l3lab/ntp-mathlib huggingface dataset:
theorem rootMultiplicity_mul_X_sub_C_pow {p : R[X]} {a : R} {n : ℕ} (h : p ≠ 0) :
    (p * (X - C a) ^ n).rootMultiplicity a = p.rootMultiplicity a + n := by
  have h2 := monic_X_sub_C a |>.pow n |>.mul_left_ne_zero h
  refine le_antisymm ?_ ?_
  · rw [rootMultiplicity_le_iff h2, add_assoc, add_comm n, ← add_assoc, pow_add,
      dvd_cancel_right_mem_nonZeroDivisors (monic_X_sub_C a |>.pow n |>.mem_nonZeroDivisors)]
    

Proof state:
case refine_1
R : Type u
S : Type v
T : Type w
a✝ b : R
n✝ : ℕ
inst✝ : CommRing R
p : R[X]
a : R
n : ℕ
h : p ≠ 0
h2 : p * (X - C a) ^ n ≠ 0
⊢ ¬(X - C a) ^ (rootMultiplicity a p + 1) ∣ p

Next tactic:
exact pow_rootMultiplicity_not_dvd h a
theorem rootMultiplicity_mul_X_sub_C_pow {p : R[X]} {a : R} {n : ℕ} (h : p ≠ 0) :
    (p * (X - C a) ^ n).rootMultiplicity a = p.rootMultiplicity a + n := by
  have h2 := monic_X_sub_C a |>.pow n |>.mul_left_ne_zero h
  refine le_antisymm ?_ ?_
  · rw [rootMultiplicity_le_iff h2, add_assoc, add_comm n, ← add_assoc, pow_add,
      dvd_cancel_right_mem_nonZeroDivisors (monic_X_sub_C a |>.pow n |>.mem_nonZeroDivisors)]
    exact pow_rootMultiplicity_not_dvd h a
  · 

Proof state:
case refine_2
R : Type u
S : Type v
T : Type w
a✝ b : R
n✝ : ℕ
inst✝ : CommRing R
p : R[X]
a : R
n : ℕ
h : p ≠ 0
h2 : p * (X - C a) ^ n ≠ 0
⊢ rootMultiplicity a p + n ≤ rootMultiplicity a (p * (X - C a) ^ n)

Next tactic:
rw [le_rootMultiplicity_iff h2, pow_add]
theorem rootMultiplicity_mul_X_sub_C_pow {p : R[X]} {a : R} {n : ℕ} (h : p ≠ 0) :
    (p * (X - C a) ^ n).rootMultiplicity a = p.rootMultiplicity a + n := by
  have h2 := monic_X_sub_C a |>.pow n |>.mul_left_ne_zero h
  refine le_antisymm ?_ ?_
  · rw [rootMultiplicity_le_iff h2, add_assoc, add_comm n, ← add_assoc, pow_add,
      dvd_cancel_right_mem_nonZeroDivisors (monic_X_sub_C a |>.pow n |>.mem_nonZeroDivisors)]
    exact pow_rootMultiplicity_not_dvd h a
  · rw [le_rootMultiplicity_iff h2, pow_add]
    

Proof state:
case refine_2
R : Type u
S : Type v
T : Type w
a✝ b : R
n✝ : ℕ
inst✝ : CommRing R
p : R[X]
a : R
n : ℕ
h : p ≠ 0
h2 : p * (X - C a) ^ n ≠ 0
⊢ (X - C a) ^ rootMultiplicity a p * (X - C a) ^ n ∣ p * (X - C a) ^ n

Next tactic:
exact mul_dvd_mul_right (pow_rootMultiplicity_dvd p a) _
/-- The multiplicity of `a` as root of `(X - a) ^ n` is `n`. -/
theorem rootMultiplicity_X_sub_C_pow [Nontrivial R] (a : R) (n : ℕ) :
    rootMultiplicity a ((X - C a) ^ n) = n := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a✝ b : R
n✝ : ℕ
inst✝¹ : CommRing R
inst✝ : Nontrivial R
a : R
n : ℕ
⊢ rootMultiplicity a ((X - C a) ^ n) = n

Next tactic:
have := rootMultiplicity_mul_X_sub_C_pow (a := a) (n := n) C.map_one_ne_zero
/-- The multiplicity of `a` as root of `(X - a) ^ n` is `n`. -/
theorem rootMultiplicity_X_sub_C_pow [Nontrivial R] (a : R) (n : ℕ) :
    rootMultiplicity a ((X - C a) ^ n) = n := by
  have := rootMultiplicity_mul_X_sub_C_pow (a := a) (n := n) C.map_one_ne_zero
  

