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α : Type u_1 β : Type u_2 inst✝² : CommMonoidWithZero α inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1 inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a b : α h : ¬Finite a b n : ℕ ⊢ a ^ n ∣ b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n
theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk]
Mathlib.RingTheory.Multiplicity.382_0.uTHZeAJqYiw3Jx8
theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Semiring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≤ multiplicity p b ⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff]
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by
Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Semiring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≤ multiplicity p b ⊢ ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ a + b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff];
Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Semiring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b ≤ multiplicity p a ⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by
rw [min_eq_right h, multiplicity_le_multiplicity_iff]
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by
Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Semiring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b ≤ multiplicity p a ⊢ ∀ (n : ℕ), p ^ n ∣ b → p ^ n ∣ a + b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff];
Mathlib.RingTheory.Multiplicity.409_0.uTHZeAJqYiw3Jx8
theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a b : α ⊢ (multiplicity a (-b)).Dom ↔ (multiplicity a b).Dom
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by
simp only [multiplicity, PartENat.find, dvd_neg]
@[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by
Mathlib.RingTheory.Multiplicity.426_0.uTHZeAJqYiw3Jx8
@[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a b : α h₁ : (multiplicity a (-b)).Dom h₂ : (multiplicity a b).Dom ⊢ ↑(Part.get (multiplicity a (-b)) h₁) = ↑(Part.get (multiplicity a b) h₂)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
@[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by
Mathlib.RingTheory.Multiplicity.426_0.uTHZeAJqYiw3Jx8
@[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a b : α h₁ : (multiplicity a (-b)).Dom h₂ : (multiplicity a b).Dom ⊢ multiplicity a (-b) = ↑(Part.get (multiplicity a b) h₂)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get]
exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))
@[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get]
Mathlib.RingTheory.Multiplicity.426_0.uTHZeAJqYiw3Jx8
@[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ ⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
case inl α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = ↑(_root_.Int.natAbs b) ⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;>
conv_rhs => rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = ↑(_root_.Int.natAbs b) | multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = ↑(_root_.Int.natAbs b) | multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = ↑(_root_.Int.natAbs b) | multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
case inr α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = -↑(_root_.Int.natAbs b) ⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;>
conv_rhs => rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = -↑(_root_.Int.natAbs b) | multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = -↑(_root_.Int.natAbs b) | multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = -↑(_root_.Int.natAbs b) | multiplicity (↑a) b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
rw [h]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs =>
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
case inl α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = ↑(_root_.Int.natAbs b) ⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity ↑a ↑(_root_.Int.natAbs b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] ·
rw [Int.coe_nat_multiplicity]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] ·
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
case inr α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : ℕ b : ℤ h : b = -↑(_root_.Int.natAbs b) ⊢ multiplicity a (_root_.Int.natAbs b) = multiplicity (↑a) (-↑(_root_.Int.natAbs b))
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] ·
rw [multiplicity.neg, Int.coe_nat_multiplicity]
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] ·
Mathlib.RingTheory.Multiplicity.436_0.uTHZeAJqYiw3Jx8
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p (a + b) = multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p (a + b) ≤ multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm ·
apply PartENat.le_of_lt_add_one
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm ·
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a.h α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p (a + b) < multiplicity p b + 1
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a.h.intro α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k ⊢ multiplicity p (a + b) < multiplicity p b + 1
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a.h.intro α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k ⊢ multiplicity p (a + b) < ↑k + 1
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk]
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a.h.intro α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k ⊢ ¬p ^ (k + 1) ∣ b case a.h.intro α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k ⊢ p ^ (k + 1) ∣ a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
intro h_dvd
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a.h.intro α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k h_dvd : p ^ (k + 1) ∣ b ⊢ False
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd ·
