awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
⊢ IsPurelyInseparable F E ↔ ∀ (x : E), natSepDegree (minpoly F x) = 1
have obtain : ∀ (q : ℕ), ExpChar F q → (IsPurelyInseparable F E ↔ ∀ (x : E), natSepDegree (minpoly F x) = 1)
Tactic: obtain ⟨q, _⟩ := ExpChar.exists F
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
q : ℕ
h : ExpChar F q
⊢ IsPurelyInseparable F E ↔ ∀ (x : E), natSepDegree (minpoly F x) = 1
have simp_rw : (∀ (x : E), ∃ (n : ℕ), Membership.mem (γ := Subring E) (RingHom.range.{u, v} (R := F) (algebraMap F E)) (x ^ q ^ n)) ↔
  ∀ (x : E), natSepDegree (minpoly F x) = 1
Tactic: simp_rw [isPurelyInseparable_iff_pow_mem F q, minpoly.natSepDegree_eq_one_iff_pow_mem q]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
q : ℕ
hF : ExpChar F q
⊢ IsPurelyInseparable F E ↔
    ∀ (x : E),
      ∃ (n : ℕ) (y : F),
        minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ n)) ((C : (a : F) → F[X]) y : F[X])
have simp_rw : (∀ (x : E), natSepDegree (minpoly F x) = 1) ↔
  ∀ (x : E),
    ∃ (n : ℕ) (y : F),
      minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ n)) ((C : (a : F) → F[X]) y : F[X])
Tactic: simp_rw [isPurelyInseparable_iff_natSepDegree_eq_one,
    minpoly.natSepDegree_eq_one_iff_eq_X_pow_sub_C q]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
q : ℕ
hF : ExpChar F q
⊢ IsPurelyInseparable F E ↔
    ∀ (x : E),
      ∃ (n : ℕ),
        Polynomial.map (algebraMap F E) (minpoly F x) =
          HSub.hSub (α := E[X]) (β := E[X]) X ((C : (a : E) → E[X]) x : E[X]) ^ q ^ n
have simp_rw : (∀ (x : E), natSepDegree (minpoly F x) = 1) ↔
  ∀ (x : E),
    ∃ (n : ℕ),
      Polynomial.map (algebraMap F E) (minpoly F x) =
        HSub.hSub (α := E[X]) (β := E[X]) X ((C : (a : E) → E[X]) x : E[X]) ^ q ^ n
Tactic: simp_rw [isPurelyInseparable_iff_natSepDegree_eq_one,
    minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_pow q]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
q : ℕ
inst_3 : ExpChar F q
inst_4 : IsPurelyInseparable F E
inst_5 : FiniteDimensional F E
⊢ ∃ (n : ℕ), finrank F E = q ^ n
have suffices : ∀ (F E : Type v) [inst : Field F] [inst_6 : Field E] [inst_7 : Algebra F E] (q : ℕ) [ExpChar F q]
  [IsPurelyInseparable F E] [FiniteDimensional F E], ∃ (n : ℕ), finrank F E = q ^ n
Tactic: suffices ∀ (F E : Type v) [Field F] [Field E] [Algebra F E] (q : ℕ) [ExpChar F q]
      [IsPurelyInseparable F E] [FiniteDimensional F E], ∃ n, finrank F E = q ^ n by
    simpa using this (⊥ : IntermediateField F E) E q
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
F E : Type v
inst_6 : Field F
inst_7 : Field E
inst_8 : Algebra F E
q : ℕ
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
⊢ ∃ (n : ℕ), finrank F E = q ^ n
have generalize : ∀ (d : ℕ) (hd : finrank F E = d), ∃ (n : ℕ), d = q ^ n
Tactic: generalize hd : finrank F E = d
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
⊢ ∃ (n : ℕ), d = q ^ n
have pos : ∀ (h : Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤), ∃ (n : ℕ), d = q ^ n
Tactic: by_cases h : (⊥ : IntermediateField F E) = ⊤