Proof state:
R : Type u
S : Type v
T : Type w
a✝ b : R
n✝ : ℕ
inst✝¹ : CommRing R
inst✝ : Nontrivial R
a : R
n : ℕ
this : rootMultiplicity a (C 1 * (X - C a) ^ n) = rootMultiplicity a (C 1) + n
⊢ rootMultiplicity a ((X - C a) ^ n) = n

Next tactic:
rwa [rootMultiplicity_C, map_one, one_mul, zero_add] at this
theorem rootMultiplicity_X_sub_C [Nontrivial R] [DecidableEq R] {x y : R} :
    rootMultiplicity x (X - C y) = if x = y then 1 else 0 := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : Nontrivial R
inst✝ : DecidableEq R
x y : R
⊢ rootMultiplicity x (X - C y) = if x = y then 1 else 0

Next tactic:
split_ifs with hxy
theorem rootMultiplicity_X_sub_C [Nontrivial R] [DecidableEq R] {x y : R} :
    rootMultiplicity x (X - C y) = if x = y then 1 else 0 := by
  split_ifs with hxy
  · 

Proof state:
case pos
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : Nontrivial R
inst✝ : DecidableEq R
x y : R
hxy : x = y
⊢ rootMultiplicity x (X - C y) = 1

Next tactic:
rw [hxy]
theorem rootMultiplicity_X_sub_C [Nontrivial R] [DecidableEq R] {x y : R} :
    rootMultiplicity x (X - C y) = if x = y then 1 else 0 := by
  split_ifs with hxy
  · rw [hxy]
    

Proof state:
case pos
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : Nontrivial R
inst✝ : DecidableEq R
x y : R
hxy : x = y
⊢ rootMultiplicity y (X - C y) = 1

Next tactic:
exact rootMultiplicity_X_sub_C_self
theorem rootMultiplicity_X_sub_C [Nontrivial R] [DecidableEq R] {x y : R} :
    rootMultiplicity x (X - C y) = if x = y then 1 else 0 := by
  split_ifs with hxy
  · rw [hxy]
    exact rootMultiplicity_X_sub_C_self
  

Proof state:
case neg
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : Nontrivial R
inst✝ : DecidableEq R
x y : R
hxy : ¬x = y
⊢ rootMultiplicity x (X - C y) = 0

Next tactic:
exact rootMultiplicity_eq_zero (mt root_X_sub_C.mp (Ne.symm hxy))
/-- The multiplicity of `p + q` is at least the minimum of the multiplicities. -/
theorem rootMultiplicity_add {p q : R[X]} (a : R) (hzero : p + q ≠ 0) :
    min (rootMultiplicity a p) (rootMultiplicity a q) ≤ rootMultiplicity a (p + q) := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a✝ b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
a : R
hzero : p + q ≠ 0
⊢ min (rootMultiplicity a p) (rootMultiplicity a q) ≤ rootMultiplicity a (p + q)

Next tactic:
rw [le_rootMultiplicity_iff hzero]
/-- The multiplicity of `p + q` is at least the minimum of the multiplicities. -/
theorem rootMultiplicity_add {p q : R[X]} (a : R) (hzero : p + q ≠ 0) :
    min (rootMultiplicity a p) (rootMultiplicity a q) ≤ rootMultiplicity a (p + q) := by
  rw [le_rootMultiplicity_iff hzero]
  

Proof state:
R : Type u
S : Type v
T : Type w
a✝ b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
a : R
hzero : p + q ≠ 0
⊢ (X - C a) ^ min (rootMultiplicity a p) (rootMultiplicity a q) ∣ p + q

Next tactic:
exact min_pow_dvd_add (pow_rootMultiplicity_dvd p a) (pow_rootMultiplicity_dvd q a)
theorem le_rootMultiplicity_mul {p q : R[X]} (x : R) (hpq : p * q ≠ 0) :
    rootMultiplicity x p + rootMultiplicity x q ≤ rootMultiplicity x (p * q) := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
x : R
hpq : p * q ≠ 0
⊢ rootMultiplicity x p + rootMultiplicity x q ≤ rootMultiplicity x (p * q)

