mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
p f n : ℕ
P : K[X]
hK : Fintype.card K = p ^ f
hn : p.Coprime n
hp : Fact (Nat.Prime p)
hP : P ∣ cyclotomic n K
hPirr : Irreducible P
hPmo : P.Monic
this✝¹ : Fact (Irreducible P)
this✝ : Module.Finite K (AdjoinRoot P)
this : Finite (AdjoinRoot P)
hζ : IsPrimitiveRoot (root P) n
pB : PowerBasis K (AdjoinRoot P) := powerBasis ⋯
h0 : n = 0
⊢ False

Tactic:
simp [h0, hp.out.ne_one] at hn

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ Real.exp (x w₀) ^ finrank ℚ K *
      ∏ y, ∏ x_1, (x_1 ((algebraMap (𝓞 K) K) ↑(fundSystem K (equivFinRank.symm y))) ^ x ↑y) ^ x_1.mult =
    Real.exp (x w₀) ^ finrank ℚ K

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ Real.exp (x w₀) ^ finrank ℚ K *
      ∏ y, ∏ x_1, (x_1 ((algebraMap (𝓞 K) K) ↑(fundSystem K (equivFinRank.symm y))) ^ x ↑y) ^ x_1.mult =
    Real.exp (x w₀) ^ finrank ℚ K
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ logMap (mixedSpaceOfRealSpace (↑expMapBasis x)) ∈
      ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)) ↔
    ∀ (w : InfinitePlace K), w ≠ w₀ → x w ∈ Set.Ico 0 1

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ logMap (mixedSpaceOfRealSpace (↑expMapBasis x)) ∈
      ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)) ↔
    ∀ (w : InfinitePlace K), w ≠ w₀ → x w ∈ Set.Ico 0 1
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
w : InfinitePlace K
hw : w ≠ w₀
⊢ ((Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)).repr fun w ↦ (∑ i, x ↑i • (completeBasis K) ↑i) ↑w)
        (equivFinRank.symm ⟨w, hw⟩) ∈
      Set.Ico 0 1 ↔
    x w ∈ Set.Ico 0 1

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
w : InfinitePlace K
hw : w ≠ w₀
⊢ ((Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)).repr fun w ↦ (∑ i, x ↑i • (completeBasis K) ↑i) ↑w)
        (equivFinRank.symm ⟨w, hw⟩) ∈
      Set.Ico 0 1 ↔
    x w ∈ Set.Ico 0 1
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: case e_a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ ∏ x_1, Real.exp ((↑x_1.mult)⁻¹ * (completeBasis K).equivFun.symm x x_1) =
    Real.exp (x w₀) ^ finrank ℚ K * (∏ w, ↑expMapBasis x ↑w)⁻¹

Tactic:
[

Goal state after:
case e_a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ ∏ x_1, Real.exp ((↑x_1.mult)⁻¹ * (completeBasis K).equivFun.symm x x_1) =
    Real.exp (x w₀) ^ finrank ℚ K * (∏ w, ↑expMapBasis x ↑w)⁻¹
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ |(fderiv_expMapBasis K x).det| =
    Real.exp (x w₀ * ↑(finrank ℚ K)) * (∏ w, ↑expMapBasis x ↑w)⁻¹ * 2⁻¹ ^ nrComplexPlaces K * ↑(finrank ℚ K) *
      regulator K

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ |(fderiv_expMapBasis K x).det| =
    Real.exp (x w₀ * ↑(finrank ℚ K)) * (∏ w, ↑expMapBasis x ↑w)⁻¹ * 2⁻¹ ^ nrComplexPlaces K * ↑(finrank ℚ K) *
      regulator K
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ |Real.exp (x w₀) ^ finrank ℚ K * (∏ w, ↑expMapBasis x ↑w)⁻¹ * 2⁻¹ ^ nrComplexPlaces K| *
      (↑(finrank ℚ K) * regulator K) =
    Real.exp (x w₀ * ↑(finrank ℚ K)) * (∏ w, ↑expMapBasis x ↑w)⁻¹ * 2⁻¹ ^ nrComplexPlaces K * ↑(finrank ℚ K) *
      regulator K

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : realSpace K
⊢ |Real.exp (x w₀) ^ finrank ℚ K * (∏ w, ↑expMapBasis x ↑w)⁻¹ * 2⁻¹ ^ nrComplexPlaces K| *
      (↑(finrank ℚ K) * regulator K) =
    Real.exp (x w₀ * ↑(finrank ℚ K)) * (∏ w, ↑expMapBasis x ↑w)⁻¹ * 2⁻¹ ^ nrComplexPlaces K * ↑(finrank ℚ K) *
      regulator K
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : Set (realSpace K)
hs : MeasurableSet s
f : (InfinitePlace K → ℝ) → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (x : realSpace K) in s, ENNReal.ofReal |(fderiv_expMapBasis K x).det| * f (↑expMapBasis x) =
    2⁻¹ ^ nrComplexPlaces K * ENNReal.ofReal (regulator K) * ↑(finrank ℚ K) *
      ∫⁻ (x : realSpace K) in s,
        ENNReal.ofReal (Real.exp (x w₀ * ↑(finrank ℚ K))) * (∏ i, ENNReal.ofReal (↑expMapBasis (fun w ↦ x w) ↑i))⁻¹ *
          f (↑expMapBasis x)

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : Set (realSpace K)
hs : MeasurableSet s
f : (InfinitePlace K → ℝ) → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (x : realSpace K) in s, ENNReal.ofReal |(fderiv_expMapBasis K x).det| * f (↑expMapBasis x) =
    2⁻¹ ^ nrComplexPlaces K * ENNReal.ofReal (regulator K) * ↑(finrank ℚ K) *
      ∫⁻ (x : realSpace K) in s,
        ENNReal.ofReal (Real.exp (x w₀ * ↑(finrank ℚ K))) * (∏ i, ENNReal.ofReal (↑expMapBasis (fun w ↦ x w) ↑i))⁻¹ *
          f (↑expMapBasis x)
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal

Goal state before:
pp: case pos
p k : ℕ
hp : Fact (Nat.Prime p)
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
hK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
h : p = 2
⊢ Associated ((Algebra.norm ℤ) (hζ.toInteger - 1)) ↑p

Tactic:
cases k with

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn✝ : 0 < n
hn : 0 < ↑n
⊢ (∫⋯∫⁻_univ.erase w₀, fun x ↦
        ∫⁻ (xᵢ : ℝ) in Set.Iic 0, ENNReal.ofReal (Real.exp (Function.update x w₀ xᵢ w₀ * ↑n)) ∂fun i ↦
        volume.restrict (if i = w₀ then Set.Iic 0 else Set.Ico 0 1))
      0 =
    (↑n)⁻¹

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
n : ℕ
hn✝ : 0 < n
hn : 0 < ↑n
⊢ (∫⋯∫⁻_univ.erase w₀, fun x ↦
        ∫⁻ (xᵢ : ℝ) in Set.Iic 0, ENNReal.ofReal (Real.exp (Function.update x w₀ xᵢ w₀ * ↑n)) ∂fun i ↦
        volume.restrict (if i = w₀ then Set.Iic 0 else Set.Ico 0 1))
      0 =
    (↑n)⁻¹
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 2 ^ nrRealPlaces K * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      (2⁻¹ ^ nrComplexPlaces K * ENNReal.ofReal (regulator K) * ↑(finrank ℚ K) *
        ∫⁻ (x : realSpace K) in paramSet K,
          ENNReal.ofReal (Real.exp (x w₀ * ↑(finrank ℚ K))) * (∏ i, ENNReal.ofReal (↑expMapBasis (fun w ↦ x w) ↑i))⁻¹ *
            ∏ w, ENNReal.ofReal (↑expMapBasis x ↑w)) =
    2 ^ nrRealPlaces K * ↑NNReal.pi ^ nrComplexPlaces K * ENNReal.ofReal (regulator K)

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 2 ^ nrRealPlaces K * ENNReal.ofReal (2 * Real.pi) ^ nrComplexPlaces K *
      (2⁻¹ ^ nrComplexPlaces K * ENNReal.ofReal (regulator K) * ↑(finrank ℚ K) *
        ∫⁻ (x : realSpace K) in paramSet K,
          ENNReal.ofReal (Real.exp (x w₀ * ↑(finrank ℚ K))) * (∏ i, ENNReal.ofReal (↑expMapBasis (fun w ↦ x w) ↑i))⁻¹ *
            ∏ w, ENNReal.ofReal (↑expMapBasis x ↑w)) =
    2 ^ nrRealPlaces K * ↑NNReal.pi ^ nrComplexPlaces K * ENNReal.ofReal (regulator K)
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Galois

Goal state before:
pp: n : ℕ
inst✝⁸ : NeZero n
K : Type u_1
inst✝⁷ : Field K
inst✝⁶ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)
inst✝³ : IsAbelianGalois ℚ K
F : IntermediateField ℚ K
m : ℕ
inst✝² : NeZero m
inst✝¹ : IsGalois ℚ ↥F
inst✝ : IsCyclotomicExtension {m} ℚ ↥F
hdiv : m ∣ n
χ : DirichletCharacter R n
⊢ χ ∈ (intermediateFieldEquivSubgroupChar n K R) F ↔ χ.conductor ∣ m

Tactic:
[

Goal state after:
n : ℕ
inst✝⁸ : NeZero n
K : Type u_1
inst✝⁷ : Field K
inst✝⁶ : NumberField K
hK : IsCyclotomicExtension {n} ℚ K
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)
inst✝³ : IsAbelianGalois ℚ K
F : IntermediateField ℚ K
m : ℕ
inst✝² : NeZero m
inst✝¹ : IsGalois ℚ ↥F
inst✝ : IsCyclotomicExtension {m} ℚ ↥F
hdiv : m ∣ n
χ : DirichletCharacter R n
⊢ χ ∈ (intermediateFieldEquivSubgroupChar n K R) F ↔ χ.conductor ∣ m
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
hJ : ClassGroup.mk0 J = C⁻¹
⊢ Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C } =
    Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s * ↑(absNorm ↑J) }

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
hJ : ClassGroup.mk0 J = C⁻¹
⊢ Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C } =
    Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s * ↑(absNorm ↑J) }
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
hJ : ClassGroup.mk0 J = C⁻¹
I : ↥(Ideal (𝓞 K))⁰
⊢ ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C ↔
    ClassGroup.mk0 ⟨↑↑((Equiv.dvd J) I), ⋯⟩ = 1 ∧ ↑(absNorm ↑↑((Equiv.dvd J) I)) ≤ s * ↑(absNorm ↑J)

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
hJ : ClassGroup.mk0 J = C⁻¹
I : ↥(Ideal (𝓞 K))⁰
⊢ ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C ↔
    ClassGroup.mk0 ⟨↑↑((Equiv.dvd J) I), ⋯⟩ = 1 ∧ ↑(absNorm ↑↑((Equiv.dvd J) I)) ≤ s * ↑(absNorm ↑J)
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
x✝ : mixedSpace K
⊢ x✝ ∈
      ⇑(↑(toMixed K)).toEquiv ''
        ({x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} ∩
          ⇑↑↑(toMixed K) ⁻¹' ↑(idealLattice K ((FractionalIdeal.mk0 K) J))) ↔
    x✝ ∈ idealSet K J ∧ mixedEmbedding.norm x✝ ≤ s

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
s : ℝ
x✝ : mixedSpace K
⊢ x✝ ∈
      ⇑(↑(toMixed K)).toEquiv ''
        ({x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} ∩
          ⇑↑↑(toMixed K) ⁻¹' ↑(idealLattice K ((FractionalIdeal.mk0 K) J))) ↔
    x✝ ∈ idealSet K J ∧ mixedEmbedding.norm x✝ ≤ s
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: case e'_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
s : ℝ
⊢ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C }) / s =
    ((fun x ↦
          ↑(Nat.card
                  ↑({x_1 | x_1 ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x_1) ≤ x} ∩
                      ↑(ZLattice.comap ℝ (idealLattice K ((FractionalIdeal.mk0 K) J)) ↑↑(toMixed K)))) /
              x *
            (↑(absNorm ↑J) * (↑(torsionOrder K))⁻¹)) ∘
        fun x ↦ id x * ↑(absNorm ↑J))
      s

Tactic:
[

Goal state after:
case e'_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
s : ℝ
⊢ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C }) / s =
    ((fun x ↦
          ↑(Nat.card
                  ↑({x_1 | x_1 ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x_1) ≤ x} ∩
                      ↑(ZLattice.comap ℝ (idealLattice K ((FractionalIdeal.mk0 K) J)) ↑↑(toMixed K)))) /
              x *
            (↑(absNorm ↑J) * (↑(torsionOrder K))⁻¹)) ∘
        fun x ↦ id x * ↑(absNorm ↑J))
      s
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: case e'_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
s : ℝ
⊢ ↑(Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s * ↑(absNorm ↑J) }) / s =
    ((fun x ↦
          ↑(Nat.card
                  ↑({x_1 | x_1 ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x_1) ≤ x} ∩
                      ↑(ZLattice.comap ℝ (idealLattice K ((FractionalIdeal.mk0 K) J)) ↑↑(toMixed K)))) /
              x *
            (↑(absNorm ↑J) * (↑(torsionOrder K))⁻¹)) ∘
        fun x ↦ x * ↑(absNorm ↑J))
      s