apply multiplicity.is_greatest _ h_dvd
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd ·
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k h_dvd : p ^ (k + 1) ∣ b ⊢ multiplicity p b < ↑(k + 1)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd
rw [hk, ← Nat.succ_eq_add_one]
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k h_dvd : p ^ (k + 1) ∣ b ⊢ ↑k < ↑(succ k)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one]
norm_cast
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one]
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k h_dvd : p ^ (k + 1) ∣ b ⊢ k < succ k
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast
apply Nat.lt_succ_self k
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a.h.intro α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k ⊢ p ^ (k + 1) ∣ a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k ·
rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k ·
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a.h.intro α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a k : ℕ hk : multiplicity p b = ↑k ⊢ multiplicity p b + 1 ≤ multiplicity p a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
exact PartENat.add_one_le_of_lt h
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p b ≤ multiplicity p (a + b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h ·
have := @min_le_multiplicity_add α _ _ p a b
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h ·
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
case a α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a this : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) ⊢ multiplicity p b ≤ multiplicity p (a + b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b
rwa [← min_eq_right (le_of_lt h)]
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b
Mathlib.RingTheory.Multiplicity.442_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p (a - b) = multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt]
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by
Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p (-b) = multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;>
rw [multiplicity.neg]
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;>
Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p (-b) < multiplicity p a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;>
rw [multiplicity.neg]
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;>
Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p b < multiplicity p a ⊢ multiplicity p b < multiplicity p a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg];
assumption
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg];
Mathlib.RingTheory.Multiplicity.460_0.uTHZeAJqYiw3Jx8
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≠ multiplicity p b ⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
Mathlib_RingTheory_Multiplicity
case inl α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≠ multiplicity p b hab : multiplicity p a < multiplicity p b ⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) ·
rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) ·
Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
Mathlib_RingTheory_Multiplicity
case inl α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≠ multiplicity p b hab : multiplicity p a < multiplicity p b ⊢ multiplicity p a ≤ multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
Mathlib_RingTheory_Multiplicity
case inr.inl α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≠ multiplicity p b hab : multiplicity p a = multiplicity p b ⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab ·
contradiction
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab ·
Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
Mathlib_RingTheory_Multiplicity
case inr.inr α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≠ multiplicity p b hab : multiplicity p b < multiplicity p a ⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction ·
rw [multiplicity_add_of_gt hab, min_eq_right]
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction ·
Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
Mathlib_RingTheory_Multiplicity
case inr.inr α : Type u_1 β : Type u_2 inst✝¹ : Ring α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α h : multiplicity p a ≠ multiplicity p b hab : multiplicity p b < multiplicity p a ⊢ multiplicity p b ≤ multiplicity p a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right]
Mathlib.RingTheory.Multiplicity.465_0.uTHZeAJqYiw3Jx8
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝ : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s ⊢ a * b = p * (p ^ (n + m) * s)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by
simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝¹ : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝ : p ∣ a x : α hx : a = p * x hn0 : n = 0 ⊢ False
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by
simp [hx, hn0] at ha
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝² : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝¹ : p ∣ a x : α hx : a = p * x hn0 : 0 < n x✝ : p ^ (n - 1 + 1) ∣ x y : α hy : x = p ^ (n - 1 + 1) * y ⊢ p * x * p = p ^ (n + 1) * y * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝² : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝¹ : p ∣ a x : α hx : a = p * x hn0 : 0 < n x✝ : p ^ (n - 1 + 1) ∣ x y : α hy : x = p ^ n * y ⊢ p * x * p = p ^ (n + 1) * y * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝¹ : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this✝ : p ∣ a * b x✝ : p ∣ a x : α hx : a = p * x hn0 : 0 < n hpx : ¬p ^ (n - 1 + 1) ∣ x this : 1 ≤ n + m ⊢ x * b * p = p ^ (n - 1 + m + 1) * s * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝¹ : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this✝ : p ∣ a * b x✝ : p ∣ a x : α hx : a = p * x hn0 : 0 < n hpx : ¬p ^ (n - 1 + 1) ∣ x this : 1 ≤ n + m ⊢ x * b * p = p ^ (n + m) * s * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝¹ : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝ : p ∣ b x : α hx : b = p * x hm0 : m = 0 ⊢ False
