@(suc d-1) = begin-equality
  (m * n)                                             % d ≡⟨ cong  h  (h * n) % d) (m≡m%n+[m/n]*n m d) 
  ((m′ + k * d) * n)                                  % d ≡⟨ cong  h  ((m′ + k * d) * h) % d) (m≡m%n+[m/n]*n n d) 
  ((m′ + k * d) * (n′ + j * d))                       % d ≡⟨ cong (_% d) lemma 
  (m′ * n′ + (m′ * j + (n′ + j * d) * k) * d)         % d ≡⟨ [m+kn]%n≡m%n (m′ * n′) (m′ * j + (n′ + j * d) * k) d 
  (m′ * n′)                                           % d ≡⟨⟩
  ((m % d) * (n % d)) % d 
  where
  m′ = m % d
  n′ = n % d
  k  = m / d
  j  = n / d
  lemma : (m′ + k * d) * (n′ + j * d)  m′ * n′ + (m′ * j + (n′ + j * d) * k) * d
  lemma = begin-equality
    (m′ + k * d) * (n′ + j * d)                       ≡⟨ *-distribʳ-+ (n′ + j * d) m′ (k * d) 
    m′ * (n′ + j * d) + (k * d) * (n′ + j * d)        ≡⟨ cong₂ _+_ (*-distribˡ-+ m′ n′ (j * d)) (*-comm (k * d) (n′ + j * d)) 
    (m′ * n′ + m′ * (j * d)) + (n′ + j * d) * (k * d) ≡⟨ +-assoc (m′ * n′) (m′ * (j * d)) ((n′ + j * d) * (k * d)) 
    m′ * n′ + (m′ * (j * d) + (n′ + j * d) * (k * d)) ≡˘⟨ cong (m′ * n′ +_) (cong₂ _+_ (*-assoc m′ j d) (*-assoc (n′ + j * d) k d)) 
    m′ * n′ + ((m′ * j) * d + ((n′ + j * d) * k) * d) ≡˘⟨ cong (m′ * n′ +_) (*-distribʳ-+ d (m′ * j) ((n′ + j * d) * k)) 
    m′ * n′ + (m′ * j + (n′ + j * d) * k) * d         

%-remove-+ˡ :  {m} n {d} .{{_ : NonZero d}}  d  m  (m + n) % d  n % d
%-remove-+ˡ {m} n {d@(suc d-1)} (divides p refl) = begin-equality
  (p * d + n) % d ≡⟨ cong (_% d) (+-comm (p * d) n) 
  (n + p * d) % d ≡⟨ [m+kn]%n≡m%n n p d 
  n           % d 

%-remove-+ʳ :  m {n d} .{{_ : NonZero d}}  d  n  (m + n) % d  m % d
%-remove-+ʳ m {n} {suc _} eq rewrite +-comm m n = %-remove-+ˡ {n} m eq

------------------------------------------------------------------------
-- Properties of _/_

/-congˡ :  {m n o : } .{{_ : NonZero o}} 
          m  n  m / o  n / o
/-congˡ refl = refl

/-congʳ :  {m n o : } .{{_ : NonZero n}} .{{_ : NonZero o}} 
          n  o  m / n  m / o
/-congʳ refl = refl

0/n≡0 :  n .{{_ : NonZero n}}  0 / n  0
0/n≡0 (suc n-1) = refl

n/1≡n :  n  n / 1  n
n/1≡n n = a[divₕ]1≡a 0 n

n/n≡1 :  n .{{_ : NonZero n}}  n / n  1
n/n≡1 (suc n-1) = n[divₕ]n≡1 n-1 n-1

m*n/n≡m :  m n .{{_ : NonZero n}}  m * n / n  m
m*n/n≡m m (suc n-1) = a*n[divₕ]n≡a 0 m n-1

m/n*n≡m :  {m n} .{{_ : NonZero n}}  n  m  m / n * n  m
m/n*n≡m {_} {n@(suc n-1)} (divides q refl) = cong (_* n) (m*n/n≡m q n)

m*[n/m]≡n :  {m n} .{{_ : NonZero m}}  m  n  m * (n / m)  n
m*[n/m]≡n {m} m∣n = trans (*-comm m (_ / m)) (m/n*n≡m m∣n)

m/n*n≤m :  m n .{{_ : NonZero n}}  (m / n) * n  m
m/n*n≤m m n@(suc n-1) = begin
  (m / n) * n          ≤⟨ m≤m+n ((m / n) * n) (m % n) 
  (m / n) * n + m % n  ≡⟨ +-comm _ (m % n) 
  m % n + (m / n) * n  ≡⟨ sym (m≡m%n+[m/n]*n m n) 
  m                    

m/n≤m :  m n .{{_ : NonZero n}}  (m / n)  m
m/n≤m m n = *-cancelʳ-≤ (m / n) m n (begin
  (m / n) * n ≤⟨ m/n*n≤m m n 
  m           ≤⟨ m≤m*n m n 
  m * n       )

m/n<m :  m n .{{_ : NonZero m}} .{{_ : NonZero n}}  n  2  m / n < m
m/n<m m n n≥2 = *-cancelʳ-< (m / n) m (begin-strict
  (m / n) * n ≤⟨ m/n*n≤m m n 
  m           <⟨ m<m*n m n n≥2 
  m * n       )

