feat(Data/Polynomial/RingDivision): improvements to `Polynomial.rootM…

…ultiplicity` (#8563)

Main changes:

- add `Monic.mem_nonZeroDivisors` and `mem_nonZeroDivisors_of_leadingCoeff` which states that a monic polynomial (resp. a polynomial whose leading coefficient is not zero divisor) is not a zero divisor.
- add `rootMultiplicity_mul_X_sub_C_pow` which states that `* (X - a) ^ n` adds the root multiplicity at `a` by `n`.
- change the conditions in `rootMultiplicity_X_sub_C_self`, `rootMultiplicity_X_sub_C` and `rootMultiplicity_X_sub_C_pow` from `IsDomain` to `Nontrivial`.
- add `rootMultiplicity_eq_natTrailingDegree` which relates `rootMultiplicity` and `natTrailingDegree`, and `eval_divByMonic_eq_trailingCoeff_comp`.
- add `le_rootMultiplicity_mul` which is similar to `le_trailingDegree_mul`.
- add `rootMultiplicity_mul'` which slightly generalizes `rootMultiplicity_mul`

In `Data/Polynomial/FieldDivision`:

- add `rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero` which slightly generalizes `rootMultiplicity_sub_one_le_derivative_rootMultiplicity`.
- add `derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors` which slightly generalizes `derivative_rootMultiplicity_of_root`.
- add several theorems relating roots of iterate derivative to `rootMultiplicity`

In addition:

- move `eq_of_monic_of_associated` from RingDivision to Monic and generalize.
- add `dvd_cancel` lemmas to NonZeroDivisors.
- add `algEquivOfCompEqX`: two polynomials that compose to X both ways induces an isomorphism of the polynomial algebra.
- add divisibility lemmas to Polynomial/Derivative.



Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

Mathlib/Algebra/Divisibility/Basic.lean

@@ -95,17 +95,18 @@ theorem dvd_of_mul_right_dvd (h : a * b ∣ c) : a ∣ c :=
 
 section map_dvd
 
-variable {M N : Type*} [Monoid M] [Monoid N]
+variable {M N : Type*}
 
-theorem map_dvd {F : Type*} [MulHomClass F M N] (f : F) {a b} : a ∣ b → f a ∣ f b
+theorem map_dvd [Semigroup M] [Semigroup N] {F : Type*} [MulHomClass F M N] (f : F) {a b} :
+    a ∣ b → f a ∣ f b
   | ⟨c, h⟩ => ⟨f c, h.symm ▸ map_mul f a c⟩
 #align map_dvd map_dvd
 
-theorem MulHom.map_dvd (f : M →ₙ* N) {a b} : a ∣ b → f a ∣ f b :=
+theorem MulHom.map_dvd [Semigroup M] [Semigroup N] (f : M →ₙ* N) {a b} : a ∣ b → f a ∣ f b :=
   _root_.map_dvd f
 #align mul_hom.map_dvd MulHom.map_dvd
 
-theorem MonoidHom.map_dvd (f : M →* N) {a b} : a ∣ b → f a ∣ f b :=
+theorem MonoidHom.map_dvd [Monoid M] [Monoid N] (f : M →* N) {a b} : a ∣ b → f a ∣ f b :=
   _root_.map_dvd f
 #align monoid_hom.map_dvd MonoidHom.map_dvd
 

Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean

@@ -141,6 +141,20 @@ theorem mul_cancel_left_coe_nonZeroDivisors {x y : R'} {c : R'⁰} : (c : R') *
   mul_cancel_left_mem_nonZeroDivisors c.prop
 #align mul_cancel_left_coe_non_zero_divisor mul_cancel_left_coe_nonZeroDivisors
 