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by
simp [hx, hm0] at hb
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝² : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝¹ : p ∣ b x : α hx : b = p * x hm0 : 0 < m x✝ : p ^ (m - 1 + 1) ∣ x y : α hy : x = p ^ (m - 1 + 1) * y ⊢ p * x * p = p ^ (m + 1) * y * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝² : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝¹ : p ∣ b x : α hx : b = p * x hm0 : 0 < m x✝ : p ^ (m - 1 + 1) ∣ x y : α hy : x = p ^ m * y ⊢ p * x * p = p ^ (m + 1) * y * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝¹ : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝ : p ∣ b x : α hx : b = p * x hm0 : 0 < m hpx : ¬p ^ (m - 1 + 1) ∣ x ⊢ a * x * p = p ^ (n + (m - 1) + 1) * s * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p : α hp : Prime p a b : α n m : ℕ ha : ¬p ^ (n + 1) ∣ a hb : ¬p ^ (m + 1) ∣ b x✝¹ : p ^ (n + m + 1) ∣ a * b s : α hs : a * b = p ^ (n + m + 1) * s this : p ∣ a * b x✝ : p ∣ b x : α hx : b = p * x hm0 : 0 < m hpx : ¬p ^ (m - 1 + 1) ∣ x ⊢ a * x * p = p ^ (n + m) * s * p
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
Mathlib.RingTheory.Multiplicity.483_0.uTHZeAJqYiw3Jx8
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p a : α hp : Prime p x✝ : Finite p a ⊢ ¬p ^ (0 + 1) ∣ a ^ 0
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by
simp [mt isUnit_iff_dvd_one.2 hp.2.1]
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by
Mathlib.RingTheory.Multiplicity.526_0.uTHZeAJqYiw3Jx8
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p a : α hp : Prime p k : ℕ ha : Finite p a ⊢ Finite p (a ^ (k + 1))
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by
rw [_root_.pow_succ]
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by
Mathlib.RingTheory.Multiplicity.526_0.uTHZeAJqYiw3Jx8
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝ : CancelCommMonoidWithZero α p a : α hp : Prime p k : ℕ ha : Finite p a ⊢ Finite p (a * a ^ k)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ];
exact finite_mul hp ha (finite_pow hp ha)
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ];
Mathlib.RingTheory.Multiplicity.526_0.uTHZeAJqYiw3Jx8
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : ¬IsUnit a ha0 : a ≠ 0 ⊢ multiplicity a a = 1
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : ¬IsUnit a ha0 : a ≠ 0 ⊢ multiplicity a a = ↑1
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one]
Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : ¬IsUnit a ha0 : a ≠ 0 ⊢ a ^ 1 ∣ a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by
simp
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by
Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : ¬IsUnit a ha0 : a ≠ 0 x✝ : a ^ (1 + 1) ∣ a b : α hb : a = a ^ (1 + 1) * b ⊢ a * 1 = a * (a * b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by
simpa [_root_.pow_succ, mul_assoc] using hb
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by
Mathlib.RingTheory.Multiplicity.533_0.uTHZeAJqYiw3Jx8
@[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : Finite a a ⊢ a ^ 1 ∣ a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by
simp
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by
Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : Finite a a x✝ : a ^ (1 + 1) ∣ a b : α hb : a = a ^ (1 + 1) * b ⊢ False
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by
Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : Finite a a x✝ : a ^ (1 + 1) ∣ a b : α hb : 1 = 1 * (a * 1 * b) ⊢ False
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb;
Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 a : α ha : Finite a a x✝ : a ^ (1 + 1) ∣ a b : α hb : 1 = 1 * (a * 1 * b) ⊢ 1 = a * b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by
simp_all
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by
Mathlib.RingTheory.Multiplicity.540_0.uTHZeAJqYiw3Jx8
@[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom ⊢ Part.get (multiplicity p (a * b)) h = Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a ⊢ Part.get (multiplicity p (a * b)) h = Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b ⊢ Part.get (multiplicity p (a * b)) h = Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add]
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b ⊢ p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
simp [pow_add]
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) ⊢ Part.get (multiplicity p (a * b)) h = Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add]
have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add]
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) ⊢ p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) ∣ a * b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by
rw [hpoweq]
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) ⊢ p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ a * b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq];
apply mul_dvd_mul
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq];
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
case a α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) ⊢ p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;>
assumption
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;>
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
case a α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) ⊢ p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;>
assumption
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;>
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) hdiv : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) ∣ a * b ⊢ Part.get (multiplicity p (a * b)) h = Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption
have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) hdiv : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) ∣ a * b hsucc : ¬p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b) + 1) ∣ a * b ⊢ Part.get (multiplicity p (a * b)) h = Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : (multiplicity p (a * b)).Dom hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b hpoweq : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) = p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) hdiv : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) ∣ a * b hsucc : ¬p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b) + 1) ∣ a * b ⊢ p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) ∣ a * b ∧ ¬p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b) + 1) ∣ a * b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff];
exact ⟨hdiv, hsucc⟩
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff];
Mathlib.RingTheory.Multiplicity.551_0.uTHZeAJqYiw3Jx8
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : Finite p a ∧ Finite p b ⊢ multiplicity p (a * b) = multiplicity p a + multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp]
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by
Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : ¬(Finite p a ∧ Finite p b) ⊢ multiplicity p (a * b) = multiplicity p a + multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by
Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h : ¬(Finite p a ∧ Finite p b) ⊢ ⊤ = multiplicity p a + multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
cases' not_and_or.1 h with h h
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b
Mathlib_RingTheory_Multiplicity
case inl α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h✝ : ¬(Finite p a ∧ Finite p b) h : ¬Finite p a ⊢ ⊤ = multiplicity p a + multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;>
simp [eq_top_iff_not_finite.2 h]
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;>
Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b
Mathlib_RingTheory_Multiplicity
case inr α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a b : α hp : Prime p h✝ : ¬(Finite p a ∧ Finite p b) h : ¬Finite p b ⊢ ⊤ = multiplicity p a + multiplicity p b
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;>
simp [eq_top_iff_not_finite.2 h]
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;>
Mathlib.RingTheory.Multiplicity.581_0.uTHZeAJqYiw3Jx8
protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b
Mathlib_RingTheory_Multiplicity
α : Type u_1 β✝ : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 β : Type u_3 p : α hp : Prime p s : Finset β f : β → α ⊢ multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β✝ : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 β : Type u_3 p : α hp : Prime p s : Finset β f : β → α ⊢ multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical
induction' s using Finset.induction with a s has ih h
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical
Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
Mathlib_RingTheory_Multiplicity
case empty α : Type u_1 β✝ : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 β : Type u_3 p : α hp : Prime p f : β → α ⊢ multiplicity p (∏ x in ∅, f x) = ∑ x in ∅, multiplicity p (f x)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h ·
simp only [Finset.sum_empty, Finset.prod_empty]
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h ·
Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
Mathlib_RingTheory_Multiplicity
case empty α : Type u_1 β✝ : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 β : Type u_3 p : α hp : Prime p f : β → α ⊢ multiplicity p 1 = 0
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty]
Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
Mathlib_RingTheory_Multiplicity
case insert α : Type u_1 β✝ : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 β : Type u_3 p : α hp : Prime p f : β → α a : β s : Finset β has : a ∉ s ih : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) ⊢ multiplicity p (∏ x in insert a s, f x) = ∑ x in insert a s, multiplicity p (f x)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit ·
simp [has, ← ih]
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit ·
Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
Mathlib_RingTheory_Multiplicity
case insert α : Type u_1 β✝ : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 β : Type u_3 p : α hp : Prime p f : β → α a : β s : Finset β has : a ∉ s ih : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) ⊢ multiplicity p (f a * ∏ x in s, f x) = multiplicity p (f a) + multiplicity p (∏ x in s, f x)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih]
convert multiplicity.mul hp
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih]
Mathlib.RingTheory.Multiplicity.593_0.uTHZeAJqYiw3Jx8
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x)
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p ha : Finite p a ⊢ ∀ {k : ℕ}, Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p ha : Finite p a k : ℕ ⊢ Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k
induction' k with k hk
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k
Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
Mathlib_RingTheory_Multiplicity
case zero α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p ha : Finite p a ⊢ Part.get (multiplicity p (a ^ zero)) (_ : Finite p (a ^ zero)) = zero * Part.get (multiplicity p a) ha
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk ·
simp [one_right hp.not_unit]
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk ·
Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
Mathlib_RingTheory_Multiplicity