/-mono-≤ :  {m n o p} .{{_ : NonZero o}} .{{_ : NonZero p}} 
           m  n  o  p  m / o  n / p
/-mono-≤ m≤n (s≤s o≥p) = divₕ-mono-≤ 0 m≤n o≥p

/-monoˡ-≤ :  {m n} o .{{_ : NonZero o}}  m  n  m / o  n / o
/-monoˡ-≤ o m≤n = /-mono-≤ m≤n (≤-refl {o})

/-monoʳ-≤ :  m {n o} .{{_ : NonZero n}} .{{_ : NonZero o}} 
            n  o  m / n  m / o
/-monoʳ-≤ m n≥o = /-mono-≤ ≤-refl n≥o

/-cancelʳ-≡ :  {m n o} .{{_ : NonZero o}} 
              o  m  o  n  m / o  n / o  m  n
/-cancelʳ-≡ {m} {n} {o} o∣m o∣n m/o≡n/o = begin-equality
  m           ≡˘⟨ m*[n/m]≡n {o} {m} o∣m 
  o * (m / o) ≡⟨  cong (o *_) m/o≡n/o 
  o * (n / o) ≡⟨  m*[n/m]≡n {o} {n} o∣n 
  n           

m<n⇒m/n≡0 :  {m n} .{{_ : NonZero n}}  m < n  m / n  0
m<n⇒m/n≡0 {m} {suc n-1} (s≤s m≤n) = divₕ-finish n-1 m n-1 m≤n

m≥n⇒m/n>0 :  {m n} .{{_ : NonZero n}}  m  n  m / n > 0
m≥n⇒m/n>0 {m@(suc _)} {n@(suc _)} m≥n = begin
  1     ≡⟨ sym (n/n≡1 m) 
  m / m ≤⟨ /-monoʳ-≤ m m≥n 
  m / n 

+-distrib-/ :  m n {d} .{{_ : NonZero d}}  m % d + n % d < d 
              (m + n) / d  m / d + n / d
+-distrib-/ m n {suc d-1} leq = +-distrib-divₕ 0 0 m n d-1 leq

+-distrib-/-∣ˡ :  {m} n {d} .{{_ : NonZero d}} 
                 d  m  (m + n) / d  m / d + n / d
+-distrib-/-∣ˡ {m} n {d} (divides p refl) = +-distrib-/ m n (begin-strict
  p * d % d + n % d ≡⟨ cong (_+ n % d) (m*n%n≡0 p d) 
  n % d             <⟨ m%n<n n d 
  d                 )

+-distrib-/-∣ʳ :  {m} n {d} .{{_ : NonZero d}} 
                 d  n  (m + n) / d  m / d + n / d
+-distrib-/-∣ʳ {m} n {d} (divides p refl) = +-distrib-/ m n (begin-strict
  m % d + p * d % d ≡⟨ cong (m % d +_) (m*n%n≡0 p d) 
  m % d + 0         ≡⟨ +-identityʳ _ 
  m % d             <⟨ m%n<n m d 
  d                 )

m/n≡1+[m∸n]/n :  {m n} .{{_ : NonZero n}}  m  n  m / n  1 + ((m  n) / n)
m/n≡1+[m∸n]/n {m@(suc m-1)} {n@(suc n-1)} m≥n = begin-equality
  m / n                              ≡⟨⟩
  div-helper 0 n-1 m n-1             ≡⟨ divₕ-restart n-1 m n-1 m≥n 
  div-helper 1 n-1 (m  n) n-1       ≡⟨ divₕ-extractAcc 1 n-1 (m  n) n-1 
  1 + (div-helper 0 n-1 (m  n) n-1) ≡⟨⟩
  1 + (m  n) / n                    

m*n/m*o≡n/o :  m n o .{{_ : NonZero o}} .{{_ : NonZero (m * o)}} 
              (m * n) / (m * o)  n / o
m*n/m*o≡n/o m@(suc m-1) n o {{o≢0}} = helper (<-wellFounded n)
  where
  helper :  {n}  Acc _<_ n  (m * n) / (m * o)  n / o
  helper {n} (acc rec) with n <? o
  ... | yes n<o = trans (m<n⇒m/n≡0 (*-monoʳ-< m n<o)) (sym (m<n⇒m/n≡0 n<o))
  ... | no  n≮o = begin-equality
    (m * n) / (m * o)             ≡⟨  m/n≡1+[m∸n]/n (*-monoʳ-≤ m (≮⇒≥ n≮o)) 
    1 + (m * n  m * o) / (m * o) ≡˘⟨ cong  v  1 + v / (m * o)) (*-distribˡ-∸ m n o) 
    1 + (m * (n  o)) / (m * o)   ≡⟨  cong suc (helper (rec (n  o) n∸o<n)) 
    1 + (n  o) / o               ≡˘⟨ cong₂ _+_ (n/n≡1 o) refl 
    o / o + (n  o) / o           ≡˘⟨ +-distrib-/-∣ˡ (n  o) (divides 1 ((sym (*-identityˡ o)))) 
    (o + (n  o)) / o             ≡⟨  cong (_/ o) (m+[n∸m]≡n (≮⇒≥ n≮o)) 
    n / o                         
    where n∸o<n = ∸-monoʳ-< (n≢0⇒n>0 (≢-nonZero⁻¹ o≢0)) (≮⇒≥ n≮o)