+theorem dvd_cancel_right_mem_nonZeroDivisors {x y r : R'} (hr : r ∈ R'⁰) : x * r ∣ y * r ↔ x ∣ y :=
+  ⟨fun ⟨z, _⟩ ↦ ⟨z, by rwa [← mul_cancel_right_mem_nonZeroDivisors hr, mul_assoc, mul_comm z r,
+    ← mul_assoc]⟩, fun ⟨z, h⟩ ↦ ⟨z, by rw [h, mul_assoc, mul_comm z r, ← mul_assoc]⟩⟩
+
+theorem dvd_cancel_right_coe_nonZeroDivisors {x y : R'} {c : R'⁰} : x * c ∣ y * c ↔ x ∣ y :=
+  dvd_cancel_right_mem_nonZeroDivisors c.prop
+
+theorem dvd_cancel_left_mem_nonZeroDivisors {x y r : R'} (hr : r ∈ R'⁰) : r * x ∣ r * y ↔ x ∣ y :=
+  ⟨fun ⟨z, _⟩ ↦ ⟨z, by rwa [← mul_cancel_left_mem_nonZeroDivisors hr, ← mul_assoc]⟩,
+    fun ⟨z, h⟩ ↦ ⟨z, by rw [h, ← mul_assoc]⟩⟩
+
+theorem dvd_cancel_left_coe_nonZeroDivisors {x y : R'} {c : R'⁰} : c * x ∣ c * y ↔ x ∣ y :=
+  dvd_cancel_left_mem_nonZeroDivisors c.prop
+
 theorem nonZeroDivisors.ne_zero [Nontrivial M] {x} (hx : x ∈ M⁰) : x ≠ 0 := fun h ↦
   one_ne_zero (hx _ <| (one_mul _).trans h)
 #align non_zero_divisors.ne_zero nonZeroDivisors.ne_zero

Mathlib/Algebra/Ring/Divisibility/Basic.lean

@@ -20,6 +20,11 @@ imports. Further results about divisibility in rings may be found in
 
 variable {α β : Type*}
 
+theorem map_dvd_iff [Semigroup α] [Semigroup β] {F : Type*} [MulEquivClass F α β] (f : F) {a b} :
+    f a ∣ f b ↔ a ∣ b :=
+  let f := MulEquivClass.toMulEquiv f
+  ⟨fun h ↦ by rw [← f.left_inv a, ← f.left_inv b]; exact map_dvd f.symm h, map_dvd f⟩
+
 section DistribSemigroup
 
 variable [Add α] [Semigroup α]

Mathlib/Data/Polynomial/AlgebraMap.lean

@@ -257,11 +257,26 @@ theorem aeval_mul : aeval x (p * q) = aeval x p * aeval x q :=
   AlgHom.map_mul _ _ _
 #align polynomial.aeval_mul Polynomial.aeval_mul
 
+theorem comp_eq_aeval : p.comp q = aeval q p := rfl
+
 theorem aeval_comp {A : Type*} [CommSemiring A] [Algebra R A] (x : A) :
     aeval x (p.comp q) = aeval (aeval x q) p :=
   eval₂_comp (algebraMap R A)
 #align polynomial.aeval_comp Polynomial.aeval_comp
 
+/-- Two polynomials `p` and `q` such that `p(q(X))=X` and `q(p(X))=X`
+  induces an automorphism of the polynomial algebra. -/
+@[simps!]
+def algEquivOfCompEqX (p q : R[X]) (hpq : p.comp q = X) (hqp : q.comp p = X) : R[X] ≃ₐ[R] R[X] := by
+  refine AlgEquiv.ofAlgHom (aeval p) (aeval q) ?_ ?_ <;>
+    exact AlgHom.ext fun _ ↦ by simp [← comp_eq_aeval, comp_assoc, hpq, hqp]
+
+/-- The automorphism of the polynomial algebra given by `p(X) ↦ p(X+t)`,
+  with inverse `p(X) ↦ p(X-t)`. -/
+@[simps!]
+def algEquivAevalXAddC {R} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :=
+  algEquivOfCompEqX (X + C t) (X - C t) (by simp) (by simp)
+
 theorem aeval_algHom (f : A →ₐ[R] B) (x : A) : aeval (f x) = f.comp (aeval x) :=
   algHom_ext <| by simp only [aeval_X, AlgHom.comp_apply]
 #align polynomial.aeval_alg_hom Polynomial.aeval_algHom

Mathlib/Data/Polynomial/Derivative.lean

@@ -475,6 +475,23 @@ theorem derivative_sq (p : R[X]) : derivative (p ^ 2) = C 2 * p * derivative p :
   rw [derivative_pow_succ, Nat.cast_one, one_add_one_eq_two, pow_one]
 #align polynomial.derivative_sq Polynomial.derivative_sq
 
+theorem pow_sub_one_dvd_derivative_of_pow_dvd {p q : R[X]} {n : ℕ}
+    (dvd : q ^ n ∣ p) : q ^ (n - 1) ∣ derivative p := by
+  obtain ⟨r, rfl⟩ := dvd
+  rw [derivative_mul, derivative_pow]
+  exact (((dvd_mul_left _ _).mul_right _).mul_right _).add ((pow_dvd_pow q n.pred_le).mul_right _)
+
+theorem pow_sub_dvd_iterate_derivative_of_pow_dvd {p q : R[X]} {n : ℕ} (m : ℕ)
+    (dvd : q ^ n ∣ p) : q ^ (n - m) ∣ derivative^[m] p := by
+  revert p
+  induction' m with m ih <;> intro p h
+  · exact h
+  · rw [Nat.sub_succ, Function.iterate_succ']
+    exact pow_sub_one_dvd_derivative_of_pow_dvd (ih h)
+
+theorem pow_sub_dvd_iterate_derivative_pow (p : R[X]) (n m : ℕ) :
+    p ^ (n - m) ∣ derivative^[m] (p ^ n) := pow_sub_dvd_iterate_derivative_of_pow_dvd m dvd_rfl
+
 theorem dvd_iterate_derivative_pow (f : R[X]) (n : ℕ) {m : ℕ} (c : R) (hm : m ≠ 0) :
     (n : R) ∣ eval c (derivative^[m] (f ^ n)) := by
   obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
@@ -658,11 +675,15 @@ set_option linter.uppercaseLean3 false in
 #align polynomial.derivative_X_sub_C_sq Polynomial.derivative_X_sub_C_sq
 
 theorem iterate_derivative_X_sub_pow (n k : ℕ) (c : R) :
-    (derivative^[k]) ((X - C c) ^ n : R[X]) = n.descFactorial k • (X - C c) ^ (n - k) := by
+    derivative^[k] ((X - C c) ^ n) = n.descFactorial k • (X - C c) ^ (n - k) := by
   rw [sub_eq_add_neg, ← C_neg, iterate_derivative_X_add_pow]
 set_option linter.uppercaseLean3 false in
 #align polynomial.iterate_derivative_X_sub_pow Polynomial.iterate_derivative_X_sub_powₓ
 
+theorem iterate_derivative_X_sub_pow_self (n : ℕ) (c : R) :
+    derivative^[n] ((X - C c) ^ n) = n.factorial := by
+  rw [iterate_derivative_X_sub_pow, n.sub_self, pow_zero, nsmul_one, n.descFactorial_self]
+
 end CommRing
 
 end Derivative

Mathlib/Data/Polynomial/Div.lean

@@ -540,6 +540,12 @@ theorem pow_mul_divByMonic_rootMultiplicity_eq (p : R[X]) (a : R) :
   simp
 #align polynomial.div_by_monic_mul_pow_root_multiplicity_eq Polynomial.pow_mul_divByMonic_rootMultiplicity_eq
 
+theorem exists_eq_pow_rootMultiplicity_mul_and_not_dvd (p : R[X]) (hp : p ≠ 0) (a : R) :
+    ∃ q : R[X], p = (X - C a) ^ p.rootMultiplicity a * q ∧ ¬ (X - C a) ∣ q := by
+  classical
+  rw [rootMultiplicity_eq_multiplicity, dif_neg hp]
+  apply multiplicity.exists_eq_pow_mul_and_not_dvd
+
 end multiplicity
 