case succ α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p ha : Finite p a k : ℕ hk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha ⊢ Part.get (multiplicity p (a ^ succ k)) (_ : Finite p (a ^ succ k)) = succ k * Part.get (multiplicity p a) ha
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] ·
have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] ·
Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p ha : Finite p a k : ℕ hk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha ⊢ multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k)
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by
rw [_root_.pow_succ]
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by
Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
Mathlib_RingTheory_Multiplicity
case succ α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p ha : Finite p a k : ℕ hk : Part.get (multiplicity p (a ^ k)) (_ : Finite p (a ^ k)) = k * Part.get (multiplicity p a) ha this : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) ⊢ Part.get (multiplicity p (a ^ succ k)) (_ : Finite p (a ^ succ k)) = succ k * Part.get (multiplicity p a) ha
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
rw [succ_eq_add_one, get_eq_get_of_eq _ _ this, multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
Mathlib.RingTheory.Multiplicity.604_0.uTHZeAJqYiw3Jx8
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p ⊢ multiplicity p (a ^ 0) = 0 • multiplicity p a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ] rw [succ_eq_add_one, get_eq_get_of_eq _ _ this, multiplicity.mul' hp, hk, add_mul, one_mul, add_comm] #align multiplicity.pow' multiplicity.pow' theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by
simp [one_right hp.not_unit]
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by
Mathlib.RingTheory.Multiplicity.614_0.uTHZeAJqYiw3Jx8
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p a : α hp : Prime p k : ℕ ⊢ multiplicity p (a ^ succ k) = succ k • multiplicity p a
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ] rw [succ_eq_add_one, get_eq_get_of_eq _ _ this, multiplicity.mul' hp, hk, add_mul, one_mul, add_comm] #align multiplicity.pow' multiplicity.pow' theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by
simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by
Mathlib.RingTheory.Multiplicity.614_0.uTHZeAJqYiw3Jx8
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p : α h0 : p ≠ 0 hu : ¬IsUnit p n : ℕ ⊢ multiplicity p (p ^ n) = ↑n
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ] rw [succ_eq_add_one, get_eq_get_of_eq _ _ this, multiplicity.mul' hp, hk, add_mul, one_mul, add_comm] #align multiplicity.pow' multiplicity.pow' theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp] #align multiplicity.pow multiplicity.pow theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by
Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n
Mathlib_RingTheory_Multiplicity
α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p : α h0 : p ≠ 0 hu : ¬IsUnit p n : ℕ ⊢ p ^ n ∣ p ^ n ∧ ¬p ^ (n + 1) ∣ p ^ n
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ] rw [succ_eq_add_one, get_eq_get_of_eq _ _ this, multiplicity.mul' hp, hk, add_mul, one_mul, add_comm] #align multiplicity.pow' multiplicity.pow' theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp] #align multiplicity.pow multiplicity.pow theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by rw [eq_coe_iff]
use dvd_rfl
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by rw [eq_coe_iff]
Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n
Mathlib_RingTheory_Multiplicity
case right α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p : α h0 : p ≠ 0 hu : ¬IsUnit p n : ℕ ⊢ ¬p ^ (n + 1) ∣ p ^ n
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ] rw [succ_eq_add_one, get_eq_get_of_eq _ _ this, multiplicity.mul' hp, hk, add_mul, one_mul, add_comm] #align multiplicity.pow' multiplicity.pow' theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp] #align multiplicity.pow multiplicity.pow theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by rw [eq_coe_iff] use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by rw [eq_coe_iff] use dvd_rfl
Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n
Mathlib_RingTheory_Multiplicity
case right α : Type u_1 β : Type u_2 inst✝¹ : CancelCommMonoidWithZero α inst✝ : DecidableRel fun x x_1 => x ∣ x_1 p : α h0 : p ≠ 0 hu : ¬IsUnit p n : ℕ ⊢ ¬n + 1 ≤ n
/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Chris Hughes -/ import Mathlib.Algebra.Associated import Mathlib.Algebra.SMulWithZero import Mathlib.Data.Nat.PartENat import Mathlib.Tactic.Linarith #align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Multiplicity of a divisor For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it. ## Main definitions * `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers `n`. * `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite. -/ variable {α β : Type*} open Nat Part open BigOperators /-- `multiplicity a b` returns the largest natural number `n` such that `a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`, then it returns `⊤`-/ def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat := PartENat.find fun n => ¬a ^ (n + 1) ∣ b #align multiplicity multiplicity namespace multiplicity section Monoid variable [Monoid α] [Monoid β] /-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/ @[reducible] def Finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b #align multiplicity.finite multiplicity.Finite theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} : Finite a b ↔ (multiplicity a b).Dom := Iff.rfl #align multiplicity.finite_iff_dom multiplicity.finite_iff_dom theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := Iff.rfl #align multiplicity.finite_def multiplicity.finite_def theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ => hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ #align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right @[norm_cast] theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by apply Part.ext' · rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ] norm_cast · intro h1 h2 apply _root_.le_antisymm <;> · apply Nat.find_mono norm_cast simp #align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b := ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) (by simpa [Finite, Classical.not_not] using h), by simp [Finite, multiplicity, Classical.not_not]; tauto⟩ #align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a := let ⟨n, hn⟩ := h hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1) #align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ => ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩ #align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)] theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} : (k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by rw [← PartENat.some_eq_natCast] exact Nat.casesOn k (fun _ => by rw [_root_.pow_zero] exact one_dvd _) fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk #align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b := pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get]) #align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) #align multiplicity.is_greatest multiplicity.is_greatest theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) : ¬a ^ m ∣ b := is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm) #align multiplicity.is_greatest' multiplicity.is_greatest' theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin := by refine' zero_lt_iff.2 fun h => _ simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h) #align multiplicity.pos_of_dvd multiplicity.pos_of_dvd theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : (k : PartENat) = multiplicity a b := le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by have : Finite a b := ⟨k, hsucc⟩ rw [PartENat.le_coe_iff] exact ⟨this, Nat.find_min' _ hsucc⟩ #align multiplicity.unique multiplicity.unique theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) : k = get (multiplicity a b) ⟨k, hsucc⟩ := by rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc] #align multiplicity.unique' multiplicity.unique' theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) : (k : PartENat) ≤ multiplicity a b := le_of_not_gt fun hk' => is_greatest hk' hk #align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} : a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b := ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ #align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} : multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le] #align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd theorem eq_coe_iff {a b : α} {n : ℕ} : multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by rw [← PartENat.some_eq_natCast] exact ⟨fun h => let ⟨h₁, h₂⟩ := eq_some_iff.1 h h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by rw [PartENat.lt_coe_iff] exact ⟨h₁, lt_succ_self _⟩)⟩, fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩ #align multiplicity.eq_coe_iff multiplicity.eq_coe_iff theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b := (PartENat.find_eq_top_iff _).trans <| by simp only [Classical.not_not] exact ⟨fun h n => Nat.casesOn n (by rw [_root_.pow_zero] exact one_dvd _) fun n => h _, fun h n => h _⟩ #align multiplicity.eq_top_iff multiplicity.eq_top_iff @[simp] theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ := eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _) #align multiplicity.is_unit_left multiplicity.isUnit_left -- @[simp] Porting note: simp can prove this theorem one_left (b : α) : multiplicity 1 b = ⊤ := isUnit_left b isUnit_one #align multiplicity.one_left multiplicity.one_left @[simp] theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero] simp [not_dvd_one_of_finite_one_right ha] #align multiplicity.get_one_right multiplicity.get_one_right -- @[simp] Porting note: simp can prove this theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ := isUnit_left a u.isUnit #align multiplicity.unit_left multiplicity.unit_left theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by rw [← Nat.cast_zero, eq_coe_iff] simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and] #align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b := multiplicity_eq_zero.not_left #align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b := Part.eq_none_iff' #align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by rw [Ne.def, eq_top_iff_not_finite, Classical.not_not] #align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by rw [lt_top_iff_ne_top, ne_top_iff_finite] #align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) : ∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin refine' ⟨c, hc, _⟩ rintro ⟨k, hk⟩ rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩ exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁ #align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} : multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := ⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h => letI := Classical.dec (Finite a b) if hab : Finite a b then by rw [← PartENat.natCast_get (finite_iff_dom.1 hab)]; exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _)) else by have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _) rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩ #align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} : multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d := ⟨fun h n => ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩, fun h => le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp) (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩ #align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} : multiplicity