l3lab/ntp-mathlib huggingface dataset:
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; 

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r : F[X]
h : α ^ m * (aeval (α ^ n)) 0 - (aeval (α ^ n)) r = 0
hzero : ¬(aeval (α ^ n)) 0 = 0
⊢ False

Next tactic:
simp only [map_zero, not_true_eq_false] at hzero
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    

Proof state:
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
⊢ IsAlgebraic F α

Next tactic:
let f : F[X] := X ^ m * expand F n s - expand F n r
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    

Proof state:
case neg
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ IsAlgebraic F α

Next tactic:
refine ⟨f, ?_, ?_⟩
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · 

Proof state:
case neg.refine_1
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ f ≠ 0

Next tactic:
have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          exact Nat.not_dvd_of_pos_of_lt hm hmn
        simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
          coeff_expand hn r, hndvd, ite_false, sub_zero]
        exact leadingCoeff_ne_zero.2 hzero
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ coeff f (n * natDegree s + m) ≠ 0

Next tactic:
have hn : 0 < n := by linarith only [hm, hmn]
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by 

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ 0 < n

Next tactic:
linarith only [hm, hmn]
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
⊢ coeff f (n * natDegree s + m) ≠ 0

Next tactic:
have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          exact Nat.not_dvd_of_pos_of_lt hm hmn
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
⊢ ¬n ∣ n * natDegree s + m

Next tactic:
rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
⊢ ¬n ∣ m

Next tactic:
exact Nat.not_dvd_of_pos_of_lt hm hmn
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          exact Nat.not_dvd_of_pos_of_lt hm hmn
        

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
hndvd : ¬n ∣ n * natDegree s + m
⊢ coeff f (n * natDegree s + m) ≠ 0

Next tactic:
simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
          coeff_expand hn r, hndvd, ite_false, sub_zero]
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          exact Nat.not_dvd_of_pos_of_lt hm hmn
        simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
          coeff_expand hn r, hndvd, ite_false, sub_zero]
        

Proof state:
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
hn : 0 < n
hndvd : ¬n ∣ n * natDegree s + m
⊢ leadingCoeff s ≠ 0

Next tactic:
exact leadingCoeff_ne_zero.2 hzero
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          exact Nat.not_dvd_of_pos_of_lt hm hmn
        simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
          coeff_expand hn r, hndvd, ite_false, sub_zero]
        exact leadingCoeff_ne_zero.2 hzero
      

Proof state:
case neg.refine_1
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
this : coeff f (n * natDegree s + m) ≠ 0
⊢ f ≠ 0

Next tactic:
intro h
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          exact Nat.not_dvd_of_pos_of_lt hm hmn
        simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
          coeff_expand hn r, hndvd, ite_false, sub_zero]
        exact leadingCoeff_ne_zero.2 hzero
      intro h
      

Proof state:
case neg.refine_1
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h✝ : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
this : coeff f (n * natDegree s + m) ≠ 0
h : f = 0
⊢ False

Next tactic:
simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
    (heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by
  wlog hmn : m < n
  · exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
  by_cases hm : m = 0
  · rw [hm] at heq hmn
    simp only [pow_zero, adjoin_one] at heq
    obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
    refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
    simp only [map_sub, map_pow, aeval_X, aeval_C, h, sub_self]
  replace hm : 0 < m := Nat.pos_of_ne_zero hm
  obtain ⟨r, s, h⟩ := (mem_adjoin_simple_iff F _).1 (heq ▸ mem_adjoin_simple_self F (α ^ m))
  by_cases hzero : aeval (α ^ n) s = 0
  · simp only [hzero, div_zero, pow_eq_zero_iff hm] at h
    exact h.symm ▸ isAlgebraic_zero
  · rw [eq_div_iff hzero, ← sub_eq_zero] at h
    replace hzero : s ≠ 0 := by rintro rfl; simp only [map_zero, not_true_eq_false] at hzero
    let f : F[X] := X ^ m * expand F n s - expand F n r
    refine ⟨f, ?_, ?_⟩
    · have : f.coeff (n * s.natDegree + m) ≠ 0 := by
        have hn : 0 < n := by linarith only [hm, hmn]
        have hndvd : ¬ n ∣ n * s.natDegree + m := by
          rw [← Nat.dvd_add_iff_right (n.dvd_mul_right s.natDegree)]
          exact Nat.not_dvd_of_pos_of_lt hm hmn
        simp only [coeff_sub, coeff_X_pow_mul, s.coeff_expand_mul' hn, coeff_natDegree,
          coeff_expand hn r, hndvd, ite_false, sub_zero]
        exact leadingCoeff_ne_zero.2 hzero
      intro h
      simp only [h, coeff_zero, ne_eq, not_true_eq_false] at this
    · 

Proof state:
case neg.refine_2
F✝ : Type u_1
E✝ : Type u_2
inst✝⁵ : Field F✝
inst✝⁴ : Field E✝
inst✝³ : Algebra F✝ E✝
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
m n : ℕ
hneq : m ≠ n
heq : F⟮α ^ m⟯ = F⟮α ^ n⟯
hmn : m < n
hm : 0 < m
r s : F[X]
h : α ^ m * (aeval (α ^ n)) s - (aeval (α ^ n)) r = 0
hzero : s ≠ 0
f : F[X] := X ^ m * (expand F n) s - (expand F n) r
⊢ (aeval α) f = 0

Next tactic:
simp only [map_sub, map_mul, map_pow, aeval_X, expand_aeval, h]
theorem finiteDimensional_of_finite_intermediateField
    [Finite (IntermediateField F E)] : FiniteDimensional F E := by
  

Proof state:
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
⊢ FiniteDimensional F E

Next tactic:
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }