{-# OPTIONS --cubical-compatible --safe #-}
module Data.Nat.Divisibility where
open import Data.Nat.Base
open import Data.Nat.DivMod
using (m≡m%n+[m/n]*n; m%n≡m∸m/n*n; m*n/n≡m; m*n%n≡0; *-/-assoc)
open import Data.Nat.Properties
open import Function.Base using (_∘′_; _$_; flip)
open import Function.Bundles using (_⇔_; mk⇔)
open import Level using (0ℓ)
open import Relation.Nullary.Decidable as Dec using (yes; no)
open import Relation.Nullary.Negation.Core using (contradiction)
open import Relation.Binary.Core using (_⇒_)
open import Relation.Binary.Bundles using (Preorder; Poset)
open import Relation.Binary.Structures
using (IsPreorder; IsPartialOrder)
open import Relation.Binary.Definitions
using (Reflexive; Transitive; Antisymmetric; Decidable)
import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; _≢_; refl; sym; cong; subst)
open import Relation.Binary.Reasoning.Syntax
open import Relation.Binary.PropositionalEquality.Properties
using (isEquivalence; module ≡-Reasoning)
private
variable d m n o p : ℕ
open import Data.Nat.Divisibility.Core public hiding (*-pres-∣)
quotient≢0 : (m∣n : m ∣ n) → .{{NonZero n}} → NonZero (quotient m∣n)
quotient≢0 m∣n rewrite _∣_.equality m∣n = m*n≢0⇒m≢0 (quotient m∣n)
m∣n⇒n≡quotient*m : (m∣n : m ∣ n) → n ≡ (quotient m∣n) * m
m∣n⇒n≡quotient*m m∣n = _∣_.equality m∣n
m∣n⇒n≡m*quotient : (m∣n : m ∣ n) → n ≡ m * (quotient m∣n)
m∣n⇒n≡m*quotient {m = m} m∣n rewrite _∣_.equality m∣n = *-comm (quotient m∣n) m
quotient-∣ : (m∣n : m ∣ n) → (quotient m∣n) ∣ n
quotient-∣ {m = m} m∣n = divides m (m∣n⇒n≡m*quotient m∣n)
quotient>1 : (m∣n : m ∣ n) → m < n → 1 < quotient m∣n
quotient>1 {m} {n} m∣n m<n = *-cancelˡ-< m 1 (quotient m∣n) $ begin-strict
m * 1 ≡⟨ *-identityʳ m ⟩
m <⟨ m<n ⟩
n ≡⟨ m∣n⇒n≡m*quotient m∣n ⟩
m * quotient m∣n ∎
where open ≤-Reasoning
quotient-< : (m∣n : m ∣ n) → .{{NonTrivial m}} → .{{NonZero n}} → quotient m∣n < n
quotient-< {m} {n} m∣n = begin-strict
quotient m∣n <⟨ m<m*n (quotient m∣n) m (nonTrivial⇒n>1 m) ⟩
quotient m∣n * m ≡⟨ _∣_.equality m∣n ⟨
n ∎
where open ≤-Reasoning; instance _ = quotient≢0 m∣n
n/m≡quotient : (m∣n : m ∣ n) .{{_ : NonZero m}} → n / m ≡ quotient m∣n
n/m≡quotient {m = m} (divides-refl q) = m*n/n≡m q m
m%n≡0⇒n∣m : ∀ m n .{{_ : NonZero n}} → m % n ≡ 0 → n ∣ m
m%n≡0⇒n∣m m n eq = divides (m / n) $ begin
m ≡⟨ m≡m%n+[m/n]*n m n ⟩
m % n + [m/n]*n ≡⟨ cong (_+ [m/n]*n) eq ⟩