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
pos : ∀ (h : Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤), ∃ (n : ℕ), d = q ^ n
⊢ ∃ (n : ℕ), d = q ^ n
have neg : ∀ (h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤), ∃ (n : ℕ), d = q ^ n
Tactic: by_cases h : (⊥ : IntermediateField F E) = ⊤
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (hypothesis)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
⊢ ∃ (n : ℕ), d = q ^ n
have hd₁ : 1 = d
Tactic: rw [← finrank_top', ← h, IntermediateField.finrank_bot] at hd
Cleared hypotheses: hd, h
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
⊢ ∃ (n : ℕ), d = q ^ n
have neg : ∀ (x : E) (hx : x ∉ ⊥), ∃ (n : ℕ), d = q ^ n
Tactic: obtain ⟨x, -, hx⟩ := SetLike.exists_of_lt (lt_of_le_of_ne bot_le h :)
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
⊢ ∃ (n : ℕ), d = q ^ n
have neg : ∀ (m : ℕ) (y : F)
  (e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])),
  ∃ (n : ℕ), d = q ^ n
Tactic: obtain ⟨m, y, e⟩ := IsPurelyInseparable.minpoly_eq_X_pow_sub_C F q x
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
⊢ finrank F ↥F⟮x⟯ = q ^ m
have rw : natDegree (minpoly F x) = q ^ m
Tactic: rw [adjoin.finrank (Algebra.IsIntegral.isIntegral x), e, natDegree_sub_C, natDegree_X_pow]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
rw : natDegree (minpoly F x) = q ^ m
⊢ finrank F ↥F⟮x⟯ = q ^ m
have rw₁ : (HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])).natDegree = q ^ m
Tactic: rw [adjoin.finrank (Algebra.IsIntegral.isIntegral x), e, natDegree_sub_C, natDegree_X_pow]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
rw : natDegree (minpoly F x) = q ^ m
rw₁ : (HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])).natDegree = q ^ m
⊢ finrank F ↥F⟮x⟯ = q ^ m
have rw₂ : (HPow.hPow (α := F[X]) X (q ^ m)).natDegree = q ^ m
Tactic: rw [adjoin.finrank (Algebra.IsIntegral.isIntegral x), e, natDegree_sub_C, natDegree_X_pow]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
rw : natDegree (minpoly F x) = q ^ m
rw₁ : (HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])).natDegree = q ^ m
rw₂ : (HPow.hPow (α := F[X]) X (q ^ m)).natDegree = q ^ m
⊢ finrank F ↥F⟮x⟯ = q ^ m
have rw₃ : q ^ m = q ^ m
Tactic: rw [adjoin.finrank (Algebra.IsIntegral.isIntegral x), e, natDegree_sub_C, natDegree_X_pow]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (hypothesis)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
⊢ ∃ (n : ℕ), d = q ^ n
have this : finrank F ↥F⟮x⟯ = q ^ m
Tactic: have : finrank F F⟮x⟯ = q ^ m := by
      rw [adjoin.finrank (Algebra.IsIntegral.isIntegral x), e, natDegree_sub_C, natDegree_X_pow]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
⊢ finrank (↥F⟮x⟯) E < d
have rw : 1 < q ^ m
Tactic: rw [← hd, ← finrank_mul_finrank F F⟮x⟯, Nat.lt_mul_iff_one_lt_left finrank_pos, this]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
⊢ 1 < q ^ m
have by_contra! : ∀ (H : q ^ m ≤ 1), False
Tactic: by_contra! H
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
H : q ^ m ≤ 1
