feat: port Data.Nat.Factors (#1664)

Co-authored-by: qawbecrdtey <qawbecrdtey@kaist.ac.kr>
Co-authored-by: qawbecrdtey <qawbecrdtey@naver.com>
Co-authored-by: ChrisHughes24 <chrishughes24@gmail.com>

Mathlib.lean

@@ -375,6 +375,7 @@ import Mathlib.Data.Nat.Choose.Dvd
 import Mathlib.Data.Nat.Dist
 import Mathlib.Data.Nat.EvenOddRec
 import Mathlib.Data.Nat.Factorial.Basic
+import Mathlib.Data.Nat.Factors
 import Mathlib.Data.Nat.Fib
 import Mathlib.Data.Nat.ForSqrt
 import Mathlib.Data.Nat.GCD.Basic

Mathlib/Data/Nat/Factors.lean

@@ -0,0 +1,313 @@
+/-
+Copyright (c) 2015 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
+
+! This file was ported from Lean 3 source module data.nat.factors
+! leanprover-community/mathlib commit 008205aa645b3f194c1da47025c5f110c8406eab
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Data.Nat.Prime
+import Mathlib.Data.List.Prime
+import Mathlib.Data.List.Sort
+import Mathlib.Tactic.NthRewrite
+
+/-!
+# Prime numbers
+
+This file deals with the factors of natural numbers.
+
+## Important declarations
+
+- `Nat.factors n`: the prime factorization of `n`
+- `Nat.factors_unique`: uniqueness of the prime factorisation
+
+-/
+
+set_option autoImplicit false
+
+open Bool Subtype
+
+open Nat
+
+namespace Nat
+
+attribute [instance 0] instBEqNat
+
+/-- `factors n` is the prime factorization of `n`, listed in increasing order. -/
+def factors : ℕ → List ℕ
+  | 0 => []
+  | 1 => []
+  | k + 2 =>
+    let m := minFac (k + 2)
+    have : (k + 2) / m < (k + 2) := factors_lemma
+    m :: factors ((k + 2) / m)
+#align nat.factors Nat.factors
+
+@[simp]
+theorem factors_zero : factors 0 = [] := by rw [factors]
+#align nat.factors_zero Nat.factors_zero
+
+@[simp]
+theorem factors_one : factors 1 = [] := by rw [factors]
+#align nat.factors_one Nat.factors_one
+
+theorem prime_of_mem_factors {n : ℕ} : ∀ {p : ℕ}, (h : p ∈ factors n) → Prime p := by
+  match n with
+  | 0 => simp
+  | 1 => simp
+  | k + 2 =>
+      intro p h
+      let m := minFac (k + 2)
+      have : (k + 2) / m < (k + 2) := factors_lemma
+      have h₁ : p = m ∨ p ∈ factors ((k + 2) / m) :=
+        List.mem_cons.1 (by rwa [factors] at h)
+      exact Or.casesOn h₁ (fun h₂ => h₂.symm ▸ minFac_prime (by simp)) prime_of_mem_factors
+#align nat.prime_of_mem_factors Nat.prime_of_mem_factors
+
+theorem pos_of_mem_factors {n p : ℕ} (h : p ∈ factors n) : 0 < p :=
+  Prime.pos (prime_of_mem_factors h)
+#align nat.pos_of_mem_factors Nat.pos_of_mem_factors
+
+theorem prod_factors : ∀ {n}, n ≠ 0 → List.prod (factors n) = n
+  | 0 => by simp
+  | 1 => by simp
+  | k + 2 => fun _ =>
+    let m := minFac (k + 2)
+    have : (k + 2) / m < (k + 2) := factors_lemma
+    show (factors (k + 2)).prod = (k + 2) by
+      have h₁ : (k + 2) / m ≠ 0 := fun h => by
+        have : (k + 2) = 0 * m := (Nat.div_eq_iff_eq_mul_left (minFac_pos _) (minFac_dvd _)).1 h
+        rw [zero_mul] at this; exact (show k + 2 ≠ 0 by simp) this
+      rw [factors, List.prod_cons, prod_factors h₁, Nat.mul_div_cancel' (minFac_dvd _)]
+#align nat.prod_factors Nat.prod_factors
+
+theorem factors_prime {p : ℕ} (hp : Nat.Prime p) : p.factors = [p] := by
+  have : p = p - 2 + 2 := (tsub_eq_iff_eq_add_of_le hp.two_le).mp rfl
