awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
K : Type u
inst_3 : Field K
inst_4 : Algebra F K
L : Type u_3
inst_5 : Field L
inst_6 : Algebra K L
x : L
hx : IsIntegral K x
i : Fin (natDegree (minpoly K x))
rw :
  (((AdjoinRoot.powerBasis ⋯).basis.map (↑(adjoinRootEquivAdjoin K hx) : AdjoinRoot (minpoly K x) ≃ₗ[K] ↥K⟮x⟯)).reindex
          (finCongr ⋯) :
        (a : Fin (minpoly K x).natDegree) → ↥K⟮x⟯)
      i =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
rw₁ :
  ((AdjoinRoot.powerBasis ⋯).basis.map (↑(adjoinRootEquivAdjoin K hx) : AdjoinRoot (minpoly K x) ≃ₗ[K] ↥K⟮x⟯) :
        (a : Fin (AdjoinRoot.powerBasis ⋯).dim) → ↥K⟮x⟯)
      (((finCongr ⋯).symm : (a : Fin (minpoly K x).natDegree) → Fin (AdjoinRoot.powerBasis ⋯).dim) i :
        Fin (AdjoinRoot.powerBasis ⋯).dim) =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
rw₂ :
  ((↑(adjoinRootEquivAdjoin K hx) : AdjoinRoot (minpoly K x) ≃ₗ[K] ↥K⟮x⟯) : (a : AdjoinRoot (minpoly K x)) → ↥K⟮x⟯)
      (((AdjoinRoot.powerBasis ⋯).basis : (a : Fin (AdjoinRoot.powerBasis ⋯).dim) → AdjoinRoot (minpoly K x))
        (((finCongr ⋯).symm : (a : Fin (minpoly K x).natDegree) → Fin (AdjoinRoot.powerBasis ⋯).dim) i :
          Fin (AdjoinRoot.powerBasis ⋯).dim)) =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
rw₃ :
  ((↑(adjoinRootEquivAdjoin K hx) : AdjoinRoot (minpoly K x) ≃ₗ[K] ↥K⟮x⟯) : (a : AdjoinRoot (minpoly K x)) → ↥K⟮x⟯)
      ((AdjoinRoot.powerBasis ⋯).gen ^
        (↑(((finCongr ⋯).symm : (a : Fin (minpoly K x).natDegree) → Fin (AdjoinRoot.powerBasis ⋯).dim) i) : ℕ)) =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
rw₄ :
  ((↑(adjoinRootEquivAdjoin K hx) : AdjoinRoot (minpoly K x) ≃ₗ[K] ↥K⟮x⟯) : (a : AdjoinRoot (minpoly K x)) → ↥K⟮x⟯)
      ((AdjoinRoot.powerBasis ⋯).gen ^
        (↑((finCongr ⋯ : (a : Fin (minpoly K x).natDegree) → Fin (AdjoinRoot.powerBasis ⋯).dim) i) : ℕ)) =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
rw₅ :
  ((↑(adjoinRootEquivAdjoin K hx) : AdjoinRoot (minpoly K x) ≃ₗ[K] ↥K⟮x⟯) : (a : AdjoinRoot (minpoly K x)) → ↥K⟮x⟯)
      ((AdjoinRoot.powerBasis ⋯).gen ^ (↑(Fin.cast ⋯ i) : ℕ)) =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
rw₆ :
  ((↑(adjoinRootEquivAdjoin K hx) : AdjoinRoot (minpoly K x) ≃ₗ[K] ↥K⟮x⟯) : (a : AdjoinRoot (minpoly K x)) → ↥K⟮x⟯)
      ((AdjoinRoot.powerBasis ⋯).gen ^ (↑i : ℕ)) =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
⊢ (powerBasisAux hx : (a : Fin (minpoly K x).natDegree) → ↥K⟮x⟯) i =
    HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
have rw₇ : (adjoinRootEquivAdjoin K hx : (a : AdjoinRoot (minpoly K x)) → ↥K⟮x⟯) ((AdjoinRoot.powerBasis sorry).gen ^ (↑i : ℕ)) =
  HPow.hPow (α := ↥K⟮x⟯) (AdjoinSimple.gen K x) (↑i : ℕ)
Tactic: rw [powerBasisAux, Basis.reindex_apply, Basis.map_apply, PowerBasis.basis_eq_pow,
      finCongr_symm, finCongr_apply, Fin.cast_eq_self, AlgEquiv.toLinearEquiv_apply,
      map_pow, AdjoinRoot.powerBasis_gen, adjoinRootEquivAdjoin_apply_root]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
K : Type u
inst : Field K
L : Type u_3
inst_1 : Field L