[m/n]*n ∎
where open ≡-Reasoning; [m/n]*n = m / n * n
n∣m⇒m%n≡0 : ∀ m n .{{_ : NonZero n}} → n ∣ m → m % n ≡ 0
n∣m⇒m%n≡0 .(q * n) n (divides-refl q) = m*n%n≡0 q n
m%n≡0⇔n∣m : ∀ m n .{{_ : NonZero n}} → m % n ≡ 0 ⇔ n ∣ m
m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)
∣⇒≤ : .{{_ : NonZero n}} → m ∣ n → m ≤ n
∣⇒≤ {n@.(q * m)} {m} (divides-refl q@(suc p)) = m≤m+n m (p * m)
>⇒∤ : .{{_ : NonZero n}} → m > n → m ∤ n
>⇒∤ {n@(suc _)} n<m@(s<s _) m∣n = contradiction (∣⇒≤ m∣n) (<⇒≱ n<m)
∣-reflexive : _≡_ ⇒ _∣_
∣-reflexive {n} refl = divides 1 (sym (*-identityˡ n))
∣-refl : Reflexive _∣_
∣-refl = ∣-reflexive refl
∣-trans : Transitive _∣_
∣-trans (divides-refl p) (divides-refl q) =
divides (q * p) (sym (*-assoc q p _))
∣-antisym : Antisymmetric _≡_ _∣_
∣-antisym {m} {zero} _ q∣p = m∣n⇒n≡m*quotient q∣p
∣-antisym {zero} {n} p∣q _ = sym (m∣n⇒n≡m*quotient p∣q)
∣-antisym {suc m} {suc n} p∣q q∣p = ≤-antisym (∣⇒≤ p∣q) (∣⇒≤ q∣p)
infix 4 _∣?_
_∣?_ : Decidable _∣_
zero ∣? zero = yes (divides-refl 0)
zero ∣? suc m = no ((λ()) ∘′ ∣-antisym (divides-refl 0))
n@(suc _) ∣? m = Dec.map (m%n≡0⇔n∣m m n) (m % n ≟ 0)
∣-isPreorder : IsPreorder _≡_ _∣_
∣-isPreorder = record
{ isEquivalence = isEquivalence
; reflexive = ∣-reflexive
; trans = ∣-trans
}
∣-isPartialOrder : IsPartialOrder _≡_ _∣_
∣-isPartialOrder = record
{ isPreorder = ∣-isPreorder
; antisym = ∣-antisym
}
∣-preorder : Preorder 0ℓ 0ℓ 0ℓ
∣-preorder = record
{ isPreorder = ∣-isPreorder
}
∣-poset : Poset 0ℓ 0ℓ 0ℓ
∣-poset = record
{ isPartialOrder = ∣-isPartialOrder
}
module ∣-Reasoning where
private module Base = ≲-Reasoning ∣-preorder
open Base public
hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)
renaming (≲-go to ∣-go)
open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∣-go public
infix 10 _∣0 1∣_
_∣0 : ∀ n → n ∣ 0
n ∣0 = divides-refl 0
0∣⇒≡0 : 0 ∣ n → n ≡ 0
0∣⇒≡0 {n} 0∣n = ∣-antisym (n ∣0) 0∣n
1∣_ : ∀ n → 1 ∣ n
1∣ n = divides n (sym (*-identityʳ n))
∣1⇒≡1 : n ∣ 1 → n ≡ 1
∣1⇒≡1 {n} n∣1 = ∣-antisym n∣1 (1∣ n)
n∣n : n ∣ n
n∣n = ∣-refl
∣m∣n⇒∣m+n : d ∣ m → d ∣ n → d ∣ m + n
∣m∣n⇒∣m+n (divides-refl p) (divides-refl q) =
divides (p + q) (sym (*-distribʳ-+ _ p q))
∣m+n∣m⇒∣n : d ∣ m + n → d ∣ m → d ∣ n
∣m+n∣m⇒∣n {d} {m} {n} (divides p m+n≡p*d) (divides-refl q) =
divides (p ∸ q) $ begin-equality
n ≡⟨ m+n∸n≡m n m ⟨
n + m ∸ m ≡⟨ cong (_∸ m) (+-comm n m) ⟩
