l3lab/ntp-mathlib huggingface dataset:
lemma coeff_minpolyDiv_mem_adjoin (x : S) (i) :
    coeff (minpolyDiv R x) i ∈ Algebra.adjoin R {x} := by
  by_contra H
  have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
  · intro j; induction j with
    | zero => exact H
    | succ j IH =>
      intro H; apply IH
      rw [coeff_minpolyDiv]
      refine add_mem ?_ (mul_mem H (Algebra.self_mem_adjoin_singleton R x))
      exact Subalgebra.algebraMap_mem _ _
  apply this (natDegree (minpolyDiv R x) + 1)
  

Proof state:
R : Type u_2
K : Type ?u.362955
L : Type ?u.362958
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x✝ : S
hx : IsIntegral R x✝
x : S
i : ℕ
H : coeff (minpolyDiv R x) i ∉ Algebra.adjoin R {x}
this : ∀ (j : ℕ), coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
⊢ coeff (minpolyDiv R x) (i + (natDegree (minpolyDiv R x) + 1)) ∈ Algebra.adjoin R {x}

Next tactic:
rw [coeff_eq_zero_of_natDegree_lt]
lemma coeff_minpolyDiv_mem_adjoin (x : S) (i) :
    coeff (minpolyDiv R x) i ∈ Algebra.adjoin R {x} := by
  by_contra H
  have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
  · intro j; induction j with
    | zero => exact H
    | succ j IH =>
      intro H; apply IH
      rw [coeff_minpolyDiv]
      refine add_mem ?_ (mul_mem H (Algebra.self_mem_adjoin_singleton R x))
      exact Subalgebra.algebraMap_mem _ _
  apply this (natDegree (minpolyDiv R x) + 1)
  rw [coeff_eq_zero_of_natDegree_lt]
  · 

Proof state:
R : Type u_2
K : Type ?u.362955
L : Type ?u.362958
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x✝ : S
hx : IsIntegral R x✝
x : S
i : ℕ
H : coeff (minpolyDiv R x) i ∉ Algebra.adjoin R {x}
this : ∀ (j : ℕ), coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
⊢ 0 ∈ Algebra.adjoin R {x}

Next tactic:
exact zero_mem _
lemma coeff_minpolyDiv_mem_adjoin (x : S) (i) :
    coeff (minpolyDiv R x) i ∈ Algebra.adjoin R {x} := by
  by_contra H
  have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
  · intro j; induction j with
    | zero => exact H
    | succ j IH =>
      intro H; apply IH
      rw [coeff_minpolyDiv]
      refine add_mem ?_ (mul_mem H (Algebra.self_mem_adjoin_singleton R x))
      exact Subalgebra.algebraMap_mem _ _
  apply this (natDegree (minpolyDiv R x) + 1)
  rw [coeff_eq_zero_of_natDegree_lt]
  · exact zero_mem _
  · 

Proof state:
R : Type u_2
K : Type ?u.362955
L : Type ?u.362958
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x✝ : S
hx : IsIntegral R x✝
x : S
i : ℕ
H : coeff (minpolyDiv R x) i ∉ Algebra.adjoin R {x}
this : ∀ (j : ℕ), coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
⊢ natDegree (minpolyDiv R x) < i + (natDegree (minpolyDiv R x) + 1)

Next tactic:
refine (Nat.le_add_left _ i).trans_lt ?_
lemma coeff_minpolyDiv_mem_adjoin (x : S) (i) :
    coeff (minpolyDiv R x) i ∈ Algebra.adjoin R {x} := by
  by_contra H
  have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
  · intro j; induction j with
    | zero => exact H
    | succ j IH =>
      intro H; apply IH
      rw [coeff_minpolyDiv]
      refine add_mem ?_ (mul_mem H (Algebra.self_mem_adjoin_singleton R x))
      exact Subalgebra.algebraMap_mem _ _
  apply this (natDegree (minpolyDiv R x) + 1)
  rw [coeff_eq_zero_of_natDegree_lt]
  · exact zero_mem _
  · refine (Nat.le_add_left _ i).trans_lt ?_
    

Proof state:
R : Type u_2
K : Type ?u.362955
L : Type ?u.362958
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x✝ : S
hx : IsIntegral R x✝
x : S
i : ℕ
H : coeff (minpolyDiv R x) i ∉ Algebra.adjoin R {x}
this : ∀ (j : ℕ), coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
⊢ i + natDegree (minpolyDiv R x) < i + (natDegree (minpolyDiv R x) + 1)

Next tactic:
rw [← add_assoc]
lemma coeff_minpolyDiv_mem_adjoin (x : S) (i) :
    coeff (minpolyDiv R x) i ∈ Algebra.adjoin R {x} := by
  by_contra H
  have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
  · intro j; induction j with
    | zero => exact H
    | succ j IH =>
      intro H; apply IH
      rw [coeff_minpolyDiv]
      refine add_mem ?_ (mul_mem H (Algebra.self_mem_adjoin_singleton R x))
      exact Subalgebra.algebraMap_mem _ _
  apply this (natDegree (minpolyDiv R x) + 1)
  rw [coeff_eq_zero_of_natDegree_lt]
  · exact zero_mem _
  · refine (Nat.le_add_left _ i).trans_lt ?_
    rw [← add_assoc]
    

