awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (goal)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsReal} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
this_1 : CharZero.{u_2} ↥K
this : NumberField.{u_2} ↥K
w₀ :
  InfinitePlace
    ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
        IntermediateField ℚ A)
hw₀ :
  Membership.mem (γ :=
    Set
      (InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsReal}
    w₀
minkowskiBound : (↑B : ℝ≥0∞) = 1 * (↑B : ℝ≥0∞)
⊢ mixedEmbedding.minkowskiBound.{u_2} (↥K) 1 < (↑(convexBodyLTFactor.{u_2} ↥K) : ℝ≥0∞) * (↑B : ℝ≥0∞)
have minkowskiBound₁ : 1 * (↑B : ℝ≥0∞) ≤ (↑(convexBodyLTFactor.{u_2} ↥K) : ℝ≥0∞) * (↑B : ℝ≥0∞)
Tactic: calc
    minkowskiBound K 1 < B := minkowskiBound_lt_boundOfDiscBdd hK₂
    _ = 1 * B := by rw [one_mul]
    _ ≤ convexBodyLTFactor K * B := by gcongr; exact mod_cast one_le_convexBodyLTFactor K
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (goal)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
⊢ Set.Finite.{u_2} (α := { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
have refine : ∀ (x : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }) (K : IntermediateField ℚ A)
  (hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K),
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
      {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
        Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
            {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
          |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
      (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) →
    ∀
      (hK₁ :
        Set.Nonempty.{u_2} (α :=
          InfinitePlace
            ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
                IntermediateField ℚ A))
          {w :
            InfinitePlace
              ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
                  IntermediateField ℚ A) |
            w.IsComplex})
      (hK₂ :
        |discr.{u_2}
              ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
                  IntermediateField ℚ A)| ≤
          (↑N : ℤ)),
      ∃ (x : A),
        Membership.mem (γ := Set A)
            (⋃ (f : ℤ[X]),
              ⋃ (_ :
                f.natDegree ≤ D ∧ ∀ (i : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (f.coeff i)),
                (↑(roots (Polynomial.map (algebraMap ℤ A) f)).toFinset : Set A))
            x ∧
          (↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
              IntermediateField ℚ A) =
            ℚ⟮x⟯
Tactic: refine finite_of_finite_generating_set A _ (bUnion_roots_finite (algebraMap ℤ A) D
      (Set.finite_Icc (-C : ℤ) C)) (fun ⟨K, hK₀⟩ ⟨hK₁, hK₂⟩ ↦ ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (goal)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₁ :
  Set.Nonempty.{u_2} (α :=
    InfinitePlace
      ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
          IntermediateField ℚ A))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
⊢ ∃ (x : A),
    Membership.mem (γ := Set A)
        (⋃ (f : ℤ[X]),
          ⋃ (_ : f.natDegree ≤ D ∧ ∀ (i : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (f.coeff i)),
            (↑(roots (Polynomial.map (algebraMap ℤ A) f)).toFinset : Set A))
        x ∧
      (↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
          IntermediateField ℚ A) =
        ℚ⟮x⟯
have simp_rw : ∃ (x : A),
  (∃ (i : ℤ[X]) (_ :
      i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
      x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
    K = ℚ⟮x⟯
Tactic: simp_rw [Set.mem_iUnion]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (hypothesis)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₁ :
  Set.Nonempty.{u_2} (α :=
    InfinitePlace
      ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
          IntermediateField ℚ A))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
⊢ ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
have this : CharZero.{u_2} ↥K
Tactic: have : CharZero K := SubsemiringClass.instCharZero K
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (hypothesis)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₁ :
  Set.Nonempty.{u_2} (α :=
    InfinitePlace
      ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
          IntermediateField ℚ A))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
this : CharZero.{u_2} ↥K
⊢ ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
have this₁ : NumberField.{u_2} ↥K
Tactic: haveI : NumberField K := @NumberField.mk _ _ inferInstance hK₀
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (goal)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₁ :
  Set.Nonempty.{u_2} (α :=
    InfinitePlace
      ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
          IntermediateField ℚ A))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
this_1 : CharZero.{u_2} ↥K
this : NumberField.{u_2} ↥K
⊢ ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
have obtain : ∀
  (w₀ :
    InfinitePlace
      ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
          IntermediateField ℚ A))
  (hw₀ :
    Membership.mem (γ :=
      Set
        (InfinitePlace
          ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
              IntermediateField ℚ A)))
      {w :