 end Ring

Mathlib/Data/Polynomial/FieldDivision.lean

@@ -27,34 +27,125 @@ universe u v w y z
 
 variable {R : Type u} {S : Type v} {k : Type y} {A : Type z} {a b : R} {n : ℕ}
 
+section CommRing
+
+variable [CommRing R]
+
+theorem rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero
+    (p : R[X]) (t : R) (hnezero : derivative p ≠ 0) :
+    p.rootMultiplicity t - 1 ≤ p.derivative.rootMultiplicity t :=
+  (le_rootMultiplicity_iff hnezero).2 <|
+    pow_sub_one_dvd_derivative_of_pow_dvd (p.pow_rootMultiplicity_dvd t)
+
+theorem derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors
+    {p : R[X]} {t : R} (hpt : Polynomial.IsRoot p t)
+    (hnzd : (p.rootMultiplicity t : R) ∈ nonZeroDivisors R) :
+    (derivative p).rootMultiplicity t = p.rootMultiplicity t - 1 := by
+  by_cases h : p = 0
+  · simp only [h, map_zero, rootMultiplicity_zero]
+  obtain ⟨g, hp, hndvd⟩ := p.exists_eq_pow_rootMultiplicity_mul_and_not_dvd h t
+  set m := p.rootMultiplicity t
+  have hm : m - 1 + 1 = m := Nat.sub_add_cancel <| (rootMultiplicity_pos h).2 hpt
+  have hndvd : ¬(X - C t) ^ m ∣ derivative p := by
+    rw [hp, derivative_mul, dvd_add_left (dvd_mul_right _ _),
+      derivative_X_sub_C_pow, ← hm, pow_succ', hm, mul_comm (C _), mul_assoc,
+      dvd_cancel_left_mem_nonZeroDivisors (monic_X_sub_C t |>.pow _ |>.mem_nonZeroDivisors)]
+    rw [dvd_iff_isRoot, IsRoot] at hndvd ⊢
+    rwa [eval_mul, eval_C, mul_left_mem_nonZeroDivisors_eq_zero_iff hnzd]
+  have hnezero : derivative p ≠ 0 := fun h ↦ hndvd (by rw [h]; exact dvd_zero _)
+  exact le_antisymm (by rwa [rootMultiplicity_le_iff hnezero, hm])
+    (rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero _ t hnezero)
+
+theorem isRoot_iterate_derivative_of_lt_rootMultiplicity {p : R[X]} {t : R} {n : ℕ}
+    (hn : n < p.rootMultiplicity t) : (derivative^[n] p).IsRoot t :=
+  dvd_iff_isRoot.mp <| (dvd_pow_self _ <| Nat.sub_ne_zero_of_lt hn).trans
+    (pow_sub_dvd_iterate_derivative_of_pow_dvd _ <| p.pow_rootMultiplicity_dvd t)
+
+open Finset in
+theorem eval_iterate_derivative_rootMultiplicity {p : R[X]} {t : R} :
+    (derivative^[p.rootMultiplicity t] p).eval t =
+      (p.rootMultiplicity t).factorial • (p /ₘ (X - C t) ^ p.rootMultiplicity t).eval t := by
+  set m := p.rootMultiplicity t with hm
+  conv_lhs => rw [← p.pow_mul_divByMonic_rootMultiplicity_eq t, ← hm]
+  rw [iterate_derivative_mul, eval_finset_sum, sum_eq_single_of_mem _ (mem_range.mpr m.succ_pos)]
+  · rw [m.choose_zero_right, one_smul, eval_mul, m.sub_zero, iterate_derivative_X_sub_pow_self,
+      eval_nat_cast, nsmul_eq_mul]; rfl
+  · intro b hb hb0
+    rw [iterate_derivative_X_sub_pow, eval_smul, eval_mul, eval_smul, eval_pow,
+      Nat.sub_sub_self (mem_range_succ_iff.mp hb), eval_sub, eval_X, eval_C, sub_self,
+      zero_pow' b hb0, smul_zero, zero_mul, smul_zero]
+
