------------------------------------------------------------------------
-- The Agda standard library
--
-- Core lemmas for division and modulus operations
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Data.Nat.DivMod.Core where
open import Agda.Builtin.Nat using ()
renaming (div-helper to divₕ; mod-helper to modₕ)
open import Data.Nat.Base using (zero; suc; _+_; _*_; _∸_; _≤_; _<_;
_≥_; z≤n; s≤s)
open import Data.Nat.Properties
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
open import Data.Product.Base using (_×_; _,_)
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; refl; cong; sym; subst; trans)
open import Relation.Nullary.Decidable using (yes; no)
open import Relation.Nullary.Negation using (contradiction)
open ≤-Reasoning
------------------------------------------------------------------------
-- Helper lemmas that have no interpretation for _%_, only for modₕ
private
mod-cong₃ : ∀ {c n a₁ a₂ b} → a₁ ≡ a₂ → modₕ c n a₁ b ≡ modₕ c n a₂ b
mod-cong₃ refl = refl
modₕ-skipTo0 : ∀ acc n a b → modₕ acc n (b + a) a ≡ modₕ (a + acc) n b 0
modₕ-skipTo0 acc n zero b = cong (λ v → modₕ acc n v 0) (+-identityʳ b)
modₕ-skipTo0 acc n (suc a) b = begin-equality
modₕ acc n (b + suc a) (suc a) ≡⟨ mod-cong₃ (+-suc b a) ⟩
modₕ acc n (suc b + a) (suc a) ≡⟨⟩
modₕ (suc acc) n (b + a) a ≡⟨ modₕ-skipTo0 (suc acc) n a b ⟩
modₕ (a + suc acc) n b 0 ≡⟨ cong (λ v → modₕ v n b 0) (+-suc a acc) ⟩
modₕ (suc a + acc) n b 0 ∎
------------------------------------------------------------------------
-- Lemmas for modₕ that also have an interpretation for _%_
a[modₕ]1≡0 : ∀ a → modₕ 0 0 a 0 ≡ 0
a[modₕ]1≡0 zero = refl
a[modₕ]1≡0 (suc a) = a[modₕ]1≡0 a
n[modₕ]n≡0 : ∀ acc v → modₕ acc (acc + v) (suc v) v ≡ 0
n[modₕ]n≡0 acc v = modₕ-skipTo0 acc (acc + v) v 1
a[modₕ]n<n : ∀ acc d n → modₕ acc (acc + n) d n ≤ acc + n
a[modₕ]n<n acc zero n = m≤m+n acc n
a[modₕ]n<n acc (suc d) zero = a[modₕ]n<n zero d (acc + 0)
a[modₕ]n<n acc (suc d) (suc n) rewrite +-suc acc n = a[modₕ]n<n (suc acc) d n
a[modₕ]n≤a : ∀ acc a n → modₕ acc (acc + n) a n ≤ acc + a
a[modₕ]n≤a acc zero n = ≤-reflexive (sym (+-identityʳ acc))
a[modₕ]n≤a acc (suc a) (suc n) = begin
modₕ acc (acc + suc n) (suc a) (suc n) ≡⟨ cong (λ v → modₕ acc v (suc a) (suc n)) (+-suc acc n) ⟩
modₕ acc (suc acc + n) (suc a) (suc n) ≤⟨ a[modₕ]n≤a (suc acc) a n ⟩
suc acc + a ≡⟨ sym (+-suc acc a) ⟩
acc + suc a ∎
a[modₕ]n≤a acc (suc a) zero = begin
modₕ acc (acc + 0) (suc a) 0 ≡⟨ cong (λ v → modₕ acc v (suc a) 0) (+-identityʳ acc) ⟩
modₕ acc acc (suc a) 0 ≤⟨ a[modₕ]n≤a 0 a acc ⟩
a ≤⟨ n≤1+n a ⟩
suc a ≤⟨ m≤n+m (suc a) acc ⟩
acc + suc a ∎
a≤n⇒a[modₕ]n≡a : ∀ acc n a b → modₕ acc n a (a + b) ≡ acc + a
a≤n⇒a[modₕ]n≡a acc n zero b = sym (+-identityʳ acc)
a≤n⇒a[modₕ]n≡a acc n (suc a) b = begin-equality