Tactic:
Nat.card_congr (Ideal.tendsto_norm_le_and_mk_eq_div_atTop_aux₂ K),

Goal state after:
case e'_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
h₁ :
  ∀ (s : ℝ),
    {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =
      ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}
h₂ : {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ 1} = normLeOne K
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
h₃ : ↑(absNorm ↑J) ≠ 0
s : ℝ
⊢ ↑(Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s * ↑(absNorm ↑J) }) / s =
    ((fun x ↦ ↑(Nat.card { a // mixedEmbedding.norm ↑a ≤ x }) / x * (↑(absNorm ↑J) * (↑(torsionOrder K))⁻¹)) ∘ fun x ↦
        x * ↑(absNorm ↑J))
      s
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: ⊢ Tendsto (fun k ↦ ↑k ^ (↑k)⁻¹) atTop (𝓝 1)

Tactic:
[

Goal state after:
⊢ Tendsto (fun k ↦ ↑k ^ (↑k)⁻¹) atTop (𝓝 1)
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
T : Type u_2
l : List T
y x : R
⊢ (List.mapIdx (fun i x_1 ↦ x * y ^ i) l).sum = x * (List.mapIdx (fun i x ↦ y ^ i) l).sum

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : Semiring R
T : Type u_2
l : List T
y x : R
⊢ (List.mapIdx (fun i x_1 ↦ x * y ^ i) l).sum = x * (List.mapIdx (fun i x ↦ y ^ i) l).sum
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
inst✝ : Fintype β
hmulvec0 : asiegel K a *ᵥ x = 0
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r, ∑ l, ↑(a' K a k l r u) * ↑(x (l, r)) = 0
this : 0 = ∑ y, ∑ x_1, (∑ l, ↑(a' K a k l y x_1) * ↑(x (l, y))) * (newBasis K) x_1
⊢ 0 = ∑ r, ∑ l, ↑(x (l, r)) * ∑ u, ↑(a' K a k l r u) * (newBasis K) u

Tactic:
this

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
inst✝ : Fintype β
hmulvec0 : asiegel K a *ᵥ x = 0
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r, ∑ l, ↑(a' K a k l r u) * ↑(x (l, r)) = 0
this : 0 = ∑ y, ∑ x_1, (∑ l, ↑(a' K a k l y x_1) * ↑(x (l, y))) * (newBasis K) x_1
⊢ ∑ y, ∑ x_1, (∑ l, ↑(a' K a k l y x_1) * ↑(x (l, y))) * (newBasis K) x_1 =
    ∑ r, ∑ l, ↑(x (l, r)) * ∑ u, ↑(a' K a k l r u) * (newBasis K) u
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : ℝ
hs : 0 ≤ s
⊢ Fintype { I // ↑(absNorm ↑I) ≤ s }

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : ℝ
hs : 0 ≤ s
⊢ Fintype { I // ↑(absNorm ↑I) ≤ s }
Module: Mathlib.NumberTheory.SiegelsLemma

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : Fintype β
A : Matrix α β ℤ
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
hn : m < n
hm : 0 < m
i : α
x✝ : i ∈ univ
h1 : ↑n * max 1 ‖A‖ * ↑B = ∑ x, max 1 ‖A‖ * ↑B
⊢ ∑ x_1, ↑B * ↑(A i x_1)⁺ + ∑ x, ↑B * ↑(A i x)⁻ ≤ ↑n * max 1 ‖A‖ * ↑B