inst_2 : Algebra K L
x : L
hx : IsIntegral K x
⊢ Module.finrank.{u, u_3} K ↥K⟮x⟯ = natDegree (minpoly K x)
have rw : (powerBasis hx).dim = natDegree (minpoly K x)
Tactic: rw [PowerBasis.finrank (adjoin.powerBasis hx :)]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
S : Set K
hprim : Eq (α := IntermediateField F K) (adjoin F S) ⊤
⊢ Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) (restrictScalars.{u_1, u, u_2} F (L' := E) ⊤)
have rw : Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) ⊤
Tactic: rw [restrictScalars_top, ← top_le_iff, ← hprim, adjoin_le_iff,
      coe_restrictScalars, ← adjoin_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
S : Set K
hprim : Eq (α := IntermediateField F K) (adjoin F S) ⊤
rw : Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) ⊤
⊢ Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) (restrictScalars.{u_1, u, u_2} F (L' := E) ⊤)
have rw₁ : LE.le (α := IntermediateField F K) ⊤ (restrictScalars F (adjoin E S))
Tactic: rw [restrictScalars_top, ← top_le_iff, ← hprim, adjoin_le_iff,
      coe_restrictScalars, ← adjoin_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
S : Set K
hprim : Eq (α := IntermediateField F K) (adjoin F S) ⊤
rw : Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) ⊤
rw₁ : LE.le (α := IntermediateField F K) ⊤ (restrictScalars F (adjoin E S))
⊢ Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) (restrictScalars.{u_1, u, u_2} F (L' := E) ⊤)
have rw₂ : LE.le (α := IntermediateField F K) (adjoin F S) (restrictScalars F (adjoin E S))
Tactic: rw [restrictScalars_top, ← top_le_iff, ← hprim, adjoin_le_iff,
      coe_restrictScalars, ← adjoin_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
S : Set K
hprim : Eq (α := IntermediateField F K) (adjoin F S) ⊤
rw : Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) ⊤
rw₁ : LE.le (α := IntermediateField F K) ⊤ (restrictScalars F (adjoin E S))
rw₂ : LE.le (α := IntermediateField F K) (adjoin F S) (restrictScalars F (adjoin E S))
⊢ Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) (restrictScalars.{u_1, u, u_2} F (L' := E) ⊤)
have rw₃ : S ⊆ (↑(restrictScalars F (adjoin E S)) : Set K)
Tactic: rw [restrictScalars_top, ← top_le_iff, ← hprim, adjoin_le_iff,
      coe_restrictScalars, ← adjoin_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
S : Set K
hprim : Eq (α := IntermediateField F K) (adjoin F S) ⊤
rw : Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) ⊤
rw₁ : LE.le (α := IntermediateField F K) ⊤ (restrictScalars F (adjoin E S))
rw₂ : LE.le (α := IntermediateField F K) (adjoin F S) (restrictScalars F (adjoin E S))
rw₃ : S ⊆ (↑(restrictScalars F (adjoin E S)) : Set K)
⊢ Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) (restrictScalars.{u_1, u, u_2} F (L' := E) ⊤)
have rw₄ : S ⊆ (↑(adjoin E S) : Set K)
Tactic: rw [restrictScalars_top, ← top_le_iff, ← hprim, adjoin_le_iff,
      coe_restrictScalars, ← adjoin_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
S : Set K
hprim : Eq (α := IntermediateField F K) (adjoin F S) ⊤
rw : Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) ⊤