+  rw [this, Nat.factors]
+  simp only [Eq.symm this]
+  have : Nat.minFac p = p := (Nat.prime_def_minFac.mp hp).2
+  simp only [this, Nat.factors, Nat.div_self (Nat.Prime.pos hp)]
+#align nat.factors_prime Nat.factors_prime
+
+theorem factors_chain {n : ℕ} :
+    ∀ {a}, (∀ p, Prime p → p ∣ n → a ≤ p) → List.Chain (· ≤ ·) a (factors n) := by
+  match n with
+  | 0 => simp
+  | 1 => simp
+  | k + 2 =>
+      intro a h
+      let m := minFac (k + 2)
+      have : (k + 2) / m < (k + 2) := factors_lemma
+      rw [factors]
+      refine' List.Chain.cons ((le_minFac.2 h).resolve_left (by simp)) (factors_chain _)
+      exact fun p pp d => minFac_le_of_dvd pp.two_le (d.trans <| div_dvd_of_dvd <| minFac_dvd _)
+#align nat.factors_chain Nat.factors_chain
+
+theorem factors_chain_2 (n) : List.Chain (· ≤ ·) 2 (factors n) :=
+  factors_chain fun _ pp _ => pp.two_le
+#align nat.factors_chain_2 Nat.factors_chain_2
+
+theorem factors_chain' (n) : List.Chain' (· ≤ ·) (factors n) :=
+  @List.Chain'.tail _ _ (_ :: _) (factors_chain_2 _)
+#align nat.factors_chain' Nat.factors_chain'
+
+theorem factors_sorted (n : ℕ) : List.Sorted (· ≤ ·) (factors n) :=
+  List.chain'_iff_pairwise.1 (factors_chain' _)
+#align nat.factors_sorted Nat.factors_sorted
+
+/-- `factors` can be constructed inductively by extracting `minFac`, for sufficiently large `n`. -/
+theorem factors_add_two (n : ℕ) :
+    factors (n + 2) = minFac (n + 2) :: factors ((n + 2) / minFac (n + 2)) := by rw [factors]
+#align nat.factors_add_two Nat.factors_add_two
+
+@[simp]
+theorem factors_eq_nil (n : ℕ) : n.factors = [] ↔ n = 0 ∨ n = 1 := by
+  constructor <;> intro h
+  · rcases n with (_ | _ | n)
+    · exact Or.inl rfl
+    · exact Or.inr rfl
+    · rw [factors] at h
+      injection h
+  · rcases h with (rfl | rfl)
+    · exact factors_zero
+    · exact factors_one
+#align nat.factors_eq_nil Nat.factors_eq_nil
+
+theorem eq_of_perm_factors {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) (h : a.factors ~ b.factors) :
+    a = b := by simpa [prod_factors ha, prod_factors hb] using List.Perm.prod_eq h
+#align nat.eq_of_perm_factors Nat.eq_of_perm_factors
+
+section
+
+open List
+
+theorem mem_factors_iff_dvd {n p : ℕ} (hn : n ≠ 0) (hp : Prime p) : p ∈ factors n ↔ p ∣ n :=
+  ⟨fun h => prod_factors hn ▸ List.dvd_prod h, fun h =>
+    mem_list_primes_of_dvd_prod (prime_iff.mp hp) (fun _ h => prime_iff.mp (prime_of_mem_factors h))
+      ((prod_factors hn).symm ▸ h)⟩
+#align nat.mem_factors_iff_dvd Nat.mem_factors_iff_dvd
+
+theorem dvd_of_mem_factors {n p : ℕ} (h : p ∈ n.factors) : p ∣ n := by
+  rcases n.eq_zero_or_pos with (rfl | hn)
+  · exact dvd_zero p
+  · rwa [← mem_factors_iff_dvd hn.ne' (prime_of_mem_factors h)]
+#align nat.dvd_of_mem_factors Nat.dvd_of_mem_factors
+
+theorem mem_factors {n p} (hn : n ≠ 0) : p ∈ factors n ↔ Prime p ∧ p ∣ n :=
+  ⟨fun h => ⟨prime_of_mem_factors h, dvd_of_mem_factors h⟩, fun ⟨hprime, hdvd⟩ =>
+    (mem_factors_iff_dvd hn hprime).mpr hdvd⟩
+#align nat.mem_factors Nat.mem_factors
+
+theorem le_of_mem_factors {n p : ℕ} (h : p ∈ n.factors) : p ≤ n := by
+  rcases n.eq_zero_or_pos with (rfl | hn)
+  · rw [factors_zero] at h
+    cases h
+  · exact le_of_dvd hn (dvd_of_mem_factors h)