        InfinitePlace
          ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
              IntermediateField ℚ A) |
        w.IsComplex}
      w₀),
  ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
Tactic: obtain ⟨w₀, hw₀⟩ := hK₁
Cleared hypotheses: hK₁
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (goal)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
this_1 : CharZero.{u_2} ↥K
this_2 : NumberField.{u_2} ↥K
w₀ :
  InfinitePlace
    ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
        IntermediateField ℚ A)
hw₀ :
  Membership.mem (γ :=
    Set
      (InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
    w₀
this : minkowskiBound.{u_2} (↥K) 1 < (↑(convexBodyLT'Factor.{u_2} ↥K) : ℝ≥0∞) * (↑(boundOfDiscBdd N) : ℝ≥0∞)
⊢ ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
have obtain : ∀ (x : 𝓞 ↥K) (hx₁ : Eq (α := IntermediateField ℚ ↥K) ℚ⟮(↑x : ↥K)⟯ ⊤)
  (hx₂ :
    ∀ (w : InfinitePlace ↥K), (w : (a : ↥K) → ℝ) (↑x : ↥K) < √(1 + HPow.hPow (β := ℕ) (↑(boundOfDiscBdd N) : ℝ) 2)),
  ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
Tactic: obtain ⟨x, hx₁, hx₂⟩ := exists_primitive_element_lt_of_isComplex K hw₀ this
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (hypothesis)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x_1 : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
this_1 : CharZero.{u_2} ↥K
this_2 : NumberField.{u_2} ↥K
w₀ :
  InfinitePlace
    ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
        IntermediateField ℚ A)
hw₀ :
  Membership.mem (γ :=
    Set
      (InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
    w₀
this : minkowskiBound.{u_2} (↥K) 1 < (↑(convexBodyLT'Factor.{u_2} ↥K) : ℝ≥0∞) * (↑(boundOfDiscBdd N) : ℝ≥0∞)
x : 𝓞 ↥K
hx₁ : Eq (α := IntermediateField ℚ ↥K) ℚ⟮(↑x : ↥K)⟯ ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), (w : (a : ↥K) → ℝ) (↑x : ↥K) < √(1 + HPow.hPow (β := ℕ) (↑(boundOfDiscBdd N) : ℝ) 2)
⊢ ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
have hx : IsIntegral ℤ (A := ↥K) ((algebraMap (𝓞 ↥K) ↥K : (a : 𝓞 ↥K) → ↥K) x : ↥K)
Tactic: have hx := x.isIntegral_coe
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (goal)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x_1 : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
this_1 : CharZero.{u_2} ↥K
this_2 : NumberField.{u_2} ↥K
w₀ :
  InfinitePlace
    ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
        IntermediateField ℚ A)
hw₀ :
  Membership.mem (γ :=
    Set
      (InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
    w₀
this : minkowskiBound.{u_2} (↥K) 1 < (↑(convexBodyLT'Factor.{u_2} ↥K) : ℝ≥0∞) * (↑(boundOfDiscBdd N) : ℝ≥0∞)
x : 𝓞 ↥K
hx₁ : Eq (α := IntermediateField ℚ ↥K) ℚ⟮(↑x : ↥K)⟯ ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), (w : (a : ↥K) → ℝ) (↑x : ↥K) < √(1 + HPow.hPow (β := ℕ) (↑(boundOfDiscBdd N) : ℝ) 2)
hx : IsIntegral ℤ (A := ↥K) ((algebraMap (𝓞 ↥K) ↥K : (a : 𝓞 ↥K) → ↥K) x : ↥K)
⊢ ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
have refine_1 : natDegree (minpoly ℤ (↑x : ↥K)) ≤ D
Tactic: refine ⟨x, ⟨⟨minpoly ℤ (x : K), ⟨?_, fun i ↦ ?_⟩, ?_⟩, ?_⟩⟩
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Discriminant.Basic (goal)
Goal:
A : Type u_2
inst : Field A
inst_1 : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ :=
  ⌈HPow.hPow (α := ℝ≥0) (max ((sqrt : (a : ℝ≥0) → ℝ≥0) (1 + HPow.hPow (β := ℕ) B 2)) 1 : ℝ≥0) D *
      (↑(D.choose (D / 2)) : ℝ≥0)⌉₊
x_1 : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional.{0, u_2} ℚ ↥K
h :
  Membership.mem (γ := Set { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F })
    {K : { F : IntermediateField ℚ A // FiniteDimensional.{0, u_2} ℚ ↥F } |
      Set.Nonempty.{u_2} (α := InfinitePlace ↥(↑K : IntermediateField ℚ A))
          {w : InfinitePlace ↥(↑K : IntermediateField ℚ A) | w.IsComplex} ∧
        |discr.{u_2} ↥(↑K : IntermediateField ℚ A)| ≤ (↑N : ℤ)}
    (Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀)
hK₂ :
  |discr.{u_2}
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)| ≤
    (↑N : ℤ)
this_1 : CharZero.{u_2} ↥K
this_2 : NumberField.{u_2} ↥K
w₀ :
  InfinitePlace
    ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
        IntermediateField ℚ A)
hw₀ :
  Membership.mem (γ :=
    Set
      (InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A)))
    {w :
      InfinitePlace
        ↥(↑(Subtype.mk (p := fun (F : IntermediateField ℚ A) ↦ FiniteDimensional.{0, u_2} ℚ ↥F) K hK₀) :
            IntermediateField ℚ A) |
      w.IsComplex}
    w₀
this : minkowskiBound.{u_2} (↥K) 1 < (↑(convexBodyLT'Factor.{u_2} ↥K) : ℝ≥0∞) * (↑(boundOfDiscBdd N) : ℝ≥0∞)
x : 𝓞 ↥K
hx₁ : Eq (α := IntermediateField ℚ ↥K) ℚ⟮(↑x : ↥K)⟯ ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), (w : (a : ↥K) → ℝ) (↑x : ↥K) < √(1 + HPow.hPow (β := ℕ) (↑(boundOfDiscBdd N) : ℝ) 2)
hx : IsIntegral ℤ (A := ↥K) ((algebraMap (𝓞 ↥K) ↥K : (a : 𝓞 ↥K) → ↥K) x : ↥K)
refine_1 : natDegree (minpoly ℤ (↑x : ↥K)) ≤ D
⊢ ∃ (x : A),
    (∃ (i : ℤ[X]) (_ :
        i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (i.coeff i_1)),
        x ∈ (↑(roots (Polynomial.map (algebraMap ℤ A) i)).toFinset : Set A)) ∧
      K = ℚ⟮x⟯
have refine_2 : ∀ (i : ℕ), Membership.mem (γ := Set ℤ) (Set.Icc (-(↑C : ℤ)) (↑C : ℤ)) (coeff (minpoly ℤ (↑x : ↥K)) i)
Tactic: refine ⟨x, ⟨⟨minpoly ℤ (x : K), ⟨?_, fun i ↦ ?_⟩, ?_⟩, ?_⟩⟩