a b ≤ multiplicity (f a) (f b) := multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} : multiplicity (f a) (f b) = multiplicity a b := multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) : multiplicity a b ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right theorem eq_of_associated_right {a b c : α} (h : Associated b c) : multiplicity a b = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd) (multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) #align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by rw [← pow_one a] apply pow_dvd_of_le_multiplicity simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h #align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b := ⟨dvd_of_multiplicity_pos, fun hdvd => lt_of_le_of_ne (zero_le _) fun heq => is_greatest (show multiplicity a b < ↑1 by simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one) (by rwa [pow_one a])⟩ #align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt, le_zero_iff] exact ⟨fun h => or_iff_not_imp_right.2 fun hb => have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1 Classical.by_contradiction fun ha1 : a ≠ 1 => have ha_gt_one : 1 < a := lt_of_not_ge fun _ => match a with | 0 => ha rfl | 1 => ha1 rfl | b+2 => by linarith not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b), fun h => by cases h <;> simp [*]⟩ #align multiplicity.finite_nat_iff multiplicity.finite_nat_iff alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos end Monoid section CommMonoid variable [CommMonoid α] theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by rw [mul_comm]; exact finite_of_finite_mul_right #align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 := eq_coe_iff.2 ⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by rw [pow_one] exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩ #align multiplicity.is_unit_right multiplicity.isUnit_right theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 := isUnit_right ha isUnit_one #align multiplicity.one_right multiplicity.one_right theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 := isUnit_right ha u.isUnit #align multiplicity.unit_right multiplicity.unit_right open Classical theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) : multiplicity b c ≤ multiplicity a c := multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h #align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left theorem eq_of_associated_left {a b c : α} (h : Associated a b) : multiplicity b c = multiplicity a c := le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd) (multiplicity_le_multiplicity_of_dvd_left h.symm.dvd) #align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left -- Porting note: this was doing nothing in mathlib3 also -- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos end CommMonoid section MonoidWithZero variable [MonoidWithZero α] theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 := let ⟨n, hn⟩ := h fun hb => by simp [hb] at hn #align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem zero (a : α) : multiplicity a 0 = ⊤ := Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _) #align multiplicity.zero multiplicity.zero @[simp] theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 := multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha #align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero end MonoidWithZero section CommMonoidWithZero variable [CommMonoidWithZero α] variable [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem multiplicity_mk_eq_multiplicity [DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} : multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by by_cases h : Finite a b · rw [← PartENat.natCast_get (finite_iff_dom.mp h)] refine' (multiplicity.unique (show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;> rw [← Associates.mk_pow, Associates.mk_dvd_mk] · exact pow_multiplicity_dvd h · exact is_greatest ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) · suffices ¬Finite (Associates.mk a) (Associates.mk b) by rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this rw [h, this] refine' not_finite_iff_forall.mpr fun n => by rw [← Associates.mk_pow, Associates.mk_dvd_mk] exact not_finite_iff_forall.mp h n #align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity end CommMonoidWithZero section Semiring variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] theorem min_le_multiplicity_add {p a b : α} : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) := (le_total (multiplicity p a) (multiplicity p b)).elim (fun h => by rw [min_eq_left h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn)) fun h => by rw [min_eq_right h, multiplicity_le_multiplicity_iff]; exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn #align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add end Semiring section Ring variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b := Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ => PartENat.natCast_inj.1 (by rw [PartENat.natCast_get] exact Eq.symm (unique (pow_multiplicity_dvd _).neg_right (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))) #align multiplicity.neg multiplicity.neg theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h] · rw [Int.coe_nat_multiplicity] · rw [multiplicity.neg, Int.coe_nat_multiplicity] #align multiplicity.int.nat_abs multiplicity.Int.natAbs theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a + b) = multiplicity p b := by apply le_antisymm · apply PartENat.le_of_lt_add_one cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk rw [hk] rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right] intro h_dvd · apply multiplicity.is_greatest _ h_dvd rw [hk, ← Nat.succ_eq_add_one] norm_cast apply Nat.lt_succ_self k · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one] exact