Next tactic:
rw [le_rootMultiplicity_iff hpq, pow_add]
theorem le_rootMultiplicity_mul {p q : R[X]} (x : R) (hpq : p * q ≠ 0) :
    rootMultiplicity x p + rootMultiplicity x q ≤ rootMultiplicity x (p * q) := by
  rw [le_rootMultiplicity_iff hpq, pow_add]
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
x : R
hpq : p * q ≠ 0
⊢ (X - C x) ^ rootMultiplicity x p * (X - C x) ^ rootMultiplicity x q ∣ p * q

Next tactic:
exact mul_dvd_mul (pow_rootMultiplicity_dvd p x) (pow_rootMultiplicity_dvd q x)
theorem rootMultiplicity_mul' {p q : R[X]} {x : R}
    (hpq : (p /ₘ (X - C x) ^ p.rootMultiplicity x).eval x *
      (q /ₘ (X - C x) ^ q.rootMultiplicity x).eval x ≠ 0) :
    rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
x : R
hpq : eval x (p /ₘ (X - C x) ^ rootMultiplicity x p) * eval x (q /ₘ (X - C x) ^ rootMultiplicity x q) ≠ 0
⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q

Next tactic:
simp_rw [eval_divByMonic_eq_trailingCoeff_comp] at hpq
theorem rootMultiplicity_mul' {p q : R[X]} {x : R}
    (hpq : (p /ₘ (X - C x) ^ p.rootMultiplicity x).eval x *
      (q /ₘ (X - C x) ^ q.rootMultiplicity x).eval x ≠ 0) :
    rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q := by
  simp_rw [eval_divByMonic_eq_trailingCoeff_comp] at hpq
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
x : R
hpq : trailingCoeff (comp p (X + C x)) * trailingCoeff (comp q (X + C x)) ≠ 0
⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q

Next tactic:
simp_rw [rootMultiplicity_eq_natTrailingDegree, mul_comp, natTrailingDegree_mul' hpq]
theorem prime_X_sub_C (r : R) : Prime (X - C r) :=
  ⟨X_sub_C_ne_zero r, not_isUnit_X_sub_C r, fun _ _ => by
    

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
r : R
x✝¹ x✝ : R[X]
⊢ X - C r ∣ x✝¹ * x✝ → X - C r ∣ x✝¹ ∨ X - C r ∣ x✝

Next tactic:
simp_rw [dvd_iff_isRoot, IsRoot.def, eval_mul, mul_eq_zero]
theorem prime_X_sub_C (r : R) : Prime (X - C r) :=
  ⟨X_sub_C_ne_zero r, not_isUnit_X_sub_C r, fun _ _ => by
    simp_rw [dvd_iff_isRoot, IsRoot.def, eval_mul, mul_eq_zero]
    

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
r : R
x✝¹ x✝ : R[X]
⊢ eval r x✝¹ = 0 ∨ eval r x✝ = 0 → eval r x✝¹ = 0 ∨ eval r x✝ = 0

Next tactic:
exact id
theorem prime_X : Prime (X : R[X]) := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
⊢ Prime X

Next tactic:
convert prime_X_sub_C (0 : R)
theorem prime_X : Prime (X : R[X]) := by
  convert prime_X_sub_C (0 : R)
  

Proof state:
case h.e'_3
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
⊢ X = X - C 0

Next tactic:
simp
theorem Monic.prime_of_degree_eq_one (hp1 : degree p = 1) (hm : Monic p) : Prime p :=
  have : p = X - C (-p.coeff 0) := by 

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
hp1 : degree p = 1
hm : Monic p
⊢ p = X - C (-coeff p 0)

Next tactic:
simpa [hm.leadingCoeff] using eq_X_add_C_of_degree_eq_one hp1
theorem rootMultiplicity_mul {p q : R[X]} {x : R} (hpq : p * q ≠ 0) :
    rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p✝ q✝ p q : R[X]
x : R
hpq : p * q ≠ 0
⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q

Next tactic:
classical
  have hp : p ≠ 0 := left_ne_zero_of_mul hpq
  have hq : q ≠ 0 := right_ne_zero_of_mul hpq
  rw [rootMultiplicity_eq_multiplicity (p * q), dif_neg hpq, rootMultiplicity_eq_multiplicity p,
    dif_neg hp, rootMultiplicity_eq_multiplicity q, dif_neg hq,
    multiplicity.mul' (prime_X_sub_C x)]
theorem rootMultiplicity_mul {p q : R[X]} {x : R} (hpq : p * q ≠ 0) :
    rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q := by
  classical
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p✝ q✝ p q : R[X]
x : R
hpq : p * q ≠ 0
⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q

