awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : IsMaxRank u
this :
  LE.le (α := Submodule ℤ (logSpace K))
    (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K)))
    (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hv) : (a : Fin (rank K)) → logSpace K)))
pos :
  ZLattice.covolume (E := logSpace K)
        (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K)))
        (volume : Measure (logSpace K)) /
      regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
pos₁ :
  ZLattice.covolume (E := logSpace K)
        (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K)))
        (volume : Measure (logSpace K)) /
      ZLattice.covolume (E := logSpace K)
        (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hv) : (a : Fin (rank K)) → logSpace K)))
        (volume : Measure (logSpace K)) =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
⊢ regOfFamily u / regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
have pos₂ : Eq (α := ℝ)
  (↑(AddSubgroup.relIndex (G := { w : InfinitePlace K // w ≠ w₀ } → ℝ)
        (toAddSubgroup (module_M := AddCommGroup.toIntModule ({ w : InfinitePlace K // w ≠ w₀ } → ℝ))
          (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K))))
        (toAddSubgroup (module_M := AddCommGroup.toIntModule ({ w : InfinitePlace K // w ≠ w₀ } → ℝ))
          (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hv) : (a : Fin (rank K)) → logSpace K))))) :
    ℝ)
  (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
Tactic: rw [regOfFamily_of_isMaxRank hu, regOfFamily_of_isMaxRank hv,
      covolume_div_covolume_eq_relIndex _ _ this, span_basisOfIsMaxRank hu,
      span_basisOfIsMaxRank hv, AddSubgroup.relIndex_map_map, logEmbedding_ker,
      ← OrderIso.map_sup, ← OrderIso.map_sup, ← Subgroup.relIndex_toAddSubgroup]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : IsMaxRank u
this :
  LE.le (α := Submodule ℤ (logSpace K))
    (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K)))
    (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hv) : (a : Fin (rank K)) → logSpace K)))
pos :
  ZLattice.covolume (E := logSpace K)
        (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K)))
        (volume : Measure (logSpace K)) /
      regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
pos₁ :
  ZLattice.covolume (E := logSpace K)
        (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K)))
        (volume : Measure (logSpace K)) /
      ZLattice.covolume (E := logSpace K)
        (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hv) : (a : Fin (rank K)) → logSpace K)))
        (volume : Measure (logSpace K)) =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
pos₂ :
  Eq (α := ℝ)
    (↑(AddSubgroup.relIndex (G := { w : InfinitePlace K // w ≠ w₀ } → ℝ)
          (toAddSubgroup (module_M := AddCommGroup.toIntModule ({ w : InfinitePlace K // w ≠ w₀ } → ℝ))
            (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K))))
          (toAddSubgroup (module_M := AddCommGroup.toIntModule ({ w : InfinitePlace K // w ≠ w₀ } → ℝ))
            (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hv) : (a : Fin (rank K)) → logSpace K))))) :
      ℝ)
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
⊢ regOfFamily u / regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
have pos₃ : Eq (α := ℝ)
  (↑(AddSubgroup.relIndex (G := { w : InfinitePlace K // w ≠ w₀ } → ℝ)
        (AddSubgroup.map (logEmbedding K)
          ((Subgroup.toAddSubgroup : (a : Subgroup (𝓞 K)ˣ) → AddSubgroup (Additive (𝓞 K)ˣ))
            (Subgroup.closure (Set.range u))))
        (toAddSubgroup (module_M := AddCommGroup.toIntModule ({ w : InfinitePlace K // w ≠ w₀ } → ℝ))
          (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hv) : (a : Fin (rank K)) → logSpace K))))) :
    ℝ)
  (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
Tactic: rw [regOfFamily_of_isMaxRank hu, regOfFamily_of_isMaxRank hv,
      covolume_div_covolume_eq_relIndex _ _ this, span_basisOfIsMaxRank hu,
      span_basisOfIsMaxRank hv, AddSubgroup.relIndex_map_map, logEmbedding_ker,
      ← OrderIso.map_sup, ← OrderIso.map_sup, ← Subgroup.relIndex_toAddSubgroup]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : ¬IsMaxRank u