modₕ (suc acc) n a (a + b) ≡⟨ a≤n⇒a[modₕ]n≡a (suc acc) n a b ⟩
suc acc + a ≡⟨ sym (+-suc acc a) ⟩
acc + suc a ∎
modₕ-idem : ∀ acc a n → modₕ 0 (acc + n) (modₕ acc (acc + n) a n) (acc + n) ≡ modₕ acc (acc + n) a n
modₕ-idem acc zero n = a≤n⇒a[modₕ]n≡a 0 (acc + n) acc n
modₕ-idem acc (suc a) zero rewrite +-identityʳ acc = modₕ-idem 0 a acc
modₕ-idem acc (suc a) (suc n) rewrite +-suc acc n = modₕ-idem (suc acc) a n
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 : ∀ acc l n → modₕ acc (acc + l) (suc n) l ≡ 0 → modₕ acc (acc + l) n l ≡ acc + l
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 acc zero zero eq rewrite +-identityʳ acc = refl
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 acc zero (suc n) eq rewrite +-identityʳ acc = a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 acc n eq
a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 acc (suc l) (suc n) eq rewrite +-suc acc l = a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 (suc acc) l n eq
k<1+a[modₕ]n⇒k≤a[modₕ]n : ∀ acc k n l → suc k ≤ modₕ acc (acc + l) (suc n) l → k ≤ modₕ acc (acc + l) n l
k<1+a[modₕ]n⇒k≤a[modₕ]n acc k zero (suc l) (s≤s leq) = leq
k<1+a[modₕ]n⇒k≤a[modₕ]n acc k (suc n) zero leq rewrite +-identityʳ acc = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 k n acc leq
k<1+a[modₕ]n⇒k≤a[modₕ]n acc k (suc n) (suc l) leq rewrite +-suc acc l = k<1+a[modₕ]n⇒k≤a[modₕ]n (suc acc) k n l leq
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k : ∀ acc k n l → 0 < modₕ acc (acc + l) (suc n) l →
modₕ acc (acc + l) (suc n) l ≤ suc k → modₕ acc (acc + l) n l ≤ k
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k acc k zero (suc l) 0<mod (s≤s leq) = leq
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k acc k (suc n) zero 0<mod leq rewrite +-identityʳ acc = 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 k n acc 0<mod leq
1+a[modₕ]n≤1+k⇒a[modₕ]n≤k acc k (suc n) (suc l) 0<mod leq rewrite +-suc acc l = 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k (suc acc) k n l 0<mod leq
a+n[modₕ]n≡a[modₕ]n : ∀ acc a n → modₕ acc (acc + n) (acc + a + suc n) n ≡ modₕ acc (acc + n) a n
a+n[modₕ]n≡a[modₕ]n acc zero n rewrite +-identityʳ acc = begin-equality
modₕ acc (acc + n) (acc + suc n) n ≡⟨ mod-cong₃ (+-suc acc n) ⟩
modₕ acc (acc + n) (suc acc + n) n ≡⟨ modₕ-skipTo0 acc (acc + n) n (suc acc) ⟩
modₕ (acc + n) (acc + n) (suc acc) 0 ≡⟨⟩
modₕ 0 (acc + n) acc (acc + n) ≡⟨ a≤n⇒a[modₕ]n≡a 0 (acc + n) acc n ⟩
acc ∎
a+n[modₕ]n≡a[modₕ]n acc (suc a) zero rewrite +-identityʳ acc = begin-equality
modₕ acc acc (acc + suc a + 1) 0 ≡⟨ mod-cong₃ (+-comm (acc + suc a) 1) ⟩
modₕ acc acc (1 + (acc + suc a)) 0 ≡⟨⟩
modₕ 0 acc (acc + suc a) acc ≡⟨ mod-cong₃ (+-comm acc (suc a)) ⟩
modₕ 0 acc (suc a + acc) acc ≡⟨ mod-cong₃ (sym (+-suc a acc)) ⟩
