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Add prime factorization and its properties (#1969)
* Add prime factorization and its properties * Add missing header comment I'd missed this when copy-pasting from my old code in a separate repo * Remove completely trivial lemma * Use equational reasoning in quotient|n proof * Fix typo in module header * Factorization => Factorisation * Use Nat lemma in extend-|/ * [ cleanup ] part of the proof * [ cleanup ] finishing up the job * [ cleanup ] a little bit more * [ cleanup ] the import list * [ fix ] header style * [ fix ] broken merge: missing import * Move Data.Nat.Rough to Data.Nat.Primality.Rough * Rename productPreserves↭⇒≡ to product-↭ * Use proof of Prime=>NonZero * Open reasoning once in factoriseRec * Rename Factorisation => PrimeFactorisation * Move wheres around * Tidy up Rough a bit * Move quotient|n to top of file * Replace factorisationPullToFront with slightly more generally useful factorisationHasAllPrimeFactors and a bit of logic * Fix import after merge * Clean up proof of 2-rough-n * Make argument to 2-rough-n implicit * Rename 2-rough-n to 2-rough * Complete merge, rewrite factorisation logic a bit Rewrite partially based on suggestions from James McKinna * Short circuit when candidate is greater than square root of product * Remove redefined lemma * Minor simplifications * Remove private pattern synonyms * Change name of lemma * Typo * Remove using list from import It feels like we're importing half the module anyway * Clean up proof * Fixes after merge * Addressed some feedback * Fix previous merge --------- Co-authored-by: Guillaume Allais <[email protected]> Co-authored-by: MatthewDaggitt <[email protected]>
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------------------------------------------------------------------------ | ||
-- The Agda standard library | ||
-- | ||
-- Prime factorisation of natural numbers and its properties | ||
------------------------------------------------------------------------ | ||
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{-# OPTIONS --cubical-compatible --safe #-} | ||
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module Data.Nat.Primality.Factorisation where | ||
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open import Data.Empty using (⊥-elim) | ||
open import Data.Nat.Base | ||
open import Data.Nat.Divisibility | ||
open import Data.Nat.Properties | ||
open import Data.Nat.Induction using (<-Rec; <-rec; <-recBuilder) | ||
open import Data.Nat.Primality | ||
open import Data.Product as Π using (∃-syntax; _×_; _,_; proj₁; proj₂) | ||
open import Data.List.Base using (List; []; _∷_; _++_; product) | ||
open import Data.List.Membership.Propositional using (_∈_) | ||
open import Data.List.Membership.Propositional.Properties using (∈-∃++) | ||
open import Data.List.Relation.Unary.All as All using (All; []; _∷_) | ||
open import Data.List.Relation.Unary.Any using (here; there) | ||
open import Data.List.Relation.Binary.Permutation.Propositional | ||
using (_↭_; prep; swap; ↭-reflexive; ↭-refl; ↭-trans; refl; module PermutationReasoning) | ||
open import Data.List.Relation.Binary.Permutation.Propositional.Properties using (product-↭; All-resp-↭; shift) | ||
open import Data.Sum.Base using (inj₁; inj₂) | ||
open import Function.Base using (_$_; _∘_; _|>_; flip) | ||
open import Induction using (build) | ||
open import Induction.Lexicographic using (_⊗_; [_⊗_]) | ||
open import Relation.Nullary.Decidable using (yes; no) | ||
open import Relation.Nullary.Negation using (contradiction) | ||
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; cong; module ≡-Reasoning) | ||
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private | ||
variable | ||
n : ℕ | ||
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------------------------------------------------------------------------ | ||