rw₁ : LE.le (α := IntermediateField F K) ⊤ (restrictScalars F (adjoin E S))
rw₂ : LE.le (α := IntermediateField F K) (adjoin F S) (restrictScalars F (adjoin E S))
rw₃ : S ⊆ (↑(restrictScalars F (adjoin E S)) : Set K)
rw₄ : S ⊆ (↑(adjoin E S) : Set K)
⊢ Eq (α := IntermediateField F K) (restrictScalars F (adjoin E S)) (restrictScalars.{u_1, u, u_2} F (L' := E) ⊤)
have rw₅ : LE.le (α := IntermediateField E K) (adjoin E S) (adjoin E S)
Tactic: rw [restrictScalars_top, ← top_le_iff, ← hprim, adjoin_le_iff,
      coe_restrictScalars, ← adjoin_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
⊢ K' ≤ K
have refine : ∀ x ∈ (↑g.coeffs : Set E), x ∈ (↑K : Set E)
Tactic: refine adjoin_le_iff.mpr fun x ↦ ?_
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
x : E
⊢ x ∈ (↑g.coeffs : Set E) → x ∈ (↑K : Set E)
have rw : (∃ n ∈ g.support, x = g.coeff n) → x ∈ (↑K : Set E)
Tactic: rw [Finset.mem_coe, mem_coeffs_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
n : ℕ
⊢ g.coeff n ∈ (↑K : Set E)
have rw : Membership.mem (α := E) (↑K : Set E) ((algebraMap (↥K) E : (a : ↥K) → E) (coeff (minpoly (↥K) α) n) : E)
Tactic: rw [coeff_map]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
⊢ K' = K
have hsub : K' ≤ K
Tactic: have hsub : K' ≤ K := by
    refine adjoin_le_iff.mpr fun x ↦ ?_
    rw [Finset.mem_coe, mem_coeffs_iff]
    rintro ⟨n, -, rfl⟩
    rw [coeff_map]
    apply Subtype.mem
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
⊢ Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
have hp : Eq (α := E) ((aeval α : (a : (↥K')[X]) → E) (g.toSubring K'.toSubring sorry) : E) (0 : E)
Tactic: apply minpoly.dvd
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
⊢ Eq (α := E) ((aeval α : (a : (↥K')[X]) → E) (g.toSubring K'.toSubring ⋯) : E) (0 : E)
have hp : eval α (Polynomial.map (R := ↥K') (algebraMap (↥K') E) (g.toSubring K'.toSubring sorry)) = 0
Tactic: rw [aeval_def, eval₂_eq_eval_map]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
⊢ eval α (Polynomial.map (R := ↥K') (algebraMap (↥K') E) (g.toSubring K'.toSubring ⋯)) = 0
have hp : eval α g = 0
Tactic: erw [g.map_toSubring K'.toSubring]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
⊢ eval α g = 0
have hp : Eq (α := E) ((aeval α : (a : (↥K)[X]) → E) (minpoly (↥K) α) : E) (0 : E)
Tactic: rw [eval_map_algebraMap]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
⊢ K' = K
have dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring sorry)
Tactic: have dvd_g : minpoly K' α ∣ g.toSubring K'.toSubring (subset_adjoin F _) := by
    apply minpoly.dvd
    rw [aeval_def, eval₂_eq_eval_map]
    erw [g.map_toSubring K'.toSubring]
    rw [eval_map_algebraMap]
    exact minpoly.aeval K α
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K_1 : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K_1) (algebraMap (↥K_1) E) (minpoly (↥K_1) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K_1
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
K : IntermediateField F E
⊢ finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