modₕ 0 acc (a + suc acc) acc ≡⟨ a+n[modₕ]n≡a[modₕ]n 0 a acc ⟩
modₕ 0 acc a acc ∎
a+n[modₕ]n≡a[modₕ]n acc (suc a) (suc n) rewrite +-suc acc n = begin-equality
mod₁ (acc + suc a + (2 + n)) (suc n) ≡⟨ cong (λ v → mod₁ (v + suc (suc n)) (suc n)) (+-suc acc a) ⟩
mod₁ (suc acc + a + (2 + n)) (suc n) ≡⟨⟩
mod₂ (acc + a + (2 + n)) n ≡⟨ mod-cong₃ (sym (+-assoc (acc + a) 1 (suc n))) ⟩
mod₂ (acc + a + 1 + suc n) n ≡⟨ mod-cong₃ (cong (_+ suc n) (+-comm (acc + a) 1)) ⟩
mod₂ (suc acc + a + suc n) n ≡⟨ a+n[modₕ]n≡a[modₕ]n (suc acc) a n ⟩
mod₂ a n ∎
where
mod₁ = modₕ acc (suc acc + n)
mod₂ = modₕ (suc acc) (suc acc + n)
------------------------------------------------------------------------
-- Helper lemmas that have no interpretation for `_/_`, only for `divₕ`
private
div-cong₃ : ∀ {c n a₁ a₂ b} → a₁ ≡ a₂ → divₕ c n a₁ b ≡ divₕ c n a₂ b
div-cong₃ refl = refl
acc≤divₕ[acc] : ∀ {acc} d n j → acc ≤ divₕ acc d n j
acc≤divₕ[acc] {acc} d zero j = ≤-refl
acc≤divₕ[acc] {acc} d (suc n) zero = ≤-trans (n≤1+n acc) (acc≤divₕ[acc] d n d)
acc≤divₕ[acc] {acc} d (suc n) (suc j) = acc≤divₕ[acc] d n j
pattern inj₂′ x = inj₂ (inj₁ x)
pattern inj₃ x = inj₂ (inj₂ x)
-- This hideous lemma details the conditions needed for two divisions to
-- be equal when the two offsets (i.e. the 4ᵗʰ parameters) are different.
-- It may be that this triple sum has an elegant simplification to a
-- set of inequalities involving the modulus but I can't find it.
divₕ-offsetEq : ∀ {acc₁ acc₂} d n j k → j ≤ d → k ≤ d →
(acc₁ ≡ acc₂ × j ≤ k × k < modₕ 0 d n d) ⊎
(acc₁ ≡ acc₂ × modₕ 0 d n d ≤ j × j ≤ k) ⊎
(acc₁ ≡ suc acc₂ × k < modₕ 0 d n d × modₕ 0 d n d ≤ j) →
divₕ acc₁ d n j ≡ divₕ acc₂ d n k
divₕ-offsetEq d zero j k j≤d k≤d (inj₁ (refl , _)) = refl
divₕ-offsetEq d zero j k j≤d k≤d (inj₂′ (refl , _)) = refl
divₕ-offsetEq d zero j k j≤d k≤d (inj₃ (eq , () , _))
-- (0 , 0) cases
divₕ-offsetEq d (suc n) zero zero j≤d k≤d (inj₁ (refl , _)) =
divₕ-offsetEq d n d d ≤-refl ≤-refl (inj₂′ (refl , a[modₕ]n<n 0 n d , ≤-refl))
divₕ-offsetEq d (suc n) zero zero j≤d k≤d (inj₂′ (refl , _)) =
divₕ-offsetEq d n d d ≤-refl ≤-refl (inj₂′ (refl , a[modₕ]n<n 0 n d , ≤-refl))
divₕ-offsetEq d (suc n) zero zero j≤d k≤d (inj₃ (_ , 0<mod , mod≤0)) =
contradiction (<-≤-trans 0<mod mod≤0) λ()
-- (0 , suc) cases
divₕ-offsetEq d (suc n) zero (suc k) j≤d k≤d (inj₁ (refl , _ , 1+k<mod)) =
divₕ-offsetEq d n d k ≤-refl (<⇒≤ k≤d) (inj₃ (refl , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d 1+k<mod , a[modₕ]n<n 0 n d))
divₕ-offsetEq d (suc n) zero (suc k) j≤d k≤d (inj₂′ (refl , mod≤0 , _)) =
divₕ-offsetEq d n d k ≤-refl (<⇒≤ k≤d) (inj₃ (refl , subst (k <_) (sym (a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 d n (n≤0⇒n≡0 mod≤0))) k≤d , a[modₕ]n<n 0 n d))