-- Core definition | ||
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record PrimeFactorisation (n : ℕ) : Set where | ||
field | ||
factors : List ℕ | ||
isFactorisation : n ≡ product factors | ||
factorsPrime : All Prime factors | ||
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open PrimeFactorisation public using (factors) | ||
open PrimeFactorisation | ||
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------------------------------------------------------------------------ | ||
-- Finding a factorisation | ||
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primeFactorisation[1] : PrimeFactorisation 1 | ||
primeFactorisation[1] = record | ||
{ factors = [] | ||
; isFactorisation = refl | ||
; factorsPrime = [] | ||
} | ||
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primeFactorisation[p] : Prime n → PrimeFactorisation n | ||
primeFactorisation[p] {n} pr = record | ||
{ factors = n ∷ [] | ||
; isFactorisation = sym (*-identityʳ n) | ||
; factorsPrime = pr ∷ [] | ||
} | ||
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-- This builds up three important things: | ||
-- * a proof that every number we've gotten to so far has increasingly higher | ||
-- possible least prime factor, so we don't have to repeat smaller factors | ||
-- over and over (this is the "m" and "rough" parameters) | ||
-- * a witness that this limit is getting closer to the number of interest, in a | ||
-- way that helps the termination checker (the "k" and "eq" parameters) | ||
-- * a proof that we can factorise any smaller number, which is useful when we | ||
-- encounter a factor, as we can then divide by that factor and continue from | ||
-- there without termination issues | ||
factorise : ∀ n → .{{NonZero n}} → PrimeFactorisation n | ||
factorise 1 = primeFactorisation[1] | ||
factorise n₀@(2+ _) = build [ <-recBuilder ⊗ <-recBuilder ] P facRec (n₀ , suc n₀ ∸ 4) 2-rough refl | ||
where | ||
P : ℕ × ℕ → Set | ||
P (n , k) = ∀ {m} → .{{NonTrivial n}} → .{{NonTrivial m}} → m Rough n → suc n ∸ m * m ≡ k → PrimeFactorisation n | ||
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facRec : ∀ n×k → (<-Rec ⊗ <-Rec) P n×k → P n×k | ||
facRec (n , zero) _ rough eq = | ||
-- Case 1: m * m > n, ∴ Prime n | ||
primeFactorisation[p] (rough∧square>⇒prime rough (m∸n≡0⇒m≤n eq)) | ||
facRec (n@(2+ _) , suc k) (recFactor , recQuotient) {m@(2+ _)} rough eq with m ∣? n | ||
-- Case 2: m ∤ n, try larger m, reducing k accordingly | ||
... | no m∤n = recFactor (≤-<-trans (m∸n≤m k (m + m)) (n<1+n k)) {suc m} (∤⇒rough-suc m∤n rough) $ begin | ||
suc n ∸ (suc m + m * suc m) ≡⟨ cong (λ # → suc n ∸ (suc m + #)) (*-suc m m) ⟩ | ||
suc n ∸ (suc m + (m + m * m)) ≡⟨ cong (suc n ∸_) (+-assoc (suc m) m (m * m)) ⟨ | ||
suc n ∸ (suc (m + m) + m * m) ≡⟨ cong (suc n ∸_) (+-comm (suc (m + m)) (m * m)) ⟩ | ||
suc n ∸ (m * m + suc (m + m)) ≡⟨ ∸-+-assoc (suc n) (m * m) (suc (m + m)) ⟨ | ||
(suc n ∸ m * m) ∸ suc (m + m) ≡⟨ cong (_∸ suc (m + m)) eq ⟩ | ||
suc k ∸ suc (m + m) ∎ | ||
where open ≡-Reasoning | ||
-- Case 3: m ∣ n, record m and recurse on the quotient | ||
... | yes m∣n = record | ||
{ factors = m ∷ ps | ||
; isFactorisation = sym m*Πps≡n | ||
; factorsPrime = rough∧∣⇒prime rough m∣n ∷ primes | ||
} | ||
where | ||
m<n : m < n | ||
m<n = begin-strict | ||
m <⟨ s≤s (≤-trans (m≤n+m m _) (+-monoʳ-≤ _ (m≤m+n m _))) ⟩ | ||
pred (m * m) <⟨ s<s⁻¹ (m∸n≢0⇒n<m λ eq′ → 0≢1+n (trans (sym eq′) eq)) ⟩ | ||
n ∎ | ||
where open ≤-Reasoning | ||
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q = quotient m∣n | ||
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instance _ = n>1⇒nonTrivial (quotient>1 m∣n m<n) | ||
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factorisation[q] : PrimeFactorisation q | ||
