------------------------------------------------------------------------ -- The Agda standard library -- -- Natural number division ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --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 (nonZeroIndex; toℕ-fromℕ<) open import Data.Nat.Base open import Data.Nat.DivMod.Core open import Data.Nat.Divisibility.Core open import Data.Nat.Induction open import Data.Nat.Properties open import Data.Product.Base using (_,_) open import Data.Sum.Base using (inj₁; inj₂) open import Function.Base using (_$_; _∘_) open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; cong₂; refl; trans; _≢_; sym) open import Relation.Nullary.Negation using (contradiction) open ≤-Reasoning private variable m n o p : ℕ ------------------------------------------------------------------------ -- Definitions open import Data.Nat.Base public using (_%_; _/_) ------------------------------------------------------------------------ -- Relationship between _%_ and _/_ 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 _%_ %-congˡ : .{{_ : NonZero o}} → m ≡ n → m % o ≡ n % o %-congˡ refl = refl %-congʳ : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ≡ n → o % m ≡ o % n %-congʳ refl = refl 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) (+-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∸m]%m≡n%m : .{{_ : NonZero m}} → m ≤ n → (n ∸ m) % m ≡ n % m m≤n⇒[n∸m]%m≡n%m {m} {n} m≤n = begin-equality (n ∸ m) % m ≡⟨ [m+n]%n≡m%n (n ∸ m) m ⟨ (n ∸ m + m) % m ≡⟨ cong (_% m) (m∸n+n≡m m≤n) ⟩ n % m ∎ m*n≤o⇒[o∸m*n]%n≡o%n : ∀ m {n o} .{{_ : NonZero n}} → m * n ≤ o → (o ∸ m * n) % n ≡ o % n m*n≤o⇒[o∸m*n]%n≡o%n m {n} {o} m*n≤o = begin-equality (o ∸ m * n) % n ≡⟨ [m+kn]%n≡m%n (o ∸ m * n) m n ⟨ (o ∸ m * n + m * n) % n ≡⟨ cong (_% n) (m∸n+n≡m m*n≤o) ⟩ o % n ∎ m*n%n≡0 : ∀ m n .{{_ : NonZero n}} → (m * n) % n ≡ 0 m*n%n≡0 m n@(suc _) = [m+kn]%n≡m%n 0 m n 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 % suc n ≡ m m≤n⇒m%n≡m {m = m} m≤n with k , refl ← m≤n⇒∃[o]m+o≡n m≤n = a≤n⇒a[modₕ]n≡a 0 (m + k) m k m<n⇒m%n≡m : .{{_ : NonZero n}} → m < n → m % n ≡ m m<n⇒m%n≡m {n = suc _} m<n = m≤n⇒m%n≡m (<⇒≤pred m<n) %-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 (λ 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 (λ v → (m % d + v) % d) (+-comm _ n) ⟩ (m % d + (n + m / d * d)) % d ≡⟨ cong (_% d) (+-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 (λ v → (m % d + v) % d) (m≡m%n+[m/n]*n n d) ⟩ (m % d + (n % d + (n / d) * d)) % d ≡⟨ 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 = begin-equality (m * n) % d ≡⟨ cong (λ h → (h * n) % d) (m≡m%n+[m/n]*n m d) ⟩ ((m′ + k * d) * n) % d ≡⟨ cong (λ 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@.(p * d)} n {d} (divides-refl p) = begin-equality (m + n) % d ≡⟨⟩ (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} rewrite +-comm m n = %-remove-+ˡ {n} m ------------------------------------------------------------------------ -- Properties of _/_ /-congˡ : .