divₕ-offsetEq d (suc n) zero (suc k) j≤d k≤d (inj₃ (_ , 1+k<mod , mod≤0)) =
contradiction (<-≤-trans 1+k<mod mod≤0) λ()
-- (suc , 0) cases
divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₁ (_ , () , _))
divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₂′ (_ , _ , ()))
divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₃ (eq , 0<mod , mod≤1+j)) =
divₕ-offsetEq d n j d (<⇒≤ j≤d) ≤-refl (inj₂′ (eq , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d 0<mod mod≤1+j , <⇒≤ j≤d))
-- (suc , suc) cases
divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₁ (eq , s≤s j≤k , 1+k<mod)) =
divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₁ (eq , j≤k , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d 1+k<mod))
divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₂′ (eq , mod≤1+j , (s≤s j≤k))) with modₕ 0 d (suc n) d ≟ 0
... | yes mod≡0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₁ (eq , j≤k , subst (k <_) (sym (a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 d n mod≡0)) k≤d))
... | no mod≢0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₂′ (eq , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (n≢0⇒n>0 mod≢0) mod≤1+j , j≤k))
divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₃ (eq , k<mod , mod≤1+j)) =
divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₃ (eq , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d k<mod , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (≤-<-trans z≤n k<mod) mod≤1+j))
------------------------------------------------------------------------
-- Lemmas for divₕ that also have an interpretation for _/_
-- The quotient and remainder are related to the dividend and
-- divisor in the right way.
div-mod-lemma : ∀ accᵐ accᵈ d n →
accᵐ + accᵈ * suc (accᵐ + n) + d ≡
modₕ accᵐ (accᵐ + n) d n + divₕ accᵈ (accᵐ + n) d n * suc (accᵐ + n)
div-mod-lemma accᵐ accᵈ zero n = +-identityʳ _
div-mod-lemma accᵐ accᵈ (suc d) zero rewrite +-identityʳ accᵐ = begin-equality
accᵐ + accᵈ * suc accᵐ + suc d ≡⟨ +-suc _ d ⟩
suc accᵈ * suc accᵐ + d ≡⟨ div-mod-lemma zero (suc accᵈ) d accᵐ ⟩
modₕ 0 accᵐ d accᵐ +
divₕ (suc accᵈ) accᵐ d accᵐ * suc accᵐ ≡⟨⟩
modₕ accᵐ accᵐ (suc d) 0 +
divₕ accᵈ accᵐ (suc d) 0 * suc accᵐ ∎
div-mod-lemma accᵐ accᵈ (suc d) (suc n) rewrite +-suc accᵐ n = begin-equality
accᵐ + accᵈ * m + suc d ≡⟨ +-suc _ d ⟩
suc (accᵐ + accᵈ * m + d) ≡⟨ div-mod-lemma (suc accᵐ) accᵈ d n ⟩
modₕ _ _ d n + divₕ accᵈ _ d n * m ∎
where
m = 2 + accᵐ + n
divₕ-restart : ∀ {acc} d n j → j < n → divₕ acc d n j ≡ divₕ (suc acc) d (n ∸ suc j) d
divₕ-restart d (suc n) zero j<n = refl
divₕ-restart d (suc n) (suc j) (s≤s j<n) = divₕ-restart d n j j<n
divₕ-extractAcc : ∀ acc d n j → divₕ acc d n j ≡ acc + divₕ 0 d n j
divₕ-extractAcc acc d zero j = sym (+-identityʳ acc)