Tactic:
[

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : Fintype β
A : Matrix α β ℤ
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
hn : m < n
hm : 0 < m
i : α
x✝ : i ∈ univ
h1 : ↑n * max 1 ‖A‖ * ↑B = ∑ x, max 1 ‖A‖ * ↑B
⊢ ∑ x_1, ↑B * ↑(A i x_1)⁺ + ∑ x, ↑B * ↑(A i x)⁻ ≤ ↑n * max 1 ‖A‖ * ↑B
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : ℝ
hs : 0 ≤ s
this : Fintype { I // ↑(absNorm ↑I) ≤ s }
e : (C : ClassGroup (𝓞 K)) → { x // ClassGroup.mk0 ↑x = C } ≃ { x // ↑(absNorm ↑x) ≤ s ∧ ClassGroup.mk0 x = C } :=
  fun C ↦ Equiv.subtypeSubtypeEquivSubtypeInter (fun I ↦ ↑(absNorm ↑I) ≤ s) fun I ↦ ClassGroup.mk0 I = C
⊢ ∑ c, ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = c }) / s = ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s }) / s

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
s : ℝ
hs : 0 ≤ s
this : Fintype { I // ↑(absNorm ↑I) ≤ s }
e : (C : ClassGroup (𝓞 K)) → { x // ClassGroup.mk0 ↑x = C } ≃ { x // ↑(absNorm ↑x) ≤ s ∧ ClassGroup.mk0 x = C } :=
  fun C ↦ Equiv.subtypeSubtypeEquivSubtypeInter (fun I ↦ ↑(absNorm ↑I) ≤ s) fun I ↦ ClassGroup.mk0 I = C
⊢ ∑ c, ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = c }) / s = ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s }) / s
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
inst✝ : Fintype β
hmulvec0 : asiegel K a *ᵥ x = 0
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r, ∑ l, ↑(a' K a k l r u) * ↑(x (l, r)) = 0
this✝ : 0 = ∑ u, (∑ r, ∑ l, ↑(a' K a k l r u) * ↑(x (l, r))) * (newBasis K) u
this : 0 = ∑ y, ∑ x_1, ↑(x (y, x_1)) * ∑ u, ↑(a' K a k y x_1 u) * (newBasis K) u
⊢ 0 = (a *ᵥ ξ K x) k

Tactic:
this

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
inst✝ : Fintype β
hmulvec0 : asiegel K a *ᵥ x = 0
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r, ∑ l, ↑(a' K a k l r u) * ↑(x (l, r)) = 0
this✝ : 0 = ∑ u, (∑ r, ∑ l, ↑(a' K a k l r u) * ↑(x (l, r))) * (newBasis K) u
this : 0 = ∑ y, ∑ x_1, ↑(x (y, x_1)) * ∑ u, ↑(a' K a k y x_1 u) * (newBasis K) u
⊢ ∑ y, ∑ x_1, ↑(x (y, x_1)) * ∑ u, ↑(a' K a k y x_1 u) * (newBasis K) u = (a *ᵥ ξ K x) k
Module: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this :
  Tendsto (fun x ↦ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ x }) / x + x⁻¹) atTop
    (𝓝
      (2 ^ nrRealPlaces K * (2 * π) ^ nrComplexPlaces K * regulator K * ↑(classNumber K) /
        (↑(torsionOrder K) * √|↑(discr K)|)))
s : ℝ
hs : 0 ≤ s
⊢ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s }) / s + s⁻¹ = ↑(Nat.card { I // ↑(absNorm I) ≤ s }) / s

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this :
  Tendsto (fun x ↦ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ x }) / x + x⁻¹) atTop
    (𝓝
      (2 ^ nrRealPlaces K * (2 * π) ^ nrComplexPlaces K * regulator K * ↑(classNumber K) /
        (↑(torsionOrder K) * √|↑(discr K)|)))
s : ℝ
hs : 0 ≤ s
⊢ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ s }) / s + s⁻¹ = ↑(Nat.card { I // ↑(absNorm I) ≤ s }) / s
Module: Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion

Goal state before:
pp: case mpr.left
α : Type u_1
β : Type u_2
inst✝² : AddCommGroup α
inst✝¹ : UniformSpace α
inst✝ : IsUniformAddGroup α
f : β → α
h_cauchy : CauchySeq fun s ↦ ∑ b ∈ s, f b
h_tsum : ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl
⊢ CauchySeq fun s ↦ ∑ b ∈ s, (⇑toCompl ∘ f) b

Tactic:
[

Goal state after:
case mpr.left
α : Type u_1
β : Type u_2
inst✝² : AddCommGroup α
inst✝¹ : UniformSpace α
inst✝ : IsUniformAddGroup α
f : β → α
h_cauchy : CauchySeq fun s ↦ ∑ b ∈ s, f b
h_tsum : ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl
⊢ CauchySeq fun s ↦ ∑ b ∈ s, (⇑toCompl ∘ f) b
Module: Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : AddCommGroup α
inst✝¹ : UniformSpace α
inst✝ : IsUniformAddGroup α
f : β → α
⊢ Summable f ↔ (CauchySeq fun s ↦ ∑ b ∈ s, f b) ∧ ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl

Tactic:
classical
  constructor
  · rintro ⟨a, ha⟩
    exact ⟨ha.cauchySeq, ((summable_iff_summable_compl_and_tsum_mem f).mp ⟨a, ha⟩).2⟩
  · rintro ⟨h_cauchy, h_tsum⟩
    apply (summable_iff_summable_compl_and_tsum_mem f).mpr
    constructor
    · apply summable_iff_cauchySeq_finset.mpr
      simp_rw [Function.comp_apply, ← map_sum]
      exact h_cauchy.map (uniformContinuous_coe α)
    · exact h_tsum

Goal state after:
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: f : AbsoluteValue ℚ ℝ
hf_nontriv : f.IsNontrivial
bdd : ∀ (n : ℕ), f ↑n ≤ 1
p : ℕ
hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m
hp0 : 0 < f ↑p
hp1 : f ↑p < 1
hp : Nat.Prime p
this : Fact (Nat.Prime p)
t : ℝ
ht : 0 < t
hpt : f ↑p = ↑p ^ (-t)
⊢ ∃! p, ∃ (x : Fact (Nat.Prime p)), f.IsEquiv (padic p)

Tactic:
[

Goal state after:
f : AbsoluteValue ℚ ℝ
hf_nontriv : f.IsNontrivial
bdd : ∀ (n : ℕ), f ↑n ≤ 1
p : ℕ
hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m
hp0 : 0 < f ↑p
hp1 : f ↑p < 1
hp : Nat.Prime p
this : Fact (Nat.Prime p)
t : ℝ
ht : 0 < t
hpt : f ↑p = ↑p ^ (-t)
⊢ ∃! p, ∃ (x : Fact (Nat.Prime p)), f.IsEquiv (padic p)
Module: Mathlib.NumberTheory.NumberField.House

Goal state before:
pp: case calc_1
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
p q : ℕ
hpq : p < q
x : β × (K →+* ℂ) → ℤ
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖asiegel K a‖) ^ (↑p / (↑q - ↑p))
h : ℕ := finrank ℚ K
l : β
H₀ : 0 ≤ supOfBasis K
H₁ : 0 < ↑q - ↑p
⊢ house ((algebraMap (𝓞 K) K) (∑ r, ↑(x (l, r)) * (newBasis K) r)) ≤
    ∑ r, house ((algebraMap (𝓞 K) K) ↑(x (l, r)) * (algebraMap (𝓞 K) K) ((newBasis K) r))

Tactic:
[

Goal state after:
case calc_1
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
p q : ℕ
hpq : p < q
x : β × (K →+* ℂ) → ℤ
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖asiegel K a‖) ^ (↑p / (↑q - ↑p))
h : ℕ := finrank ℚ K
l : β
H₀ : 0 ≤ supOfBasis K
H₁ : 0 < ↑q - ↑p
⊢ house ((algebraMap (𝓞 K) K) (∑ r, ↑(x (l, r)) * (newBasis K) r)) ≤
    ∑ r, house ((algebraMap (𝓞 K) K) ↑(x (l, r)) * (algebraMap (𝓞 K) K) ((newBasis K) r))
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℕ
x : ℤ_[p]
f : ℤ_[p] → ℤ_[p] := fun x ↦ Polynomial.eval x (ascPochhammer ℤ_[p] k)
hC : ↑k.factorial ≠ 0
hf : ContinuousAt f x
n : ℕ
hn : ‖f x - f ↑n‖ ≤ ‖↑k.factorial‖
⊢ ‖f ↑n‖ ≤ ‖↑k.factorial‖

Tactic:
[

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
k : ℕ
x : ℤ_[p]
f : ℤ_[p] → ℤ_[p] := fun x ↦ Polynomial.eval x (ascPochhammer ℤ_[p] k)
hC : ↑k.factorial ≠ 0
hf : ContinuousAt f x
n : ℕ
hn : ‖f x - f ↑n‖ ≤ ‖↑k.factorial‖
⊢ ‖f ↑n‖ ≤ ‖↑k.factorial‖