have this : finrank.{u_2, u_2} ↥K ↥(↥K)⟮α⟯ = natDegree (R := ↥K) (minpoly (↥K) α)
Tactic: have := adjoin.finrank (.of_finite K α)
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K_1 : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K_1) (algebraMap (↥K_1) E) (minpoly (↥K_1) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K_1
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
K : IntermediateField F E
this : finrank.{u_2, u_2} ↥K ↥(↥K)⟮α⟯ = natDegree (R := ↥K) (minpoly (↥K) α)
⊢ finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
have this₁ : finrank.{u_2, u_2} ↥K ↥⊤ = natDegree (R := ↥K) (minpoly (↥K) α)
Tactic: rw [adjoin_eq_top_of_adjoin_eq_top F hprim] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
⊢ K' = K
have finrank_eq : ∀ (K : IntermediateField F E), finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
Tactic: have finrank_eq : ∀ K : IntermediateField F E, finrank K E = natDegree (minpoly K α) := by
    intro K
    have := adjoin.finrank (.of_finite K α)
    rw [adjoin_eq_top_of_adjoin_eq_top F hprim] at this
    simp_all
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
finrank_eq : ∀ (K : IntermediateField F E), finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
⊢ K' = K
have refine : finrank (↥K') E ≤ finrank (↥K) E
Tactic: refine eq_of_le_of_finrank_le' hsub ?_
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
finrank_eq : ∀ (K : IntermediateField F E), finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
⊢ finrank (↥K') E ≤ finrank (↥K) E
have simp_rw : natDegree (R := ↥K') (minpoly (↥K') α) ≤ natDegree (R := ↥K) (minpoly (↥K) α)
Tactic: simp_rw [finrank_eq]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
finrank_eq : ∀ (K : IntermediateField F E), finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
⊢ natDegree (R := ↥K') (minpoly (↥K') α) ≤ natDegree (R := ↥K) (minpoly (↥K) α)
have h : natDegree (R := ↥K) (minpoly (↥K) α) = natDegree (R := ↥K') (g.toSubring K'.toSubring sorry)
Tactic: convert!
    natDegree_le_of_dvd dvd_g
      ((g.monic_toSubring _ _).mpr <| (minpoly.monic <| .of_finite K α).map _).ne_zero using 1
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
finrank_eq : ∀ (K : IntermediateField F E), finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
⊢ natDegree (R := ↥K) (minpoly (↥K) α) = natDegree (R := ↥K') (g.toSubring K'.toSubring ⋯)
have h : natDegree (R := ↥K) (minpoly (↥K) α) = g.natDegree
Tactic: rw [natDegree_toSubring, natDegree_map]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
α : E
inst_3 : FiniteDimensional F E
hprim : Eq (α := IntermediateField F E) F⟮α⟯ ⊤
K : IntermediateField F E
g : E[X] := Polynomial.map (R := ↥K) (algebraMap (↥K) E) (minpoly (↥K) α)
K' : IntermediateField F E := adjoin F (↑g.coeffs : Set E)
hsub : K' ≤ K
dvd_g : Dvd.dvd (α := (↥K')[X]) (minpoly (↥K') α) (g.toSubring K'.toSubring ⋯)
finrank_eq : ∀ (K : IntermediateField F E), finrank (↥K) E = natDegree (R := ↥K) (minpoly (↥K) α)
h : natDegree (R := ↥K) (minpoly (↥K) α) = g.natDegree
⊢ natDegree (R := ↥K) (minpoly (↥K) α) = natDegree (R := ↥K') (g.toSubring K'.toSubring ⋯)