{{_ : NonZero o}} → m ≡ n → m / o ≡ n / o /-congˡ refl = refl /-congʳ : .{{_ : 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 _) = 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 = n} (divides-refl q) = cong (_* n) (m*n/n≡m q 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 = 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 ≡⟨ 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}} → 1 < n → m / n < m m/n<m m n 1<n = *-cancelʳ-< _ (m / n) m $ begin-strict m / n * n ≤⟨ m/n*n≤m m n ⟩ m <⟨ m<m*n m n 1<n ⟩ m * n ∎ /-mono-≤ : .{{_ : 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-≤ : ∀ 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 ≡⟨ n/n≡1 m ⟨ m / m ≤⟨ /-monoʳ-≤ m m≥n ⟩ m / n ∎ m/n≡0⇒m<n : ∀ {m n} .{{_ : NonZero n}} → m / n ≡ 0 → m < n m/n≡0⇒m<n {m} {n@(suc _)} m/n≡0 with <-≤-connex m n ... | inj₁ m<n = m<n ... | inj₂ n≤m = contradiction m/n≡0 (≢-nonZero⁻¹ _) where instance _ = >-nonZero (m≥n⇒m/n>0 n≤m) m/n≢0⇒n≤m : ∀ {m n} .{{_ : NonZero n}} → m / n ≢ 0 → n ≤ m m/n≢0⇒n≤m {m} {n@(suc _)} m/n≢0 with <-≤-connex m n ... | inj₁ m<n = contradiction (m<n⇒m/n≡0 m<n) m/n≢0 ... | inj₂ n≤m = n≤m +-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@.(p * d)} n {d} (divides-refl p) = +-distrib-/ m n $ begin-strict m % d + n % d ≡⟨⟩ 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@.(p * d)} {d} (divides-refl p) = +-distrib-/ m n $ begin-strict m % d + n % d ≡⟨⟩ 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]/n≡m/n∸1 : ∀ m n .{{_ : NonZero n}} → (m ∸ n) / n ≡ pred (m / n) [m∸n]/n≡m/n∸1 m n with <-≤-connex m n ... | inj₁ m<n = begin-equality (m ∸ n) / n ≡⟨ m<n⇒m/n≡0 (≤-<-trans (m∸n≤m m n) m<n) ⟩ 0 ≡⟨⟩ pred 0 ≡⟨ cong pred (m<n⇒m/n≡0 m<n) ⟨ pred (m / n) ∎ ... | inj₂ n≥m = begin-equality (m ∸ n) / n ≡⟨⟩ pred (1 + (m ∸ n) / n) ≡⟨ cong pred (m/n≡1+[m∸n]/n n≥m) ⟨ pred (m / n) ∎ m∣n⇒o%n%m≡o%m : ∀ m n o .{{_ : NonZero m}} .{{_ : NonZero n}} → m ∣ n → o % n % m ≡ o % m m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality o % n % m ≡⟨⟩ o % pm % m ≡⟨ %-congˡ (m%n≡m∸m/n*n o pm) ⟩ (o ∸ o / pm * pm) % m ≡⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟨ (o ∸ o / pm * p * m) % m ≡⟨ m*n≤o⇒[o∸m*n]%n≡o%n (o / pm * p) lem ⟩ o % m ∎ where pm = p * m lem : o / pm * p * m ≤ o lem = begin o / pm * p * m ≡⟨ *-assoc (o / pm) p m ⟩ o / pm * pm ≤⟨ m/n*n≤m o pm ⟩ o ∎ 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 n o = helper (<-wellFounded n) where instance _ = m*n≢0 m o helper : ∀ {n} → Acc _<_ n → (m * n) / (m * o) ≡ n / o helper {n} (acc rec) with <-≤-connex n o ... | inj₁ n<o = trans (m<n⇒m/n≡0 (*-monoʳ-< m n<o)) (sym (m<n⇒m/n≡0 n<o)) where instance _ = m*n≢0⇒m≢0 m ... | inj₂ 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 (suc ∘ (_/ (m * o))) (*-distribˡ-∸ m n o) ⟨ 1 + (m * (n ∸ o)) / (m * o) ≡⟨ cong suc (helper (rec n∸o<n)) ⟩ 1 + (n ∸ o) / o ≡⟨ m/n≡1+[m∸n]/n n≥o ⟨ n / o ∎ where n∸o<n = ∸-monoʳ-< (n≢0⇒n>0 (≢-nonZero⁻¹ o)) n≥o m*n/o*n≡m/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (o * n)}} → m * n / (o * n) ≡ m / o m*n/o*n≡m/o m n o = begin-equality m * n / (o * n) ≡⟨ /-congˡ (*-comm m n) ⟩ n * m / (o * n) ≡⟨ /-congʳ (*-comm o n) ⟩ n * m / (n * o) ≡⟨ m*n/m*o≡n/o n m o ⟩ m / o ∎ where instance _ : NonZero n _ = m*n≢0⇒n≢0 o _ : NonZero (n * o) _ = m*n≢0 n o m<n*o⇒m/o<n : ∀ {m n o} .