awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
p q : K[X]
hq : q ≠ 0
hf :
  HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
      ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯) ≠
    0
⊢ ∃ (z : ℤ),
    (v : (a : K⟮X⟯) → Γ)
        (HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
          ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯)) =
      HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) z
have h : (v : (a : K⟮X⟯) → Γ)
    (HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
      ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯)) =
  HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ)
    (HSub.hSub (α := ℤ)
      (↑(Associates.count (Associates.mk Pᵥ.asIdeal) (Associates.factors (Associates.mk (Ideal.span {p})))) : ℤ)
      (↑(Associates.count (Associates.mk Pᵥ.asIdeal) (Associates.factors (Associates.mk (Ideal.span {q})))) : ℤ))
Tactic: use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -
      (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
f : K⟮X⟯
hf : f ≠ 0
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
⊢ ∃ (z : ℤ), (v : (a : K⟮X⟯) → Γ) f = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) z
have f₁ : ∀ (p q : K[X]) (hq : q ≠ 0)
  (hf :
    HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
        ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯) ≠
      0),
  ∃ (z : ℤ),
    (v : (a : K⟮X⟯) → Γ)
        (HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
          ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯)) =
      HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) z
Tactic: induction f using RatFunc.induction_on with
  | f p q hq =>
    use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -
      (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors
    simp only [map_div₀, map_pow, zpow_sub₀ h0, zpow_natCast,
      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hq,
      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle
        (p := p) (by aesop)]
Cleared hypotheses: f, hf
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (hypothesis)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
⊢ v.IsUniformizer (↑πᵥ : K⟮X⟯)
have h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
Tactic: have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
⊢ v.IsUniformizer (↑πᵥ : K⟮X⟯)
have rw : sorry.unit = LinearOrderedCommGroup.Subgroup.genLTOne (MonoidWithZeroHom.valueGroup v)
Tactic: rw [IsUniformizer, ← hv.valueGroup_genLTOne_eq_generator, ← h0.isUnit.unit_spec, Units.val_inj]
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
⊢ ⋯.unit = LinearOrderedCommGroup.Subgroup.genLTOne (MonoidWithZeroHom.valueGroup v)
have hg : sorry.unit < 1
Tactic: apply LinearOrderedCommGroup.Subgroup.genLTOne_unique
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
hg : ⋯.unit < 1
⊢ ⋯.unit = LinearOrderedCommGroup.Subgroup.genLTOne (MonoidWithZeroHom.valueGroup v)
have hH : Subgroup.zpowers sorry.unit = MonoidWithZeroHom.valueGroup v
Tactic: apply LinearOrderedCommGroup.Subgroup.genLTOne_unique
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
⊢ ⋯.unit < 1
have hg : LT.lt.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (1 : Γ)
Tactic: rw [← Units.val_lt_val, h0.isUnit.unit_spec, Units.val_one]
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
⊢ Subgroup.zpowers ⋯.unit = MonoidWithZeroHom.valueGroup v
have hH : ∀ (γ : Γˣ),
  Membership.mem (γ := Subgroup Γˣ) (Subgroup.zpowers sorry.unit) γ ↔
    Membership.mem (γ := Subgroup Γˣ) (MonoidWithZeroHom.valueGroup v) γ
Tactic: ext γ
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
⊢ Membership.mem (γ := Subgroup Γˣ) (Subgroup.zpowers ⋯.unit) γ ↔
    Membership.mem (γ := Subgroup Γˣ) (MonoidWithZeroHom.valueGroup v) γ
have hH : (∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ k = γ) ↔
  ∃ (a : K⟮X⟯), ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
Tactic: simp only [coePolynomial_eq_algebraMap, MonoidWithZeroHom.mem_valueGroup_iff_of_comm, ne_eq,
      map_eq_zero, Subgroup.mem_zpowers_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
⊢ (∃ (k : ℤ),
      IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ) ↔
    ∃ (a : K⟮X⟯),
      ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
have hH : (∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ k = γ) →
  ∀ (k : ℤ)
    (hk : IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ k = γ),
    ∃ (a : K⟮X⟯),
      ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
Tactic: refine ⟨fun ⟨k, hk⟩ ↦ ?_, fun ⟨a, ha, b, hab⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