Proof state:
R : Type u_2
K : Type ?u.362955
L : Type ?u.362958
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x✝ : S
hx : IsIntegral R x✝
x : S
i : ℕ
H : coeff (minpolyDiv R x) i ∉ Algebra.adjoin R {x}
this : ∀ (j : ℕ), coeff (minpolyDiv R x) (i + j) ∉ Algebra.adjoin R {x}
⊢ i + natDegree (minpolyDiv R x) < i + natDegree (minpolyDiv R x) + 1

Next tactic:
exact Nat.lt.base _
lemma minpolyDiv_eq_of_isIntegrallyClosed [IsDomain R] [IsIntegrallyClosed R] [IsDomain S]
    [Algebra R K] [Algebra K S] [IsScalarTower R K S] [IsFractionRing R K] :
    minpolyDiv R x = minpolyDiv K x := by
  

Proof state:
R : Type u_1
K : Type u_3
L : Type ?u.367515
S : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : Field K
inst✝¹⁰ : Field L
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra K L
x : S
hx : IsIntegral R x
inst✝⁶ : IsDomain R
inst✝⁵ : IsIntegrallyClosed R
inst✝⁴ : IsDomain S
inst✝³ : Algebra R K
inst✝² : Algebra K S
inst✝¹ : IsScalarTower R K S
inst✝ : IsFractionRing R K
⊢ minpolyDiv R x = minpolyDiv K x

Next tactic:
delta minpolyDiv
lemma minpolyDiv_eq_of_isIntegrallyClosed [IsDomain R] [IsIntegrallyClosed R] [IsDomain S]
    [Algebra R K] [Algebra K S] [IsScalarTower R K S] [IsFractionRing R K] :
    minpolyDiv R x = minpolyDiv K x := by
  delta minpolyDiv
  

Proof state:
R : Type u_1
K : Type u_3
L : Type ?u.367515
S : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : Field K
inst✝¹⁰ : Field L
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra K L
x : S
hx : IsIntegral R x
inst✝⁶ : IsDomain R
inst✝⁵ : IsIntegrallyClosed R
inst✝⁴ : IsDomain S
inst✝³ : Algebra R K
inst✝² : Algebra K S
inst✝¹ : IsScalarTower R K S
inst✝ : IsFractionRing R K
⊢ map (algebraMap R S) (minpoly R x) /ₘ (X - C x) = map (algebraMap K S) (minpoly K x) /ₘ (X - C x)

Next tactic:
rw [IsScalarTower.algebraMap_eq R K S, ← map_map,
    ← minpoly.isIntegrallyClosed_eq_field_fractions' _ hx]
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  

Proof state:
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
hi : i ≤ natDegree (minpolyDiv R x)
⊢ coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈ Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)

Next tactic:
induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
    · rw [Nat.succ_eq_add_one, ← tsub_tsub, tsub_add_cancel_of_le
        (le_tsub_of_add_le_left (b := 1) hi)]
      apply SetLike.le_def.mp ?_
        (Submodule.mul_mem_mul (IH ((Nat.le_succ _).trans hi))
          (Submodule.mem_span_singleton_self x))
      rw [Submodule.span_mul_span, Set.mul_singleton, Set.image_image]
      apply Submodule.span_mono
      rintro _ ⟨j, hj : j < i, rfl⟩
      exact ⟨j + 1, Nat.add_lt_of_lt_sub hj, pow_succ' x j⟩
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  

Proof state:
case zero
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
hi : Nat.zero ≤ natDegree (minpolyDiv R x)
⊢ coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - Nat.zero) - x ^ Nat.zero ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio Nat.zero)

Next tactic:
| zero => simp [(minpolyDiv_monic hx).leadingCoeff]
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => 

Proof state:
case zero
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
hi : Nat.zero ≤ natDegree (minpolyDiv R x)
⊢ coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - Nat.zero) - x ^ Nat.zero ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio Nat.zero)

Next tactic:
simp [(minpolyDiv_monic hx).leadingCoeff]
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  

Proof state:
case succ
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - Nat.succ i) - x ^ Nat.succ i ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (Nat.succ i))

Next tactic:
| succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
    · rw [Nat.succ_eq_add_one, ← tsub_tsub, tsub_add_cancel_of_le
        (le_tsub_of_add_le_left (b := 1) hi)]
      apply SetLike.le_def.mp ?_
        (Submodule.mul_mem_mul (IH ((Nat.le_succ _).trans hi))
          (Submodule.mem_span_singleton_self x))
      rw [Submodule.span_mul_span, Set.mul_singleton, Set.image_image]
      apply Submodule.span_mono
      rintro _ ⟨j, hj : j < i, rfl⟩
      exact ⟨j + 1, Nat.add_lt_of_lt_sub hj, pow_succ' x j⟩
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    