+#align nat.le_of_mem_factors Nat.le_of_mem_factors
+
+/-- **Fundamental theorem of arithmetic**-/
+theorem factors_unique {n : ℕ} {l : List ℕ} (h₁ : prod l = n) (h₂ : ∀ p ∈ l, Prime p) :
+    l ~ factors n := by
+  refine' perm_of_prod_eq_prod _ _ _
+  · rw [h₁]
+    refine' (prod_factors _).symm
+    rintro rfl
+    rw [prod_eq_zero_iff] at h₁
+    exact Prime.ne_zero (h₂ 0 h₁) rfl
+  · simp_rw [← prime_iff]
+    exact h₂
+  · simp_rw [← prime_iff]
+    exact fun p => prime_of_mem_factors
+#align nat.factors_unique Nat.factors_unique
+
+theorem Prime.factors_pow {p : ℕ} (hp : p.Prime) (n : ℕ) :
+    (p ^ n).factors = List.replicate n p := by
+  symm
+  rw [← List.replicate_perm]
+  apply Nat.factors_unique (List.prod_replicate n p)
+  intro q hq
+  rwa [eq_of_mem_replicate hq]
+#align nat.prime.factors_pow Nat.Prime.factors_pow
+
+theorem eq_prime_pow_of_unique_prime_dvd {n p : ℕ} (hpos : n ≠ 0)
+    (h : ∀ {d}, Nat.Prime d → d ∣ n → d = p) : n = p ^ n.factors.length := by
+  set k := n.factors.length
+  rw [← prod_factors hpos, ← prod_replicate k p,
+    eq_replicate_of_mem fun d hd => h (prime_of_mem_factors hd) (dvd_of_mem_factors hd)]
+#align nat.eq_prime_pow_of_unique_prime_dvd Nat.eq_prime_pow_of_unique_prime_dvd
+
+/-- For positive `a` and `b`, the prime factors of `a * b` are the union of those of `a` and `b` -/
+theorem perm_factors_mul {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) :
+    (a * b).factors ~ a.factors ++ b.factors := by
+  refine' (factors_unique _ _).symm
+  · rw [List.prod_append, prod_factors ha, prod_factors hb]
+  · intro p hp
+    rw [List.mem_append] at hp
+    cases' hp with hp' hp' <;> exact prime_of_mem_factors hp'
+
+#align nat.perm_factors_mul Nat.perm_factors_mul
+
+/-- For coprime `a` and `b`, the prime factors of `a * b` are the union of those of `a` and `b` -/
+theorem perm_factors_mul_of_coprime {a b : ℕ} (hab : coprime a b) :
+    (a * b).factors ~ a.factors ++ b.factors := by
+  rcases a.eq_zero_or_pos with (rfl | ha)
+  · simp [(coprime_zero_left _).mp hab]
+  rcases b.eq_zero_or_pos with (rfl | hb)
+  · simp [(coprime_zero_right _).mp hab]
+  exact perm_factors_mul ha.ne' hb.ne'
+#align nat.perm_factors_mul_of_coprime Nat.perm_factors_mul_of_coprime
+
+theorem factors_sublist_right {n k : ℕ} (h : k ≠ 0) : n.factors <+ (n * k).factors := by
+  cases' n with hn
+  · simp [zero_mul]
+  apply sublist_of_subperm_of_sorted _ (factors_sorted _) (factors_sorted _)
+  simp [(perm_factors_mul (Nat.succ_ne_zero _) h).subperm_left]
+  exact (sublist_append_left _ _).subperm
+#align nat.factors_sublist_right Nat.factors_sublist_right
+
+theorem factors_sublist_of_dvd {n k : ℕ} (h : n ∣ k) (h' : k ≠ 0) : n.factors <+ k.factors := by
+  obtain ⟨a, rfl⟩ := h
+  exact factors_sublist_right (right_ne_zero_of_mul h')
+#align nat.factors_sublist_of_dvd Nat.factors_sublist_of_dvd
+
+theorem factors_subset_right {n k : ℕ} (h : k ≠ 0) : n.factors ⊆ (n * k).factors :=
+  (factors_sublist_right h).subset
+#align nat.factors_subset_right Nat.factors_subset_right
+
+theorem factors_subset_of_dvd {n k : ℕ} (h : n ∣ k) (h' : k ≠ 0) : n.factors ⊆ k.factors :=
+  (factors_sublist_of_dvd h h').subset