⊢ regOfFamily u / regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
have neg : (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) = 0
Tactic: rw [regOfFamily_eq_zero hu, zero_div, eq_comm, Nat.cast_eq_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : ¬IsMaxRank u
⊢ (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0
have rw : Subgroup.FiniteIndex (Subgroup.closure (Set.range v))
Tactic: rw [← Subgroup.finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : ¬IsMaxRank u
⊢ (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) = 0
have this : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0
Tactic: have : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0 := by
      rw [← Subgroup.finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex]
      exact isMaxRank_iff_closure_finiteIndex.mp hv
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : ¬IsMaxRank u
this : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0
⊢ (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) = 0
have neg : (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) *
    (Subgroup.closure (Set.range v) ⊔ torsion K).index =
  0
Tactic: rwa [← mul_eq_zero_iff_right this, Subgroup.relIndex_mul_index h,
      ← Subgroup.not_finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex,
      ← isMaxRank_iff_closure_finiteIndex.not]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : ¬IsMaxRank u
this : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0
neg :
  (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) *
      (Subgroup.closure (Set.range v) ⊔ torsion K).index =
    0
⊢ (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) = 0
have neg₁ : (Subgroup.closure (Set.range u) ⊔ torsion K).index = 0
Tactic: rwa [← mul_eq_zero_iff_right this, Subgroup.relIndex_mul_index h,
      ← Subgroup.not_finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex,
      ← isMaxRank_iff_closure_finiteIndex.not]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : ¬IsMaxRank u
this : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0
neg :
  (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) *
      (Subgroup.closure (Set.range v) ⊔ torsion K).index =
    0
neg₁ : (Subgroup.closure (Set.range u) ⊔ torsion K).index = 0
⊢ (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) = 0
have neg₂ : ¬(Subgroup.closure (Set.range u) ⊔ torsion K).FiniteIndex
Tactic: rwa [← mul_eq_zero_iff_right this, Subgroup.relIndex_mul_index h,
      ← Subgroup.not_finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex,
      ← isMaxRank_iff_closure_finiteIndex.not]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
hu : ¬IsMaxRank u
this : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0
neg :
  (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) *
      (Subgroup.closure (Set.range v) ⊔ torsion K).index =
    0
neg₁ : (Subgroup.closure (Set.range u) ⊔ torsion K).index = 0
neg₂ : ¬(Subgroup.closure (Set.range u) ⊔ torsion K).FiniteIndex
⊢ (Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K) = 0
have neg₃ : ¬Subgroup.FiniteIndex (Subgroup.closure (Set.range u))
Tactic: rwa [← mul_eq_zero_iff_right this, Subgroup.relIndex_mul_index h,
      ← Subgroup.not_finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex,
      ← isMaxRank_iff_closure_finiteIndex.not]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
⊢ regOfFamily u / regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
have pos : ∀ (hu : IsMaxRank u),
  regOfFamily u / regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
Tactic: classical
  by_cases hu : IsMaxRank u
  · have : span ℤ (Set.range (basisOfIsMaxRank hu)) ≤ span ℤ (Set.range (basisOfIsMaxRank hv)) := by
      rw [← toAddSubgroup_le, span_basisOfIsMaxRank hu, span_basisOfIsMaxRank hv,
        ← map_logEmbedding_sup_torsion (Subgroup.closure (Set.range u)).toAddSubgroup,
        ← map_logEmbedding_sup_torsion (Subgroup.closure (Set.range v)).toAddSubgroup,
        ← SupHomClass.map_sup, ← SupHomClass.map_sup]
      exact AddSubgroup.map_mono <| (OrderIso.le_iff_le Subgroup.toAddSubgroup).mpr h
    rw [regOfFamily_of_isMaxRank hu, regOfFamily_of_isMaxRank hv,
      covolume_div_covolume_eq_relIndex _ _ this, span_basisOfIsMaxRank hu,
      span_basisOfIsMaxRank hv, AddSubgroup.relIndex_map_map, logEmbedding_ker,
      ← OrderIso.map_sup, ← OrderIso.map_sup, ← Subgroup.relIndex_toAddSubgroup]
  · rw [regOfFamily_eq_zero hu, zero_div, eq_comm, Nat.cast_eq_zero]
    have : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0 := by
      rw [← Subgroup.finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex]
      exact isMaxRank_iff_closure_finiteIndex.mp hv
    rwa [← mul_eq_zero_iff_right this, Subgroup.relIndex_mul_index h,
      ← Subgroup.not_finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex,
      ← isMaxRank_iff_closure_finiteIndex.not]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u v : Fin (rank K) → (𝓞 K)ˣ
hv : IsMaxRank v
h : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K
pos :
  ∀ (hu : IsMaxRank u),
    regOfFamily u / regOfFamily v =
      (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
⊢ regOfFamily u / regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
have neg : ∀ (hu : ¬IsMaxRank u),
  regOfFamily u / regOfFamily v =
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex (Subgroup.closure (Set.range v) ⊔ torsion K)) : ℝ)
Tactic: classical
  by_cases hu : IsMaxRank u
  · have : span ℤ (Set.range (basisOfIsMaxRank hu)) ≤ span ℤ (Set.range (basisOfIsMaxRank hv)) := by
      rw [← toAddSubgroup_le, span_basisOfIsMaxRank hu, span_basisOfIsMaxRank hv,
        ← map_logEmbedding_sup_torsion (Subgroup.closure (Set.range u)).toAddSubgroup,
        ← map_logEmbedding_sup_torsion (Subgroup.closure (Set.range v)).toAddSubgroup,
        ← SupHomClass.map_sup, ← SupHomClass.map_sup]
      exact AddSubgroup.map_mono <| (OrderIso.le_iff_le Subgroup.toAddSubgroup).mpr h
    rw [regOfFamily_of_isMaxRank hu, regOfFamily_of_isMaxRank hv,
      covolume_div_covolume_eq_relIndex _ _ this, span_basisOfIsMaxRank hu,
      span_basisOfIsMaxRank hv, AddSubgroup.relIndex_map_map, logEmbedding_ker,
      ← OrderIso.map_sup, ← OrderIso.map_sup, ← Subgroup.relIndex_toAddSubgroup]
  · rw [regOfFamily_eq_zero hu, zero_div, eq_comm, Nat.cast_eq_zero]
    have : (Subgroup.closure (Set.range v) ⊔ torsion K).index ≠ 0 := by
      rw [← Subgroup.finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex]
      exact isMaxRank_iff_closure_finiteIndex.mp hv
    rwa [← mul_eq_zero_iff_right this, Subgroup.relIndex_mul_index h,
      ← Subgroup.not_finiteIndex_iff, ← finiteIndex_iff_sup_torsion_finiteIndex,
      ← isMaxRank_iff_closure_finiteIndex.not]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
⊢ regOfFamily u / regulator K = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
have rw : regOfFamily u / regOfFamily (fundSystem K) = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
Tactic: rw [regulator_eq_regOfFamily_fundSystem, regOfFamily_div_regOfFamily (isMaxRank_fundSystem K)
    (by simp only [closure_fundSystem_sup_torsion_eq_top, le_top]),
    closure_fundSystem_sup_torsion_eq_top, Subgroup.relIndex_top_right]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
rw : regOfFamily u / regOfFamily (fundSystem K) = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
⊢ regOfFamily u / regulator K = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
have rw₁ : Eq (α := ℝ)
  (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex
        (Subgroup.closure (Set.range (ι := Fin (rank K)) (fundSystem K)) ⊔ torsion K)) :
    ℝ)
  (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
Tactic: rw [regulator_eq_regOfFamily_fundSystem, regOfFamily_div_regOfFamily (isMaxRank_fundSystem K)
    (by simp only [closure_fundSystem_sup_torsion_eq_top, le_top]),
    closure_fundSystem_sup_torsion_eq_top, Subgroup.relIndex_top_right]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
rw : regOfFamily u / regOfFamily (fundSystem K) = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
rw₁ :
  Eq (α := ℝ)
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex
          (Subgroup.closure (Set.range (ι := Fin (rank K)) (fundSystem K)) ⊔ torsion K)) :
      ℝ)
    (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
⊢ regOfFamily u / regulator K = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
have rw₂ : Eq (α := ℝ) (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex ⊤) : ℝ)
  (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