{{_ : NonZero o}} → m < n * o → m / o < n m<n*o⇒m/o<n {m} {1} {o} m<o rewrite *-identityˡ o = begin-strict m / o ≡⟨ m<n⇒m/n≡0 m<o ⟩ 0 <⟨ z<s ⟩ 1 ∎ m<n*o⇒m/o<n {m} {suc n@(suc _)} {o} m<n*o = pred-cancel-< $ begin-strict pred (m / o) ≡⟨ [m∸n]/n≡m/n∸1 m o ⟨ (m ∸ o) / o <⟨ m<n*o⇒m/o<n (m<n+o⇒m∸n<o m o m<n*o) ⟩ n ∎ where instance _ = m*n≢0 n o [m∸n*o]/o≡m/o∸n : ∀ m n o .{{_ : NonZero o}} → (m ∸ n * o) / o ≡ m / o ∸ n [m∸n*o]/o≡m/o∸n m zero o = refl [m∸n*o]/o≡m/o∸n m (suc n) o = begin-equality (m ∸ (o + n * o)) / o ≡⟨ /-congˡ (∸-+-assoc m o (n * o)) ⟨ (m ∸ o ∸ n * o) / o ≡⟨ [m∸n*o]/o≡m/o∸n (m ∸ o) n o ⟩ (m ∸ o) / o ∸ n ≡⟨ cong (_∸ n) ([m∸n]/n≡m/n∸1 m o) ⟩ m / o ∸ 1 ∸ n ≡⟨ ∸-+-assoc (m / o) 1 n ⟩ m / o ∸ suc n ∎ m/n/o≡m/[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}} .{{_ : NonZero (n * o)}} → m / n / o ≡ m / (n * o) m/n/o≡m/[n*o] m n o = begin-equality m / n / o ≡⟨ /-congˡ {o = o} (/-congˡ (m≡m%n+[m/n]*n m n*o)) ⟩ (m % n*o + m / n*o * n*o) / n / o ≡⟨ /-congˡ (+-distrib-/-∣ʳ (m % n*o) lem₁) ⟩ (m % n*o / n + m / n*o * n*o / n) / o ≡⟨ cong (λ # → (m % n*o / n + #) / o) lem₂ ⟩ (m % n*o / n + m / n*o * o) / o ≡⟨ +-distrib-/-∣ʳ (m % n*o / n) (divides-refl (m / n*o)) ⟩ m % n*o / n / o + m / n*o * o / o ≡⟨ cong (m % n*o / n / o +_) (m*n/n≡m (m / n*o) o) ⟩ m % n*o / n / o + m / n*o ≡⟨ cong (_+ m / n*o) (m<n⇒m/n≡0 (m<n*o⇒m/o<n {n = o} lem₃)) ⟩ m / n*o ∎ where n*o = n * o o*n = o * n lem₁ : n ∣ m / n*o * n*o lem₁ = divides (m / n*o * o) $ begin-equality m / n*o * n*o ≡⟨ cong (m / n*o *_) (*-comm n o) ⟩ m / n*o * o*n ≡⟨ *-assoc (m / n*o) o n ⟨ m / n*o * o * n ∎ lem₂ : m / n*o * n*o / n ≡ m / n*o * o lem₂ = begin-equality m / n*o * n*o / n ≡⟨ cong (λ # → m / n*o * # / n) (*-comm n o) ⟩ m / n*o * o*n / n ≡⟨ /-congˡ (*-assoc (m / n*o) o n) ⟨ m / n*o * o * n / n ≡⟨ m*n/n≡m (m / n*o * o) n ⟩ m / n*o * o ∎ lem₃ : m % n*o < o*n lem₃ = begin-strict m % n*o <⟨ m%n<n m n*o ⟩ n*o ≡⟨ *-comm n o ⟩ o*n ∎ *-/-assoc : ∀ m {n d} .{{_ : NonZero d}} → d ∣ n → m * n / d ≡ m * (n / d) *-/-assoc zero {_} {d} d∣n = 0/n≡0 d *-/-assoc (suc m) {n} {d} 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 : .{{_ : NonZero o}} .{{_ : NonZero p}} .{{_ : NonZero (o * p)}} → o ∣ m → p ∣ n → (m * n) / (o * p) ≡ (m / o) * (n / p) /-*-interchange {o} {p} {m@.(q * o)} {n@.(r * p)} (divides-refl q) (divides-refl r) = begin-equality (m * n) / (o * p) ≡⟨⟩ q * o * (r * p) / (o * p) ≡⟨ /-congˡ ([m*n]*[o*p]≡[m*o]*[n*p] q o r p) ⟩ q * r * (o * p) / (o * p) ≡⟨ m*n/n≡m (q * r) (o * p) ⟩ q * r ≡⟨ cong₂ _*_ (m*n/n≡m q o) (m*n/n≡m r p) ⟨ (q * o / o) * (r * p / p) ≡⟨⟩ (m / o) * (n / p) ∎ m*n/m!