hH :
  (∃ (k : ℤ),
      IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ) →
    ∀ (k : ℤ)
      (hk : IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ),
      ∃ (a : K⟮X⟯),
        ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
⊢ (∃ (k : ℤ),
      IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ) ↔
    ∃ (a : K⟮X⟯),
      ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
have hH₁ : (∃ (a : K⟮X⟯),
    ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x) →
  ∀ (a : K⟮X⟯) (ha : ¬a = 0) (b : K⟮X⟯)
    (hab : HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) b),
    ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ k = γ
Tactic: refine ⟨fun ⟨k, hk⟩ ↦ ?_, fun ⟨a, ha, b, hab⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
h : ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
k : ℤ
hk : IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
⊢ ∃ (a : K⟮X⟯), ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
have h : HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) 1 : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) ((↑πᵥ : K⟮X⟯) ^ k)
Tactic: use 1, one_ne_zero, πᵥ ^ k
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (hypothesis)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
h : ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
k : ℤ
hk : IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
⊢ HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) 1 : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) ((↑πᵥ : K⟮X⟯) ^ k)
have hk₁ : HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) k = (↑γ : Γ)
Tactic: simp only [← Units.val_inj, Units.val_zpow_eq_zpow_val, h0.isUnit.unit_spec] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
h :
  ∃ (a : K⟮X⟯), ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
a : K⟮X⟯
ha : ¬a = 0
b : K⟮X⟯
hab : HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) b
⊢ ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
have hH : ∀ (ka : ℤ) (hka : (v : (a : K⟮X⟯) → Γ) a = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka),
  ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ k = γ
Tactic: obtain ⟨ka, hka⟩ := exists_zpow_uniformizingPolynomial hle ha
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
h :
  ∃ (a : K⟮X⟯), ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
a : K⟮X⟯
ha : ¬a = 0
b : K⟮X⟯
hab : HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) b
ka : ℤ
hka : (v : (a : K⟮X⟯) → Γ) a = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka
⊢ ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
have hH : ∀ (kb : ℤ) (hkb : (v : (a : K⟮X⟯) → Γ) b = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb),
  ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ k = γ
Tactic: obtain ⟨kb, hkb⟩ := exists_zpow_uniformizingPolynomial hle (f := b) (by aesop)
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
h :
  ∃ (a : K⟮X⟯), ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
a : K⟮X⟯
ha : ¬a = 0
b : K⟮X⟯
hab : HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) b
ka : ℤ
hka : (v : (a : K⟮X⟯) → Γ) a = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka
kb : ℤ
hkb : (v : (a : K⟮X⟯) → Γ) b = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb
⊢ ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
have hH : ∀ (hka : (v : (a : K⟮X⟯) → Γ) a = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka) (kb : ℤ)
  (hab :
    HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka * (↑γ : Γ) =
      HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb)
  (hkb : (v : (a : K⟮X⟯) → Γ) b = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb),
  ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ k = γ
Tactic: rw [hka, hkb] at hab
Cleared hypotheses: hab, hka, kb, hkb
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
h :
  ∃ (a : K⟮X⟯), ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
a : K⟮X⟯
ha : ¬a = 0
b : K⟮X⟯
ka : ℤ
hka : (v : (a : K⟮X⟯) → Γ) a = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka
kb : ℤ
hab :
  HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka * (↑γ : Γ) =
    HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb
hkb : (v : (a : K⟮X⟯) → Γ) b = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb
⊢ ∃ (k : ℤ), IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ k = γ
have h : IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) sorry ^ (kb - ka) = γ
Tactic: use kb - ka
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (hypothesis)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