Next tactic:
have hp : p ≠ 0 := left_ne_zero_of_mul hpq
theorem rootMultiplicity_mul {p q : R[X]} {x : R} (hpq : p * q ≠ 0) :
    rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q := by
  classical
  have hp : p ≠ 0 := left_ne_zero_of_mul hpq
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p✝ q✝ p q : R[X]
x : R
hpq : p * q ≠ 0
hp : p ≠ 0
⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q

Next tactic:
have hq : q ≠ 0 := right_ne_zero_of_mul hpq
theorem rootMultiplicity_mul {p q : R[X]} {x : R} (hpq : p * q ≠ 0) :
    rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q := by
  classical
  have hp : p ≠ 0 := left_ne_zero_of_mul hpq
  have hq : q ≠ 0 := right_ne_zero_of_mul hpq
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p✝ q✝ p q : R[X]
x : R
hpq : p * q ≠ 0
hp : p ≠ 0
hq : q ≠ 0
⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q

Next tactic:
rw [rootMultiplicity_eq_multiplicity (p * q), dif_neg hpq, rootMultiplicity_eq_multiplicity p,
    dif_neg hp, rootMultiplicity_eq_multiplicity q, dif_neg hq,
    multiplicity.mul' (prime_X_sub_C x)]
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h✝ : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < degree p
h : p /ₘ (X - C x) = 0
⊢ False

Next tactic:
rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; 

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
this : Decidable (∃ x, IsRoot p x)
h✝ : ∃ x, IsRoot p x
x : R
hp : 0 ≠ 0
hx : IsRoot p x
hpd : 0 < degree p
h : p /ₘ (X - C x) = 0
⊢ False

Next tactic:
exact hp rfl
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
      have wf : degree (p /ₘ (X - C x)) < degree p :=
        degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by 

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < degree p
hd0 : p /ₘ (X - C x) ≠ 0
⊢ 0 < 1

Next tactic:
decide
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
      have wf : degree (p /ₘ (X - C x)) < degree p :=
        degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by decide)
      let ⟨t, htd, htr⟩ := @exists_multiset_roots _ (p /ₘ (X - C x)) hd0
      have hdeg : degree (X - C x) ≤ degree p := by
        

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < degree p
hd0 : p /ₘ (X - C x) ≠ 0
wf : degree (p /ₘ (X - C x)) < degree p
t : Multiset R
htd : ↑(Multiset.card t) ≤ degree (p /ₘ (X - C x))
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
⊢ degree (X - C x) ≤ degree p

Next tactic:
rw [degree_X_sub_C, degree_eq_natDegree hp]
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
      have wf : degree (p /ₘ (X - C x)) < degree p :=
        degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by decide)
      let ⟨t, htd, htr⟩ := @exists_multiset_roots _ (p /ₘ (X - C x)) hd0
      have hdeg : degree (X - C x) ≤ degree p := by
        rw [degree_X_sub_C, degree_eq_natDegree hp]
        

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < degree p
hd0 : p /ₘ (X - C x) ≠ 0
wf : degree (p /ₘ (X - C x)) < degree p
t : Multiset R
htd : ↑(Multiset.card t) ≤ degree (p /ₘ (X - C x))
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
⊢ 1 ≤ ↑(natDegree p)

Next tactic:
rw [degree_eq_natDegree hp] at hpd
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
      have wf : degree (p /ₘ (X - C x)) < degree p :=
        degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by decide)
      let ⟨t, htd, htr⟩ := @exists_multiset_roots _ (p /ₘ (X - C x)) hd0
      have hdeg : degree (X - C x) ≤ degree p := by
        rw [degree_X_sub_C, degree_eq_natDegree hp]
        rw [degree_eq_natDegree hp] at hpd
        

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < ↑(natDegree p)
hd0 : p /ₘ (X - C x) ≠ 0
wf : degree (p /ₘ (X - C x)) < degree p
t : Multiset R
htd : ↑(Multiset.card t) ≤ degree (p /ₘ (X - C x))
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
⊢ 1 ≤ ↑(natDegree p)