+#align nat.factors_subset_of_dvd Nat.factors_subset_of_dvd
+
+theorem dvd_of_factors_subperm {a b : ℕ} (ha : a ≠ 0) (h : a.factors <+~ b.factors) : a ∣ b := by
+  rcases b.eq_zero_or_pos with (rfl | hb)
+  · exact dvd_zero _
+  rcases a with (_ | _ | a)
+  · exact (ha rfl).elim
+  · exact one_dvd _
+  --Porting note: previous proof
+  --use (b.factors.diff a.succ.succ.factors).prod
+  use (@List.diff _ instBEq b.factors a.succ.succ.factors).prod
+  nth_rw 1 [← Nat.prod_factors ha]
+  rw [← List.prod_append,
+    List.Perm.prod_eq <| List.subperm_append_diff_self_of_count_le <| List.subperm_ext_iff.mp h,
+    Nat.prod_factors hb.ne']
+#align nat.dvd_of_factors_subperm Nat.dvd_of_factors_subperm
+
+end
+
+theorem mem_factors_mul {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) {p : ℕ} :
+    p ∈ (a * b).factors ↔ p ∈ a.factors ∨ p ∈ b.factors := by
+  rw [mem_factors (mul_ne_zero ha hb), mem_factors ha, mem_factors hb, ← and_or_left]
+  simpa only [and_congr_right_iff] using Prime.dvd_mul
+#align nat.mem_factors_mul Nat.mem_factors_mul
+
+/-- The sets of factors of coprime `a` and `b` are disjoint -/
+theorem coprime_factors_disjoint {a b : ℕ} (hab : a.coprime b) :
+    List.Disjoint a.factors b.factors := by
+  intro q hqa hqb
+  apply not_prime_one
+  rw [← eq_one_of_dvd_coprimes hab (dvd_of_mem_factors hqa) (dvd_of_mem_factors hqb)]
+  exact prime_of_mem_factors hqa
+#align nat.coprime_factors_disjoint Nat.coprime_factors_disjoint
+
+theorem mem_factors_mul_of_coprime {a b : ℕ} (hab : coprime a b) (p : ℕ) :
+    p ∈ (a * b).factors ↔ p ∈ a.factors ∪ b.factors := by
+  rcases a.eq_zero_or_pos with (rfl | ha)
+  · simp [(coprime_zero_left _).mp hab]
+  rcases b.eq_zero_or_pos with (rfl | hb)
+  · simp [(coprime_zero_right _).mp hab]
+  rw [mem_factors_mul ha.ne' hb.ne', List.mem_union]
+#align nat.mem_factors_mul_of_coprime Nat.mem_factors_mul_of_coprime
+
+open List
+
+/-- If `p` is a prime factor of `a` then `p` is also a prime factor of `a * b` for any `b > 0` -/
+theorem mem_factors_mul_left {p a b : ℕ} (hpa : p ∈ a.factors) (hb : b ≠ 0) : p ∈ (a * b).factors :=
+  by
+  rcases eq_or_ne a 0 with (rfl | ha)
+  · simp at hpa
+  apply (mem_factors_mul ha hb).2 (Or.inl hpa)
+#align nat.mem_factors_mul_left Nat.mem_factors_mul_left
+
+/-- If `p` is a prime factor of `b` then `p` is also a prime factor of `a * b` for any `a > 0` -/
+theorem mem_factors_mul_right {p a b : ℕ} (hpb : p ∈ b.factors) (ha : a ≠ 0) :
+    p ∈ (a * b).factors := by
+  rw [mul_comm]
+  exact mem_factors_mul_left hpb ha
+#align nat.mem_factors_mul_right Nat.mem_factors_mul_right
+
+theorem eq_two_pow_or_exists_odd_prime_and_dvd (n : ℕ) :
+    (∃ k : ℕ, n = 2 ^ k) ∨ ∃ p, Nat.Prime p ∧ p ∣ n ∧ Odd p :=
+  (eq_or_ne n 0).elim (fun hn => Or.inr ⟨3, prime_three, hn.symm ▸ dvd_zero 3, ⟨1, rfl⟩⟩) fun hn =>
+    or_iff_not_imp_right.mpr fun H =>
+      ⟨n.factors.length,
+        eq_prime_pow_of_unique_prime_dvd hn fun {_} hprime hdvd =>
+          hprime.eq_two_or_odd'.resolve_right fun hodd => H ⟨_, hprime, hdvd, hodd⟩⟩
+#align nat.eq_two_pow_or_exists_odd_prime_and_dvd Nat.eq_two_pow_or_exists_odd_prime_and_dvd
+
+end Nat
+
+-- Porting note: `assert_not_exists` is not implemented yet.
+--assert_not_exists Multiset