hv : v.IsRankOneDiscrete
h0 : Ne.{u_2 + 1} (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) (0 : Γ)
γ : Γˣ
h :
  ∃ (a : K⟮X⟯), ¬a = 0 ∧ ∃ (x : K⟮X⟯), HMul.hMul (α := Γ) ((v : (a : K⟮X⟯) → Γ) a : Γ) (↑γ : Γ) = (v : (a : K⟮X⟯) → Γ) x
a : K⟮X⟯
ha : ¬a = 0
b : K⟮X⟯
ka : ℤ
hka : (v : (a : K⟮X⟯) → Γ) a = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka
kb : ℤ
hab :
  HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka * (↑γ : Γ) =
    HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb
hkb : (v : (a : K⟮X⟯) → Γ) b = HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) kb
⊢ IsUnit.unit (a := (v : (a : K⟮X⟯) → Γ) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) πᵥ : K⟮X⟯)) ⋯ ^ (kb - ka) = γ
have this : HPow.hPow (α := Γ) ((v : (a : K⟮X⟯) → Γ) (↑πᵥ : K⟮X⟯) : Γ) ka ≠ 0
Tactic: have : v ↑πᵥ ^ ka ≠ 0 := zpow_ne_zero _ h0
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
inst_4 : v.IsRankOneDiscrete
⊢ v.IsEquiv (valuation K⟮X⟯ Pᵥ)
have rw : ∀ {x : K⟮X⟯},
  LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) x : Γ) (1 : Γ) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          x :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
Tactic: rw [isEquiv_iff_val_le_one]
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
inst_4 : v.IsRankOneDiscrete
f : K⟮X⟯
⊢ LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) f : Γ) (1 : Γ) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          f :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have inl : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) 0 : Γ) (1 : Γ) ↔
  LE.le (α := WithZero (Multiplicative ℤ))
    (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) 0 :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
Tactic: rcases eq_or_ne f 0 with rfl | hf0
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (goal)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
inst_4 : v.IsRankOneDiscrete
f : K⟮X⟯
inl :
  LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) 0 : Γ) (1 : Γ) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          0 :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
⊢ LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) f : Γ) (1 : Γ) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          f :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have inr : ∀ (hf0 : f ≠ 0),
  LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) f : Γ) (1 : Γ) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          f :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
Tactic: rcases eq_or_ne f 0 with rfl | hf0
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (hypothesis)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
inst_4 : v.IsRankOneDiscrete
p q : K[X]
hq0 : q ≠ 0
hf0 :
  HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
      ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯) ≠
    0
⊢ LE.le.{u_2} (α := Γ)
      ((v : (a : K⟮X⟯) → Γ)
          (HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
            ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯)) :
        Γ)
      (1 : Γ) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          (HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
            ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯)) :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have hp0 : p ≠ 0
Tactic: have hp0 : p ≠ 0 := by simp_all
-- Lean 4 proof step from Mathlib.NumberTheory.RatFunc.Ostrowski (hypothesis)
Goal:
K : Type u_1
Γ : Type u_2
inst : Field K
inst_1 : LinearOrderedCommGroupWithZero Γ
v : Valuation.{u_1, u_2} K⟮X⟯ Γ
inst_2 : v.IsNontrivial
inst_3 : IsTrivialOn K v
hle : LE.le.{u_2} (α := Γ) ((v : (a : K⟮X⟯) → Γ) X : Γ) (1 : Γ)
inst_4 : v.IsRankOneDiscrete
p q : K[X]
hq0 : q ≠ 0
hf0 :
  HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
      ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯) ≠
    0
hp0 : p ≠ 0
⊢ LE.le.{u_2} (α := Γ)
      ((v : (a : K⟮X⟯) → Γ)
          (HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
            ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯)) :
        Γ)
      (1 : Γ) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      (((valuation K⟮X⟯ Pᵥ : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          (HDiv.hDiv (α := K⟮X⟯) (β := K⟮X⟯) ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) p : K⟮X⟯)
            ((algebraMap K[X] K⟮X⟯ : (a : K[X]) → K⟮X⟯) q : K⟮X⟯)) :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have hpi_def : let pi := πᵥ;
pi = πᵥ
Tactic: set pi := πᵥ with hpi_def