≡n/[m∸1]! : ∀ m n .{{_ : NonZero m}} → let instance _ = m !≢0 ; instance _ = (pred m) !≢0 in (m * n / m !) ≡ (n / (pred m) !) m*n/m!≡n/[m∸1]! m′@(suc m) n = m*n/m*o≡n/o m′ n (m !) where instance _ = m !≢0 _ = m′ !≢0 m%[n*o]/o≡m/o%n : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}} → {{_ : NonZero (n * o)}} → m % (n * o) / o ≡ m / o % n m%[n*o]/o≡m/o%n m n o = begin-equality m % (n * o) / o ≡⟨ /-congˡ (m%n≡m∸m/n*n m (n * o)) ⟩ (m ∸ (m / (n * o) * (n * o))) / o ≡⟨ cong (λ # → (m ∸ #) / o) (*-assoc (m / (n * o)) n o) ⟨ (m ∸ (m / (n * o) * n * o)) / o ≡⟨ [m∸n*o]/o≡m/o∸n m (m / (n * o) * n) o ⟩ m / o ∸ m / (n * o) * n ≡⟨ cong (λ # → m / o ∸ # * n) (/-congʳ (*-comm n o)) ⟩ m / o ∸ m / (o * n) * n ≡⟨ cong (λ # → m / o ∸ # * n) (m/n/o≡m/[n*o] m o n ) ⟨ m / o ∸ m / o / n * n ≡⟨ m%n≡m∸m/n*n (m / o) n ⟨ m / o % n ∎ where instance _ = m*n≢0 o n m%n*o≡m*o%[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero (n * o)}} → m % n * o ≡ m * o % (n * o) m%n*o≡m*o%[n*o] m n o = begin-equality m % n * o ≡⟨ cong (_* o) (m%n≡m∸m/n*n m n) ⟩ (m ∸ m / n * n) * o ≡⟨ *-distribʳ-∸ o m (m / n * n) ⟩ m * o ∸ m / n * n * o ≡⟨ cong (λ # → m * o ∸ # * n * o) (m*n/o*n≡m/o m o n) ⟨ m * o ∸ m * o / (n * o) * n * o ≡⟨ cong (m * o ∸_) (*-assoc (m * o / (n * o)) n o) ⟩ m * o ∸ m * o / (n * o) * (n * o) ≡⟨ m%n≡m∸m/n*n (m * o) (n * o) ⟨ m * o % (n * o) ∎ [m*n+o]%[p*n]≡[m*n]%[p*n]+o : ∀ m {n o} p .{{_ : NonZero (p * n)}} → o < n → (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o [m*n+o]%[p*n]≡[m*n]%[p*n]+o m {n} {o} p@(suc p-1) o<n = begin-equality (mn + o) % pn ≡⟨ %-distribˡ-+ mn o pn ⟩ (mn % pn + o % pn) % pn ≡⟨ cong (λ # → (mn % pn + #) % pn) (m<n⇒m%n≡m (m<n⇒m<o*n p o<n)) ⟩ (mn % pn + o) % pn ≡⟨ m<n⇒m%n≡m lem₂ ⟩ mn % pn + o ∎ where mn = m * n pn = p * n lem₁ : mn % pn ≤ p-1 * n lem₁ = begin mn % pn ≡⟨ m%n*o≡m*o%[n*o] m p n ⟨ (m % p) * n ≤⟨ *-monoˡ-≤ n (m<1+n⇒m≤n (m%n<n m p)) ⟩ p-1 * n ∎ lem₂ : mn % pn + o < pn lem₂ = begin-strict mn % pn + o <⟨ +-mono-≤-< lem₁ o<n ⟩ p-1 * n + n ≡⟨ +-comm (p-1 * n) n ⟩ pn ∎ ------------------------------------------------------------------------ -- A specification of integer division. record DivMod (dividend divisor : ℕ) : Set where constructor result field quotient : ℕ remainder : Fin divisor property : dividend ≡ toℕ remainder + quotient * divisor nonZeroDivisor : NonZero divisor nonZeroDivisor = nonZeroIndex remainder infixl 7 _div_ _mod_ _divMod_ _div_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ _div_ = _/_ _mod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → Fin divisor m mod n = fromℕ< (m%n<n m n) _divMod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → DivMod dividend divisor m divMod n = 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) ⟨ toℕ (fromℕ< [m%n]<n) + [m/n]*n ∎ where [m/n]*n = m / n * n ; [m%n]<n = m%n<n m n