factorisation[q] = recQuotient (quotient-< m∣n) (suc q ∸ m * m) (rough∧∣⇒rough rough (quotient-∣ m∣n)) refl | ||
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ps = factors factorisation[q] | ||
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primes = factorsPrime factorisation[q] | ||
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m*Πps≡n : m * product ps ≡ n | ||
m*Πps≡n = begin | ||
m * product ps ≡⟨ cong (m *_) (isFactorisation factorisation[q]) ⟨ | ||
m * q ≡⟨ m∣n⇒n≡m*quotient m∣n ⟨ | ||
n ∎ | ||
where open ≡-Reasoning | ||
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------------------------------------------------------------------------ | ||
-- Properties of a factorisation | ||
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factorisationHasAllPrimeFactors : ∀ {as} {p} → Prime p → p ∣ product as → All Prime as → p ∈ as | ||
factorisationHasAllPrimeFactors {[]} {2+ p} pPrime p∣Πas [] = contradiction (∣1⇒≡1 p∣Πas) λ () | ||
factorisationHasAllPrimeFactors {a ∷ as} {p} pPrime p∣aΠas (aPrime ∷ asPrime) with euclidsLemma a (product as) pPrime p∣aΠas | ||
... | inj₂ p∣Πas = there (factorisationHasAllPrimeFactors pPrime p∣Πas asPrime) | ||
... | inj₁ p∣a with prime⇒irreducible aPrime p∣a | ||
... | inj₁ refl = contradiction pPrime ¬prime[1] | ||
... | inj₂ refl = here refl | ||
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private | ||
factorisationUnique′ : (as bs : List ℕ) → product as ≡ product bs → All Prime as → All Prime bs → as ↭ bs | ||
factorisationUnique′ [] [] Πas≡Πbs asPrime bsPrime = refl | ||
factorisationUnique′ [] (b@(2+ _) ∷ bs) Πas≡Πbs prime[as] (_ ∷ prime[bs]) = | ||
contradiction Πas≡Πbs (<⇒≢ Πas<Πbs) | ||
where | ||
Πas<Πbs : product [] < product (b ∷ bs) | ||
Πas<Πbs = begin-strict | ||
1 ≡⟨⟩ | ||
1 * 1 <⟨ *-monoˡ-< 1 {1} {b} sz<ss ⟩ | ||
b * 1 ≤⟨ *-monoʳ-≤ b (productOfPrimes≥1 prime[bs]) ⟩ | ||
b * product bs ≡⟨⟩ | ||
product (b ∷ bs) ∎ | ||
where open ≤-Reasoning | ||
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factorisationUnique′ (a ∷ as) bs Πas≡Πbs (prime[a] ∷ prime[as]) prime[bs] = a∷as↭bs | ||
where | ||
a∣Πbs : a ∣ product bs | ||
a∣Πbs = divides (product as) $ begin | ||
product bs ≡⟨ Πas≡Πbs ⟨ | ||
product (a ∷ as) ≡⟨⟩ | ||
a * product as ≡⟨ *-comm a (product as) ⟩ | ||
product as * a ∎ | ||
where open ≡-Reasoning | ||
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shuffle : ∃[ bs′ ] bs ↭ a ∷ bs′ | ||
shuffle with ys , zs , p ← ∈-∃++ (factorisationHasAllPrimeFactors prime[a] a∣Πbs prime[bs]) | ||
= ys ++ zs , ↭-trans (↭-reflexive p) (shift a ys zs) | ||
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bs′ = proj₁ shuffle | ||
bs↭a∷bs′ = proj₂ shuffle | ||
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Πas≡Πbs′ : product as ≡ product bs′ | ||
Πas≡Πbs′ = *-cancelˡ-≡ (product as) (product bs′) a {{prime⇒nonZero prime[a]}} $ begin | ||
a * product as ≡⟨ Πas≡Πbs ⟩ | ||
product bs ≡⟨ product-↭ bs↭a∷bs′ ⟩ | ||
a * product bs′ ∎ | ||
where open ≡-Reasoning | ||
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prime[bs'] : All Prime bs′ | ||
prime[bs'] = All.tail (All-resp-↭ bs↭a∷bs′ prime[bs]) | ||
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a∷as↭bs : a ∷ as ↭ bs | ||
a∷as↭bs = begin | ||
a ∷ as <⟨ factorisationUnique′ as bs′ Πas≡Πbs′ prime[as] prime[bs'] ⟩ | ||
a ∷ bs′ ↭⟨ bs↭a∷bs′ ⟨ | ||
bs ∎ | ||
where open PermutationReasoning | ||
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factorisationUnique : (f f′ : PrimeFactorisation n) → factors f ↭ factors f′ | ||
factorisationUnique {n} f f′ = | ||
factorisationUnique′ (factors f) (factors f′) Πf≡Πf′ (factorsPrime f) (factorsPrime f′) | ||
where | ||
Πf≡Πf′ : product (factors f) ≡ product (factors f′) | ||
Πf≡Πf′ = begin | ||
product (factors f) ≡⟨ isFactorisation f ⟨ | ||
n ≡⟨ isFactorisation f′ ⟩ | ||
product (factors f′) ∎ | ||
where open ≡-Reasoning |