have h₁ : natDegree (R := ↥K) (minpoly (↥K) α) = natDegree (R := ↥K) (minpoly (↥K) α)
Tactic: rw [natDegree_toSubring, natDegree_map]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.IsAlgebraic F E
hnfd : ¬FiniteDimensional F E
n : ℕ
ih : ∃ (L : IntermediateField F E), FiniteDimensional.{u_1, u_2} F ↥L ∧ n < finrank.{u_1, u_2} F ↥L
⊢ ∃ (L : IntermediateField F E), FiniteDimensional.{u_1, u_2} F ↥L ∧ n + 1 < finrank.{u_1, u_2} F ↥L
have succ : ∀ (L : IntermediateField F E) (fin : FiniteDimensional.{u_1, u_2} F ↥L) (hn : n < finrank.{u_1, u_2} F ↥L),
  ∃ (L : IntermediateField F E), FiniteDimensional.{u_1, u_2} F ↥L ∧ n + 1 < finrank.{u_1, u_2} F ↥L
Tactic: obtain ⟨L, fin, hn⟩ := ih
Cleared hypotheses: ih
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.IsAlgebraic F E
hnfd : ¬FiniteDimensional F E
n : ℕ
L : IntermediateField F E
fin : FiniteDimensional.{u_1, u_2} F ↥L
hn : n < finrank.{u_1, u_2} F ↥L
⊢ ∃ (x : E), x ∉ L
have contrapose! : ∀ (hnfd : ∀ (x : E), x ∈ L), FiniteDimensional F E
Tactic: contrapose! hnfd
Cleared hypotheses: hnfd
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.IsAlgebraic F E
n : ℕ
L : IntermediateField F E
fin : FiniteDimensional.{u_1, u_2} F ↥L
hn : n < finrank.{u_1, u_2} F ↥L
hnfd : ∀ (x : E), x ∈ L
⊢ FiniteDimensional F E
have fin₁ : FiniteDimensional.{u_1, u_2} F ↥⊤
Tactic: rw [show L = ⊤ from eq_top_iff.2 fun x _ ↦ hnfd x] at fin
Cleared hypotheses: fin, hn, hnfd
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.IsAlgebraic F E
hnfd : ¬FiniteDimensional F E
n : ℕ
L : IntermediateField F E
fin : FiniteDimensional.{u_1, u_2} F ↥L
hn : n < finrank.{u_1, u_2} F ↥L
⊢ ∃ (L : IntermediateField F E), FiniteDimensional.{u_1, u_2} F ↥L ∧ n + 1 < finrank.{u_1, u_2} F ↥L
have succ : ∀ (x : E) (hx : x ∉ L),
  ∃ (L : IntermediateField F E), FiniteDimensional.{u_1, u_2} F ↥L ∧ n + 1 < finrank.{u_1, u_2} F ↥L
Tactic: obtain ⟨x, hx⟩ : ∃ x : E, x ∉ L := by
      contrapose! hnfd
      rw [show L = ⊤ from eq_top_iff.2 fun x _ ↦ hnfd x] at fin
      exact topEquiv.toLinearEquiv.finiteDimensional
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.IsAlgebraic F E
hnfd : ¬FiniteDimensional F E
n : ℕ
L : IntermediateField F E
fin : FiniteDimensional.{u_1, u_2} F ↥L
hn : n < finrank.{u_1, u_2} F ↥L
x : E
hx : x ∉ L
L' : IntermediateField F E := L ⊔ F⟮x⟯
⊢ ∃ (L : IntermediateField F E), FiniteDimensional.{u_1, u_2} F ↥L ∧ n + 1 < finrank.{u_1, u_2} F ↥L
have this : FiniteDimensional.{u_1, u_2} F ↥F⟮x⟯
Tactic: haveI := adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral (R := F) x)
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.IsAlgebraic F E
hnfd : ¬FiniteDimensional F E
n : ℕ
L : IntermediateField F E
fin : FiniteDimensional.{u_1, u_2} F ↥L
hn : n < finrank.{u_1, u_2} F ↥L
x : E
hx : x ∉ L
L' : IntermediateField F E := L ⊔ F⟮x⟯
this : FiniteDimensional.{u_1, u_2} F ↥F⟮x⟯
⊢ ∃ (L : IntermediateField F E), FiniteDimensional.{u_1, u_2} F ↥L ∧ n + 1 < finrank.{u_1, u_2} F ↥L
have succ : ∀ (h : ¬n + 1 < finrank.{u_1, u_2} F ↥L'), False
Tactic: refine ⟨L', inferInstance, by_contra fun h ↦ ?_⟩