+theorem lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors
+    {p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
+    (hroot : ∀ m ≤ n, (derivative^[m] p).IsRoot t)
+    (hnzd : (n.factorial : R) ∈ nonZeroDivisors R) :
+    n < p.rootMultiplicity t := by
+  by_contra! h'
+  replace hroot := hroot _ h'
+  simp only [IsRoot, eval_iterate_derivative_rootMultiplicity] at hroot
+  obtain ⟨q, hq⟩ := Nat.coe_nat_dvd (α := R) <| Nat.factorial_dvd_factorial h'
+  rw [hq, mul_mem_nonZeroDivisors] at hnzd
+  rw [nsmul_eq_mul, mul_left_mem_nonZeroDivisors_eq_zero_iff hnzd.1] at hroot
+  exact eval_divByMonic_pow_rootMultiplicity_ne_zero t h hroot
+
+theorem lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors'
+    {p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
+    (hroot : ∀ m ≤ n, (derivative^[m] p).IsRoot t)
+    (hnzd : ∀ m ≤ n, m ≠ 0 → (m : R) ∈ nonZeroDivisors R) :
+    n < p.rootMultiplicity t := by
+  apply lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors h hroot
+  clear hroot
+  induction' n with n ih
+  · simp only [Nat.zero_eq, Nat.factorial_zero, Nat.cast_one]
+    exact Submonoid.one_mem _
+  · rw [Nat.factorial_succ, Nat.cast_mul, mul_mem_nonZeroDivisors]
+    exact ⟨hnzd _ le_rfl n.succ_ne_zero, ih fun m h ↦ hnzd m (h.trans n.le_succ)⟩
+
+theorem lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors
+    {p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
+    (hnzd : (n.factorial : R) ∈ nonZeroDivisors R) :
+    n < p.rootMultiplicity t ↔ ∀ m ≤ n, (derivative^[m] p).IsRoot t :=
+  ⟨fun hn _ hm ↦ isRoot_iterate_derivative_of_lt_rootMultiplicity <| hm.trans_lt hn,
+    fun hr ↦ lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors h hr hnzd⟩
+
+theorem lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors'
+    {p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
+    (hnzd : ∀ m ≤ n, m ≠ 0 → (m : R) ∈ nonZeroDivisors R) :
+    n < p.rootMultiplicity t ↔ ∀ m ≤ n, (derivative^[m] p).IsRoot t :=
+  ⟨fun hn _ hm ↦ isRoot_iterate_derivative_of_lt_rootMultiplicity <| Nat.lt_of_le_of_lt hm hn,
+    fun hr ↦ lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors' h hr hnzd⟩
+
+theorem one_lt_rootMultiplicity_iff_isRoot_iterate_derivative
+    {p : R[X]} {t : R} (h : p ≠ 0) :
+    1 < p.rootMultiplicity t ↔ ∀ m ≤ 1, (derivative^[m] p).IsRoot t :=
+  lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors h
+    (by rw [Nat.factorial_one, Nat.cast_one]; exact Submonoid.one_mem _)
+
+theorem one_lt_rootMultiplicity_iff_isRoot
+    {p : R[X]} {t : R} (h : p ≠ 0) :
+    1 < p.rootMultiplicity t ↔ p.IsRoot t ∧ (derivative p).IsRoot t := by
+  rw [one_lt_rootMultiplicity_iff_isRoot_iterate_derivative h]
+  refine ⟨fun h ↦ ⟨h 0 (by norm_num), h 1 (by norm_num)⟩, fun ⟨h0, h1⟩ m hm ↦ ?_⟩
+  obtain (_|_|m) := m
+  exacts [h0, h1, by linarith [hm]]
+
+end CommRing
+
 section IsDomain
 