Proof state:
case succ
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - Nat.succ i) - x ^ Nat.succ i ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (Nat.succ i))

Next tactic:
rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    

Proof state:
case succ
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ coeff (minpoly R x) (natDegree (minpolyDiv R x) - Nat.succ i + 1) • 1 +
      (coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - Nat.succ i + 1) - x ^ i) * x ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (Nat.succ i))

Next tactic:
refine add_mem ?_ ?_
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · 

Proof state:
case succ.refine_1
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ coeff (minpoly R x) (natDegree (minpolyDiv R x) - Nat.succ i + 1) • 1 ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (Nat.succ i))

Next tactic:
apply Submodule.smul_mem
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      

Proof state:
case succ.refine_1.h
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ 1 ∈ Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (Nat.succ i))

Next tactic:
apply Submodule.subset_span
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      

Proof state:
case succ.refine_1.h.a
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ 1 ∈ (fun x_1 => x ^ x_1) '' Set.Iio (Nat.succ i)

Next tactic:
exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
    · 

Proof state:
case succ.refine_2
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ (coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - Nat.succ i + 1) - x ^ i) * x ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (Nat.succ i))

Next tactic:
rw [Nat.succ_eq_add_one, ← tsub_tsub, tsub_add_cancel_of_le
        (le_tsub_of_add_le_left (b := 1) hi)]
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
    · rw [Nat.succ_eq_add_one, ← tsub_tsub, tsub_add_cancel_of_le
        (le_tsub_of_add_le_left (b := 1) hi)]
      

Proof state:
case succ.refine_2
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ (coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i) * x ∈
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (i + 1))

Next tactic:
apply SetLike.le_def.mp ?_
        (Submodule.mul_mem_mul (IH ((Nat.le_succ _).trans hi))
          (Submodule.mem_span_singleton_self x))
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
    · rw [Nat.succ_eq_add_one, ← tsub_tsub, tsub_add_cancel_of_le
        (le_tsub_of_add_le_left (b := 1) hi)]
      apply SetLike.le_def.mp ?_
        (Submodule.mul_mem_mul (IH ((Nat.le_succ _).trans hi))
          (Submodule.mem_span_singleton_self x))
      

Proof state:
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i) * Submodule.span R {x} ≤
    Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (i + 1))

Next tactic:
rw [Submodule.span_mul_span, Set.mul_singleton, Set.image_image]
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
    · rw [Nat.succ_eq_add_one, ← tsub_tsub, tsub_add_cancel_of_le
        (le_tsub_of_add_le_left (b := 1) hi)]
      apply SetLike.le_def.mp ?_
        (Submodule.mul_mem_mul (IH ((Nat.le_succ _).trans hi))
          (Submodule.mem_span_singleton_self x))
      rw [Submodule.span_mul_span, Set.mul_singleton, Set.image_image]
      

Proof state:
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ Submodule.span R ((fun x_1 => x ^ x_1 * x) '' Set.Iio i) ≤ Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio (i + 1))

Next tactic:
apply Submodule.span_mono
lemma coeff_minpolyDiv_sub_pow_mem_span {i} (hi : i ≤ natDegree (minpolyDiv R x)) :
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((x ^ ·) '' Set.Iio i) := by
  induction i with
  | zero => simp [(minpolyDiv_monic hx).leadingCoeff]
  | succ i IH =>
    rw [coeff_minpolyDiv, add_sub_assoc, pow_succ', ← sub_mul, Algebra.algebraMap_eq_smul_one]
    refine add_mem ?_ ?_
    · apply Submodule.smul_mem
      apply Submodule.subset_span
      exact ⟨0, Nat.zero_lt_succ _, pow_zero _⟩
    · rw [Nat.succ_eq_add_one, ← tsub_tsub, tsub_add_cancel_of_le
        (le_tsub_of_add_le_left (b := 1) hi)]
      apply SetLike.le_def.mp ?_
        (Submodule.mul_mem_mul (IH ((Nat.le_succ _).trans hi))
          (Submodule.mem_span_singleton_self x))
      rw [Submodule.span_mul_span, Set.mul_singleton, Set.image_image]
      apply Submodule.span_mono
      

Proof state:
case h
R : Type u_2
K : Type ?u.370593
L : Type ?u.370596
S : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra K L
x : S
hx : IsIntegral R x
i : ℕ
IH :
  i ≤ natDegree (minpolyDiv R x) →
    coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) - x ^ i ∈
      Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)
hi : Nat.succ i ≤ natDegree (minpolyDiv R x)
⊢ (fun x_1 => x ^ x_1 * x) '' Set.Iio i ⊆ (fun x_1 => x ^ x_1) '' Set.Iio (i + 1)

Next tactic:
rintro _ ⟨j, hj : j < i, rfl⟩