*-/-assoc :  m {n d} .{{_ : NonZero d}}  d  n  m * n / d  m * (n / d)
*-/-assoc zero    {_} {d@(suc _)} d∣n = 0/n≡0 (suc d)
*-/-assoc (suc m) {n} {d@(suc _)} d∣n = begin-equality
  (n + m * n) / d     ≡⟨ +-distrib-/-∣ˡ _ d∣n 
  n / d + (m * n) / d ≡⟨ cong (n / d +_) (*-/-assoc m d∣n) 
  n / d + m * (n / d) 

/-*-interchange :  {m n o p} .{{_ : NonZero o}} .{{_ : NonZero p}} .{{_ : NonZero (o * p)}} 
                  o  m  p  n  (m * n) / (o * p)  (m / o) * (n / p)
/-*-interchange {m} {n} {o@(suc _)} {p@(suc _)} o∣m p∣n = *-cancelˡ-≡ (o * p) (begin-equality
  (o * p) * ((m * n) / (o * p)) ≡⟨  m*[n/m]≡n (*-pres-∣ o∣m p∣n) 
  m * n                         ≡˘⟨ cong₂ _*_ (m*[n/m]≡n o∣m) (m*[n/m]≡n p∣n) 
  (o * (m / o)) * (p * (n / p)) ≡⟨ [m*n]*[o*p]≡[m*o]*[n*p] o (m / o) p (n / p) 
  (o * p) * ((m / o) * (n / p)) )

------------------------------------------------------------------------
--  A specification of integer division.

record DivMod (dividend divisor : ) : Set where
  constructor result
  field
    quotient  : 
    remainder : Fin divisor
    property  : dividend  toℕ remainder + quotient * divisor

infixl 7 _div_ _mod_ _divMod_

_div_ : (dividend divisor : ) .{{_ : NonZero divisor}}  
_div_ = _/_

_mod_ : (dividend divisor : ) .{{_ : NonZero divisor}}  Fin divisor
m mod (suc n) = fromℕ< (m%n<n m (suc n))

_divMod_ : (dividend divisor : ) .{{_ : NonZero divisor}} 
           DivMod dividend divisor
m divMod n@(suc n-1) = result (m / n) (m mod n) (begin-equality
  m                                   ≡⟨  m≡m%n+[m/n]*n m n 
  m % n                    + [m/n]*n  ≡˘⟨ cong (_+ [m/n]*n) (toℕ-fromℕ< (m%n<n m n)) 
  toℕ (fromℕ< (m%n<n m n)) + [m/n]*n  )
  where [m/n]*n = m / n * n

------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.1

a≡a%n+[a/n]*n = m≡m%n+[m/n]*n
{-# WARNING_ON_USAGE a≡a%n+[a/n]*n
"Warning: a≡a%n+[a/n]*n was deprecated in v1.1.
Please use m≡m%n+[m/n]*n instead."
#-}
a%1≡0   = n%1≡0
{-# WARNING_ON_USAGE a%1≡0
"Warning: a%1≡0 was deprecated in v1.1.
Please use n%1≡0 instead."
#-}
a%n%n≡a%n = m%n%n≡m%n
{-# WARNING_ON_USAGE a%n%n≡a%n
"Warning: a%n%n≡a%n was deprecated in v1.1.
Please use m%n%n≡m%n instead."
#-}
[a+n]%n≡a%n = [m+n]%n≡m%n
{-# WARNING_ON_USAGE [a+n]%n≡a%n
"Warning: [a+n]%n≡a%n was deprecated in v1.1.
Please use [m+n]%n≡m%n instead."
#-}
[a+kn]%n≡a%n = [m+kn]%n≡m%n
{-# WARNING_ON_USAGE [a+kn]%n≡a%n
"Warning: [a+kn]%n≡a%n was deprecated in v1.1.
Please use [m+kn]%n≡m%n instead."
#-}
kn%n≡0 = m*n%n≡0
{-# WARNING_ON_USAGE kn%n≡0
"Warning: kn%n≡0 was deprecated in v1.1.
Please use m*n%n≡0 instead."
#-}
a%n<n = m%n<n
{-# WARNING_ON_USAGE a%n<n
"Warning: a%n<n was deprecated in v1.1.
Please use m%n<n instead."
#-}