Next tactic:
exact WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 hpd)
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
      have wf : degree (p /ₘ (X - C x)) < degree p :=
        degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by decide)
      let ⟨t, htd, htr⟩ := @exists_multiset_roots _ (p /ₘ (X - C x)) hd0
      have hdeg : degree (X - C x) ≤ degree p := by
        rw [degree_X_sub_C, degree_eq_natDegree hp]
        rw [degree_eq_natDegree hp] at hpd
        exact WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 hpd)
      have hdiv0 : p /ₘ (X - C x) ≠ 0 :=
        mt (divByMonic_eq_zero_iff (monic_X_sub_C x)).1 <| not_lt.2 hdeg
      ⟨x ::ₘ t,
        calc
          (card (x ::ₘ t) : WithBot ℕ) = Multiset.card t + 1 := by
            

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < degree p
hd0 : p /ₘ (X - C x) ≠ 0
wf : degree (p /ₘ (X - C x)) < degree p
t : Multiset R
htd : ↑(Multiset.card t) ≤ degree (p /ₘ (X - C x))
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : degree (X - C x) ≤ degree p
hdiv0 : p /ₘ (X - C x) ≠ 0
⊢ ↑(Multiset.card (x ::ₘ t)) = ↑(Multiset.card t) + 1

Next tactic:
congr
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
      have wf : degree (p /ₘ (X - C x)) < degree p :=
        degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by decide)
      let ⟨t, htd, htr⟩ := @exists_multiset_roots _ (p /ₘ (X - C x)) hd0
      have hdeg : degree (X - C x) ≤ degree p := by
        rw [degree_X_sub_C, degree_eq_natDegree hp]
        rw [degree_eq_natDegree hp] at hpd
        exact WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 hpd)
      have hdiv0 : p /ₘ (X - C x) ≠ 0 :=
        mt (divByMonic_eq_zero_iff (monic_X_sub_C x)).1 <| not_lt.2 hdeg
      ⟨x ::ₘ t,
        calc
          (card (x ::ₘ t) : WithBot ℕ) = Multiset.card t + 1 := by
            congr
            

Proof state:
case e_a
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < degree p
hd0 : p /ₘ (X - C x) ≠ 0
wf : degree (p /ₘ (X - C x)) < degree p
t : Multiset R
htd : ↑(Multiset.card t) ≤ degree (p /ₘ (X - C x))
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : degree (X - C x) ≤ degree p
hdiv0 : p /ₘ (X - C x) ≠ 0
⊢ Multiset.card (x ::ₘ t) = Add.add (↑(Multiset.card t)) 1

Next tactic:
exact mod_cast Multiset.card_cons _ _
theorem exists_multiset_roots [DecidableEq R] :
    ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
      (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
  | p, hp =>
    haveI := Classical.propDecidable (∃ x, IsRoot p x)
    if h : ∃ x, IsRoot p x then
      let ⟨x, hx⟩ := h
      have hpd : 0 < degree p := degree_pos_of_root hp hx
      have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
        rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
      have wf : degree (p /ₘ (X - C x)) < degree p :=
        degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by decide)
      let ⟨t, htd, htr⟩ := @exists_multiset_roots _ (p /ₘ (X - C x)) hd0
      have hdeg : degree (X - C x) ≤ degree p := by
        rw [degree_X_sub_C, degree_eq_natDegree hp]
        rw [degree_eq_natDegree hp] at hpd
        exact WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 hpd)
      have hdiv0 : p /ₘ (X - C x) ≠ 0 :=
        mt (divByMonic_eq_zero_iff (monic_X_sub_C x)).1 <| not_lt.2 hdeg
      ⟨x ::ₘ t,
        calc
          (card (x ::ₘ t) : WithBot ℕ) = Multiset.card t + 1 := by
            congr
            exact mod_cast Multiset.card_cons _ _
          _ ≤ degree p := by
            

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : IsDomain R
p✝ q : R[X]
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, IsRoot p x)
h : ∃ x, IsRoot p x
x : R
hx : IsRoot p x
hpd : 0 < degree p
hd0 : p /ₘ (X - C x) ≠ 0
wf : degree (p /ₘ (X - C x)) < degree p
t : Multiset R
htd : ↑(Multiset.card t) ≤ degree (p /ₘ (X - C x))
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : degree (X - C x) ≤ degree p
hdiv0 : p /ₘ (X - C x) ≠ 0
⊢ ↑(Multiset.card t) + 1 ≤ degree p

Next tactic:
rw [← degree_add_divByMonic (monic_X_sub_C x) hdeg, degree_X_sub_C, add_comm]