Tactic: rw [regulator_eq_regOfFamily_fundSystem, regOfFamily_div_regOfFamily (isMaxRank_fundSystem K)
    (by simp only [closure_fundSystem_sup_torsion_eq_top, le_top]),
    closure_fundSystem_sup_torsion_eq_top, Subgroup.relIndex_top_right]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
rw : regOfFamily u / regOfFamily (fundSystem K) = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
rw₁ :
  Eq (α := ℝ)
    (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex
          (Subgroup.closure (Set.range (ι := Fin (rank K)) (fundSystem K)) ⊔ torsion K)) :
      ℝ)
    (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
rw₂ :
  Eq (α := ℝ) (↑((Subgroup.closure (Set.range u) ⊔ torsion K).relIndex ⊤) : ℝ)
    (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
⊢ regOfFamily u / regulator K = (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
have rw₃ : Eq (α := ℝ) (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
  (↑(Subgroup.closure (Set.range u) ⊔ torsion K).index : ℝ)
Tactic: rw [regulator_eq_regOfFamily_fundSystem, regOfFamily_div_regOfFamily (isMaxRank_fundSystem K)
    (by simp only [closure_fundSystem_sup_torsion_eq_top, le_top]),
    closure_fundSystem_sup_torsion_eq_top, Subgroup.relIndex_top_right]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
κ : AddChar ℤ_[p] R
hκ : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑κ : (a : ℤ_[p]) → R)
⊢ Tendsto.{0, u_1} (β := R) (fun (n : ℕ) ↦ HSub.hSub (α := R) (β := R) ((κ : (a : ℤ_[p]) → R) 1 : R) (1 : R) ^ n) atTop
    (𝓝 0)
have refine : ∀ (n : ℕ),
  Δ_[1]^[n] (⇑{ toFun := (⇑κ : (a : ℤ_[p]) → R), continuous_toFun := hκ : C(ℤ_[p], R) } : (a : ℤ_[p]) → R) 0 =
    HSub.hSub (α := R) (β := R) ((κ : (a : ℤ_[p]) → R) 1 : R) (1 : R) ^ n
Tactic: refine (PadicInt.fwdDiff_tendsto_zero ⟨κ, hκ⟩).congr fun n ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
⊢ Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) 0 : R) (1 : R)
have rw : Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑0 : ℤ_[p]) : R) (1 : R)
Tactic: rw [← Nat.cast_zero, mahlerSeries_apply_nat hr le_rfl, zero_add, Finset.sum_range_one,
      Nat.choose_self, pow_zero, one_smul]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑0 : ℤ_[p]) : R) (1 : R)
⊢ Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) 0 : R) (1 : R)
have rw₁ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 0 1), Nat.choose 0 i • r ^ i) 1
Tactic: rw [← Nat.cast_zero, mahlerSeries_apply_nat hr le_rfl, zero_add, Finset.sum_range_one,
      Nat.choose_self, pow_zero, one_smul]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑0 : ℤ_[p]) : R) (1 : R)
rw₁ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 0 1), Nat.choose 0 i • r ^ i) 1
rw₂ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range 1, Nat.choose 0 i • r ^ i) 1
⊢ Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) 0 : R) (1 : R)
have rw₃ : Nat.choose 0 0 • HPow.hPow (β := ℕ) r 0 = 1
Tactic: rw [← Nat.cast_zero, mahlerSeries_apply_nat hr le_rfl, zero_add, Finset.sum_range_one,
      Nat.choose_self, pow_zero, one_smul]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑0 : ℤ_[p]) : R) (1 : R)
rw₁ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 0 1), Nat.choose 0 i • r ^ i) 1
rw₂ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range 1, Nat.choose 0 i • r ^ i) 1
rw₃ : Nat.choose 0 0 • HPow.hPow (β := ℕ) r 0 = 1
⊢ Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) 0 : R) (1 : R)
have rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) = 1
Tactic: rw [← Nat.cast_zero, mahlerSeries_apply_nat hr le_rfl, zero_add, Finset.sum_range_one,
      Nat.choose_self, pow_zero, one_smul]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑0 : ℤ_[p]) : R) (1 : R)
rw₁ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 0 1), Nat.choose 0 i • r ^ i) 1
rw₂ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range 1, Nat.choose 0 i • r ^ i) 1
rw₃ : Nat.choose 0 0 • HPow.hPow (β := ℕ) r 0 = 1
rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) = 1
⊢ Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) 0 : R) (1 : R)
have rw₅ : HSMul.hSMul (α := ℕ) (β := R) 1 1 = 1
Tactic: rw [← Nat.cast_zero, mahlerSeries_apply_nat hr le_rfl, zero_add, Finset.sum_range_one,
      Nat.choose_self, pow_zero, one_smul]