PartENat.add_one_le_of_lt h · have := @min_le_multiplicity_add α _ _ p a b rwa [← min_eq_right (le_of_lt h)] #align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) : multiplicity p (a - b) = multiplicity p b := by rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption #align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) : multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab) · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left] exact le_of_lt hab · contradiction · rw [multiplicity_add_of_gt hab, min_eq_right] exact le_of_lt hab #align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min end Ring section CancelCommMonoidWithZero variable [CancelCommMonoidWithZero α] /- Porting note: removed previous wf recursion hints and added termination_by Also pulled a b intro parameters since Lean parses that more easily -/ theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} : ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b | n, m => fun ha hb ⟨s, hs⟩ => have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩ (hp.2.2 a b this).elim (fun ⟨x, hx⟩ => have hn0 : 0 < n := Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ => ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m) finite_mul_aux hp hpx hb ⟨s, mul_right_cancel₀ hp.1 (by rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩) fun ⟨x, hx⟩ => have hm0 : 0 < m := Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ => hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy; simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩) finite_mul_aux hp ha hpx ⟨s, mul_right_cancel₀ hp.1 (by rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)] simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩ termination_by finite_mul_aux _ _ n m => n+m #align multiplicity.finite_mul_aux multiplicity.finite_mul_aux theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) := fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩ #align multiplicity.finite_mul multiplicity.finite_mul theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b := ⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h => finite_mul hp h.1 h.2⟩ #align multiplicity.finite_mul_iff multiplicity.finite_mul_iff theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k) | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩ | k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha) #align multiplicity.finite_pow multiplicity.finite_pow variable [DecidableRel ((· ∣ ·) : α → α → Prop)] @[simp] theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by rw [← Nat.cast_one] exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2 ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩ #align multiplicity.multiplicity_self multiplicity.multiplicity_self @[simp] theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 := PartENat.get_eq_iff_eq_coe.2 (eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc, mul_right_inj' (ne_zero_of_finite ha)] at hb; exact mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩) #align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) : get (multiplicity p (a * b)) h = get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _ have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _ have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) = p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 * p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by simp [pow_add] have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b := by rw [hpoweq]; apply mul_dvd_mul <;> assumption have hsucc : ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 + get (multiplicity p b) ((finite_mul_iff hp).1 h).2 + 1) ∣ a * b := fun h => not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _)) (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h) rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩ #align multiplicity.mul' multiplicity.mul' open Classical protected theorem mul {p a b : α} (hp : Prime p) : multiplicity p (a * b) = multiplicity p a + multiplicity p b := if h : Finite p a ∧ Finite p b then by rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ← PartENat.natCast_get (finite_iff_dom.1 h.2), ← PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add, PartENat.natCast_inj, multiplicity.mul' hp] else by rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)] cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h] #align multiplicity.mul multiplicity.mul theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) : multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by classical induction' s using Finset.induction with a s has ih h · simp only [Finset.sum_empty, Finset.prod_empty] convert one_right hp.not_unit · simp [has, ← ih] convert multiplicity.mul hp #align multiplicity.finset.prod multiplicity.Finset.prod -- Porting note: with protected could not use pow' k in the succ branch protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) : ∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by intro k induction' k with k hk · simp [one_right hp.not_unit] · have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ] rw [succ_eq_add_one, get_eq_get_of_eq _ _ this, multiplicity.mul' hp, hk, add_mul, one_mul, add_comm] #align multiplicity.pow' multiplicity.pow' theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a | 0 => by simp [one_right hp.not_unit] | succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp] #align multiplicity.pow multiplicity.pow theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by rw [eq_coe_iff] use dvd_rfl rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n := by rw [eq_coe_iff] use dvd_rfl rw [pow_dvd_pow_iff h0 hu]
Mathlib.RingTheory.Multiplicity.619_0.uTHZeAJqYiw3Jx8
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) : multiplicity p (p ^ n) = n
Mathlib_RingTheory_Multiplicity