divₕ-extractAcc acc d (suc n) (suc j) = divₕ-extractAcc acc d n j
divₕ-extractAcc acc d (suc n) zero = begin-equality
divₕ (suc acc) d n d ≡⟨ divₕ-extractAcc (suc acc) d n d ⟩
suc acc + divₕ 0 d n d ≡⟨ sym (+-suc acc _) ⟩
acc + suc (divₕ 0 d n d) ≡⟨ cong (acc +_) (sym (divₕ-extractAcc 1 d n d)) ⟩
acc + divₕ 1 d n d ∎
divₕ-finish : ∀ {acc} d n j → j ≥ n → divₕ acc d n j ≡ acc
divₕ-finish d zero j j≥n = refl
divₕ-finish d (suc n) (suc j) (s≤s j≥n) = divₕ-finish d n j j≥n
n[divₕ]n≡1 : ∀ n m → divₕ 0 n (suc m) m ≡ 1
n[divₕ]n≡1 n zero = refl
n[divₕ]n≡1 n (suc m) = n[divₕ]n≡1 n m
a[divₕ]1≡a : ∀ acc a → divₕ acc 0 a 0 ≡ acc + a
a[divₕ]1≡a acc zero = sym (+-identityʳ acc)
a[divₕ]1≡a acc (suc a) = trans (a[divₕ]1≡a (suc acc) a) (sym (+-suc acc a))
a*n[divₕ]n≡a : ∀ acc a n → divₕ acc n (a * suc n) n ≡ acc + a
a*n[divₕ]n≡a acc zero n = sym (+-identityʳ acc)
a*n[divₕ]n≡a acc (suc a) n = begin-equality
divₕ acc n (suc a * suc n) n ≡⟨ divₕ-restart n (suc a * suc n) n (m≤m+n (suc n) _) ⟩
divₕ (suc acc) n (suc a * suc n ∸ suc n) n ≡⟨⟩
divₕ (suc acc) n (suc n + a * suc n ∸ suc n) n ≡⟨ div-cong₃ (m+n∸m≡n (suc n) (a * suc n)) ⟩
divₕ (suc acc) n (a * suc n) n ≡⟨ a*n[divₕ]n≡a (suc acc) a n ⟩
suc acc + a ≡⟨ sym (+-suc acc a) ⟩
acc + suc a ∎
+-distrib-divₕ : ∀ acc k m n j → modₕ k (k + j) m j + modₕ 0 (k + j) n (k + j) < suc (k + j) →
divₕ acc (k + j) (m + n) j ≡ divₕ acc (k + j) m j + divₕ 0 (k + j) n (k + j)
+-distrib-divₕ acc k (suc m) n zero leq rewrite +-identityʳ k = +-distrib-divₕ (suc acc) 0 m n k leq
+-distrib-divₕ acc k (suc m) n (suc j) leq rewrite +-suc k j = +-distrib-divₕ acc (suc k) m n j leq
+-distrib-divₕ acc k zero n j leq = begin-equality
divₕ acc (k + j) n j ≡⟨ divₕ-extractAcc acc (k + j) n j ⟩
acc + divₕ 0 (k + j) n j ≡⟨ cong (acc +_) (divₕ-offsetEq _ n j _ (m≤n+m j k) ≤-refl case) ⟩
acc + divₕ 0 (k + j) n (k + j) ∎
where
case = inj₂′ (refl , +-cancelˡ-≤ (suc k) _ _ leq , m≤n+m j k)
divₕ-mono-≤ : ∀ {acc} k {m n o p} → m ≤ n → p ≤ o → divₕ acc (k + o) m o ≤ divₕ acc (k + p) n p
divₕ-mono-≤ {acc} k {0} {n} {_} {p} z≤n p≤o = acc≤divₕ[acc] (k + p) n p
divₕ-mono-≤ {acc} k {_} {_} {suc o} {suc p} (s≤s m≤n) (s≤s p≤o)
rewrite +-suc k o | +-suc k p = divₕ-mono-≤ (suc k) m≤n p≤o
divₕ-mono-≤ {acc} k {suc m} {suc n} {o} {0} (s≤s m≤n) z≤n with o <? suc m
... | no o≮1+m rewrite +-identityʳ k = begin
divₕ acc (k + o) (suc m) o ≡⟨ divₕ-finish (k + o) (suc m) o (≮⇒≥ o≮1+m) ⟩
acc ≤⟨ n≤1+n acc ⟩
suc acc ≤⟨ acc≤divₕ[acc] k n k ⟩
divₕ (suc acc) k n k ∎
... | yes o<1+m rewrite +-identityʳ k = begin
divₕ acc (k + o) (suc m) o ≡⟨ divₕ-restart (k + o) (suc m) o o<1+m ⟩
divₕ (suc acc) (k + o) (m ∸ o) (k + o) ≤⟨ divₕ-mono-≤ 0 (≤-trans (m∸n≤m m o) m≤n) (m≤m+n k o) ⟩
divₕ (suc acc) k n k ∎
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