⊢ False
have refine : 1 ≤ finrank F ↥F⟮x⟯
Tactic: refine hx (finrank_adjoin_simple_eq_one_iff.mp (le_antisymm (this ▸ H) ?_))
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
⊢ ∃ (n : ℕ), d = q ^ n
have neg : ∀ (n : ℕ) (hn : finrank (↥F⟮x⟯) E = q ^ n), ∃ (n : ℕ), d = q ^ n
Tactic: obtain ⟨n, hn⟩ := IH _ (by
      rw [← hd, ← finrank_mul_finrank F F⟮x⟯, Nat.lt_mul_iff_one_lt_left finrank_pos, this]
      by_contra! H
      refine hx (finrank_adjoin_simple_eq_one_iff.mp (le_antisymm (this ▸ H) ?_))
      exact Nat.one_le_iff_ne_zero.mpr Module.finrank_pos.ne') (F⟮x⟯) rfl
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
n : ℕ
hn : finrank (↥F⟮x⟯) E = q ^ n
⊢ d = q ^ (m + n)
have rw : finrank F E = q ^ (m + n)
Tactic: rw [← hd, ← finrank_mul_finrank F F⟮x⟯, hn, pow_add, this]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
n : ℕ
hn : finrank (↥F⟮x⟯) E = q ^ n
rw : finrank F E = q ^ (m + n)
⊢ d = q ^ (m + n)
have rw₁ : finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E = q ^ (m + n)
Tactic: rw [← hd, ← finrank_mul_finrank F F⟮x⟯, hn, pow_add, this]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
n : ℕ
hn : finrank (↥F⟮x⟯) E = q ^ n
rw : finrank F E = q ^ (m + n)
rw₁ : finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E = q ^ (m + n)
⊢ d = q ^ (m + n)
have rw₂ : finrank F ↥F⟮x⟯ * q ^ n = q ^ (m + n)
Tactic: rw [← hd, ← finrank_mul_finrank F F⟮x⟯, hn, pow_add, this]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
n : ℕ
hn : finrank (↥F⟮x⟯) E = q ^ n
rw : finrank F E = q ^ (m + n)
rw₁ : finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E = q ^ (m + n)
rw₂ : finrank F ↥F⟮x⟯ * q ^ n = q ^ (m + n)
⊢ d = q ^ (m + n)
have rw₃ : finrank F ↥F⟮x⟯ * q ^ n = q ^ m * q ^ n
Tactic: rw [← hd, ← finrank_mul_finrank F F⟮x⟯, hn, pow_add, this]
-- Lean 4 proof step from Mathlib.FieldTheory.PurelyInseparable.Basic (goal)
Goal:
F_1 : Type u
E_1 : Type v
inst : Field F_1
inst_1 : Field E_1
inst_2 : Algebra F_1 E_1
q_1 : ℕ
inst_3 : ExpChar F_1 q_1
inst_4 : IsPurelyInseparable F_1 E_1
inst_5 : FiniteDimensional F_1 E_1
E : Type v
inst_6 : Field E
q d : ℕ
IH :
  ∀ m < d,
    ∀ (F : Type v) [inst : Field F] [inst_12 : Algebra F E] [ExpChar F q] [IsPurelyInseparable F E]
      [FiniteDimensional F E] (hd : finrank F E = m), ∃ (n : ℕ), m = q ^ n
F : Type v
inst_7 : Field F
inst_8 : Algebra F E
inst_9 : ExpChar F q
inst_10 : IsPurelyInseparable F E
inst_11 : FiniteDimensional F E
hd : finrank F E = d
h : ¬Eq.{v + 1} (α := IntermediateField F E) ⊥ ⊤
x : E
hx : x ∉ ⊥
m : ℕ
y : F
e : minpoly F x = HSub.hSub (β := F[X]) (HPow.hPow (α := F[X]) X (q ^ m)) ((C : (a : F) → F[X]) y : F[X])
this : finrank F ↥F⟮x⟯ = q ^ m
n : ℕ
hn : finrank (↥F⟮x⟯) E = q ^ n
rw : finrank F E = q ^ (m + n)
rw₁ : finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E = q ^ (m + n)
rw₂ : finrank F ↥F⟮x⟯ * q ^ n = q ^ (m + n)
rw₃ : finrank F ↥F⟮x⟯ * q ^ n = q ^ m * q ^ n
⊢ d = q ^ (m + n)
have rw₄ : q ^ m * q ^ n = q ^ m * q ^ n
Tactic: rw [← hd, ← finrank_mul_finrank F F⟮x⟯, hn, pow_add, this]