m + n ∸ m ≡⟨ cong (_∸ m) m+n≡p*d ⟩
p * d ∸ q * d ≡⟨ *-distribʳ-∸ d p q ⟨
(p ∸ q) * d ∎
where open ∣-Reasoning
n∣m*n : ∀ m {n} → n ∣ m * n
n∣m*n m = divides m refl
m∣m*n : ∀ {m} n → m ∣ m * n
m∣m*n n = divides n (*-comm _ n)
n∣m*n*o : ∀ m {n} o → n ∣ m * n * o
n∣m*n*o m o = ∣-trans (n∣m*n m) (m∣m*n o)
∣m⇒∣m*n : ∀ n → d ∣ m → d ∣ m * n
∣m⇒∣m*n n (divides-refl q) = ∣-trans (n∣m*n q) (m∣m*n n)
∣n⇒∣m*n : ∀ m {n} → d ∣ n → d ∣ m * n
∣n⇒∣m*n m {n} rewrite *-comm m n = ∣m⇒∣m*n m
m*n∣⇒m∣ : ∀ m n → m * n ∣ d → m ∣ d
m*n∣⇒m∣ m n (divides-refl q) = ∣n⇒∣m*n q (m∣m*n n)
m*n∣⇒n∣ : ∀ m n → m * n ∣ d → n ∣ d
m*n∣⇒n∣ m n rewrite *-comm m n = m*n∣⇒m∣ n m
*-pres-∣ : o ∣ m → p ∣ n → o * p ∣ m * n
*-pres-∣ {o} {m@.(q * o)} {p} {n@.(r * p)} (divides-refl q) (divides-refl r) =
divides (q * r) ([m*n]*[o*p]≡[m*o]*[n*p] q o r p)
*-monoʳ-∣ : ∀ o → m ∣ n → o * m ∣ o * n
*-monoʳ-∣ o = *-pres-∣ (∣-refl {o})
*-monoˡ-∣ : ∀ o → m ∣ n → m * o ∣ n * o
*-monoˡ-∣ o = flip *-pres-∣ (∣-refl {o})
*-cancelˡ-∣ : ∀ o .{{_ : NonZero o}} → o * m ∣ o * n → m ∣ n
*-cancelˡ-∣ {m} {n} o o*m∣o*n = divides q $ *-cancelˡ-≡ n (q * m) o $ begin-equality
o * n ≡⟨ m∣n⇒n≡m*quotient o*m∣o*n ⟩
o * m * q ≡⟨ *-assoc o m q ⟩
o * (m * q) ≡⟨ cong (o *_) (*-comm q m) ⟨
o * (q * m) ∎
where
open ∣-Reasoning
q = quotient o*m∣o*n
*-cancelʳ-∣ : ∀ o .{{_ : NonZero o}} → m * o ∣ n * o → m ∣ n
*-cancelʳ-∣ {m} {n} o rewrite *-comm m o | *-comm n o = *-cancelˡ-∣ o
∣m∸n∣n⇒∣m : ∀ d → n ≤ m → d ∣ m ∸ n → d ∣ n → d ∣ m
∣m∸n∣n⇒∣m {n} {m} d n≤m (divides p m∸n≡p*d) (divides-refl q) =
divides (p + q) $ begin-equality
m ≡⟨ m+[n∸m]≡n n≤m ⟨
n + (m ∸ n) ≡⟨ +-comm n (m ∸ n) ⟩
m ∸ n + n ≡⟨ cong (_+ n) m∸n≡p*d ⟩
p * d + q * d ≡⟨ *-distribʳ-+ d p q ⟨
(p + q) * d ∎
where open ∣-Reasoning
m/n∣m : .{{_ : NonZero n}} → n ∣ m → m / n ∣ m
m/n∣m {n} {m} n∣m = begin
m / n ≡⟨ n/m≡quotient n∣m ⟩
quotient n∣m ∣⟨ quotient-∣ n∣m ⟩
m ∎
where open ∣-Reasoning
m*n∣o⇒m∣o/n : ∀ m n .{{_ : NonZero n}} → m * n ∣ o → m ∣ o / n
m*n∣o⇒m∣o/n m n (divides-refl p) = divides p $ begin-equality
p * (m * n) / n ≡⟨ *-/-assoc p (n∣m*n m) ⟩
p * ((m * n) / n) ≡⟨ cong (p *_) (m*n/n≡m m n) ⟩
p * m ∎
where open ∣-Reasoning
m*n∣o⇒n∣o/m : ∀ m n .{{_ : NonZero m}} → m * n ∣ o → n ∣ (o / m)
m*n∣o⇒n∣o/m m n rewrite *-comm m n = m*n∣o⇒m∣o/n n m
m∣n/o⇒m*o∣n : .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → m * o ∣ n
m∣n/o⇒m*o∣n {o} {n@.(p * o)} {m} (divides-refl p) m∣p*o/o = begin
m * o ∣⟨ *-monoˡ-∣ o (subst (m ∣_) (m*n/n≡m p o) m∣p*o/o) ⟩
p * o ∎
where open ∣-Reasoning
m∣n/o⇒o*m∣n : .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → o * m ∣ n
m∣n/o⇒o*m∣n {o} {_} {m} rewrite *-comm o m = m∣n/o⇒m*o∣n
m/n∣o⇒m∣o*n : .{{_ : NonZero n}} → n ∣ m → m / n ∣ o → m ∣ o * n
m/n∣o⇒m∣o*n {n} {m@.(p * n)} {o} (divides-refl p) p*n/n∣o = begin
p * n ∣⟨ *-monoˡ-∣ n (subst (_∣ o) (m*n/n≡m p n) p*n/n∣o) ⟩
o * n ∎
where open ∣-Reasoning
m∣n*o⇒m/n∣o : .{{_ : NonZero n}} → n ∣ m → m ∣ o * n → m / n ∣ o
m∣n*o⇒m/n∣o {n} {m@.(p * n)} {o} (divides-refl p) pn∣on = begin
m / n ≡⟨⟩
p * n / n ≡⟨ m*n/n≡m p n ⟩
p ∣⟨ *-cancelʳ-∣ n pn∣on ⟩
o ∎
where open ∣-Reasoning
∣n∣m%n⇒∣m : .{{_ : NonZero n}} → d ∣ n → d ∣ m % n → d ∣ m
∣n∣m%n⇒∣m {n@.(p * d)} {d} {m} (divides-refl p) (divides q m%n≡qd) =
divides (q + (m / n) * p) $ begin-equality
m ≡⟨ m≡m%n+[m/n]*n m n ⟩
m % n + (m / n) * n ≡⟨ cong (_+ (m / n) * n) m%n≡qd ⟩
q * d + (m / n) * n ≡⟨⟩
q * d + (m / n) * (p * d) ≡⟨ cong (q * d +_) (*-assoc (m / n) p d) ⟨
q * d + ((m / n) * p) * d ≡⟨ *-distribʳ-+ d q _ ⟨
(q + (m / n) * p) * d ∎
where open ∣-Reasoning
%-presˡ-∣ : .{{_ : NonZero n}} → d ∣ m → d ∣ n → d ∣ m % n
%-presˡ-∣ {n} {d} {m@.(p * d)} (divides-refl p) (divides q 1+n≡qd) =
divides (p ∸ m / n * q) $ begin-equality
m % n ≡⟨ m%n≡m∸m/n*n m n ⟩
m ∸ m / n * n ≡⟨ cong (λ v → m ∸ m / n * v) 1+n≡qd ⟩
m ∸ m / n * (q * d) ≡⟨ cong (m ∸_) (*-assoc (m / n) q d) ⟨
m ∸ (m / n * q) * d ≡⟨⟩
p * d ∸ (m / n * q) * d ≡⟨ *-distribʳ-∸ d p (m / n * q) ⟨
(p ∸ m / n * q) * d ∎
where open ∣-Reasoning
m≤n⇒m!∣n! : m ≤ n → m ! ∣ n !
m≤n⇒m!∣n! m≤n = help (≤⇒≤′ m≤n)
where
help : m ≤′ n → m ! ∣ n !
help ≤′-refl = ∣-refl
help {n = n} (≤′-step m≤n) = ∣n⇒∣m*n n (help m≤n)
hasNonTrivialDivisor-≢ : .{{NonTrivial d}} → .{{NonZero n}} →
d ≢ n → d ∣ n → n HasNonTrivialDivisorLessThan n
hasNonTrivialDivisor-≢ d≢n d∣n
= hasNonTrivialDivisor (≤∧≢⇒< (∣⇒≤ d∣n) d≢n) d∣n
hasNonTrivialDivisor-∣ : m HasNonTrivialDivisorLessThan n → m ∣ o →
o HasNonTrivialDivisorLessThan n
hasNonTrivialDivisor-∣ (hasNonTrivialDivisor d<n d∣m) m∣o
= hasNonTrivialDivisor d<n (∣-trans d∣m m∣o)
hasNonTrivialDivisor-≤ : m HasNonTrivialDivisorLessThan n → n ≤ o →
m HasNonTrivialDivisorLessThan o
hasNonTrivialDivisor-≤ (hasNonTrivialDivisor d<n d∣m) n≤o
= hasNonTrivialDivisor (<-≤-trans d<n n≤o) d∣m