 variable [CommRing R] [IsDomain R]
 
+theorem one_lt_rootMultiplicity_iff_isRoot_gcd
+    [GCDMonoid R[X]] {p : R[X]} {t : R} (h : p ≠ 0) :
+    1 < p.rootMultiplicity t ↔ (gcd p (derivative p)).IsRoot t := by
+  simp_rw [one_lt_rootMultiplicity_iff_isRoot h, ← dvd_iff_isRoot, dvd_gcd_iff]
+
 theorem derivative_rootMultiplicity_of_root [CharZero R] {p : R[X]} {t : R} (hpt : p.IsRoot t) :
     p.derivative.rootMultiplicity t = p.rootMultiplicity t - 1 := by
-  rcases eq_or_ne p 0 with (rfl | hp)
-  · simp
-  nth_rw 1 [← p.pow_mul_divByMonic_rootMultiplicity_eq t, mul_comm]
-  simp only [derivative_pow, derivative_mul, derivative_sub, derivative_X, derivative_C, sub_zero,
-    mul_one]
-  set n := p.rootMultiplicity t - 1
-  have hn : n + 1 = _ := tsub_add_cancel_of_le ((rootMultiplicity_pos hp).mpr hpt)
-  rw [← hn]
-  set q := p /ₘ (X - C t) ^ (n + 1) with _hq
-  convert_to rootMultiplicity t ((X - C t) ^ n * (derivative q * (X - C t) + q * C ↑(n + 1))) = n
-  · congr
-    rw [mul_add, mul_left_comm <| (X - C t) ^ n, ← pow_succ']
-    congr 1
-    rw [mul_left_comm <| (X - C t) ^ n, mul_comm <| (X - C t) ^ n]
-  have h : eval t (derivative q * (X - C t) + q * C (R := R) ↑(n + 1)) ≠ 0 := by
-    suffices eval t q * ↑(n + 1) ≠ 0 by simpa
-    refine' mul_ne_zero _ (Nat.cast_ne_zero.mpr n.succ_ne_zero)
-    convert eval_divByMonic_pow_rootMultiplicity_ne_zero t hp
-  rw [rootMultiplicity_mul, rootMultiplicity_X_sub_C_pow, rootMultiplicity_eq_zero h, add_zero]
-  refine' mul_ne_zero (pow_ne_zero n <| X_sub_C_ne_zero t) _
-  contrapose! h
-  rw [h, eval_zero]
+  by_cases h : p = 0
+  · rw [h, map_zero, rootMultiplicity_zero]
+  exact derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors hpt <|
+    mem_nonZeroDivisors_of_ne_zero <| Nat.cast_ne_zero.2 ((rootMultiplicity_pos h).2 hpt).ne'
 #align polynomial.derivative_root_multiplicity_of_root Polynomial.derivative_rootMultiplicity_of_root
 
 theorem rootMultiplicity_sub_one_le_derivative_rootMultiplicity [CharZero R] (p : R[X]) (t : R) :
@@ -65,6 +156,19 @@ theorem rootMultiplicity_sub_one_le_derivative_rootMultiplicity [CharZero R] (p
     exact zero_le _
 #align polynomial.root_multiplicity_sub_one_le_derivative_root_multiplicity Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity
 
+theorem lt_rootMultiplicity_of_isRoot_iterate_derivative
+    [CharZero R] {p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
+    (hroot : ∀ m ≤ n, (derivative^[m] p).IsRoot t) :
+    n < p.rootMultiplicity t :=
+  lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors h hroot <|
+    mem_nonZeroDivisors_of_ne_zero <| Nat.cast_ne_zero.2 <| Nat.factorial_ne_zero n
+
+theorem lt_rootMultiplicity_iff_isRoot_iterate_derivative
+    [CharZero R] {p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0) :
+    n < p.rootMultiplicity t ↔ ∀ m ≤ n, (derivative^[m] p).IsRoot t :=
+  ⟨fun hn _ hm ↦ isRoot_iterate_derivative_of_lt_rootMultiplicity <| Nat.lt_of_le_of_lt hm hn,
+    fun hr ↦ lt_rootMultiplicity_of_isRoot_iterate_derivative h hr⟩
+
 section NormalizationMonoid
 
 variable [NormalizationMonoid R]

Mathlib/Data/Polynomial/Monic.lean

@@ -254,6 +254,11 @@ theorem Monic.isUnit_iff (hm : p.Monic) : IsUnit p ↔ p = 1 :=
   ⟨hm.eq_one_of_isUnit, fun h => h.symm ▸ isUnit_one⟩
 #align polynomial.monic.is_unit_iff Polynomial.Monic.isUnit_iff
 
+theorem eq_of_monic_of_associated (hp : p.Monic) (hq : q.Monic) (hpq : Associated p q) : p = q := by
+  obtain ⟨u, rfl⟩ := hpq
+  rw [(hp.of_mul_monic_left hq).eq_one_of_isUnit u.isUnit, mul_one]
+#align polynomial.eq_of_monic_of_associated Polynomial.eq_of_monic_of_associated
+
 end Semiring
 
 section CommSemiring