mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable

Goal state before:
pp: z : ℍ
this :
  -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑((σ 1) ↑n) * cexp (2 * ↑π * I * ↑z) ^ ↑n =
    ∑' (m : ℕ), -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑↑n * cexp (2 * ↑π * I * ↑z) ^ ((m + 1) * ↑n)
⊢ Tendsto (fun N ↦ ∑ m ∈ range N, -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑↑n * cexp (2 * ↑π * I * ↑z) ^ ((m + 1) * ↑n)) atTop
    (𝓝 (-8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑((σ 1) ↑n) * cexp (2 * ↑π * I * ↑z) ^ ↑n))

Tactic:
rw [this]

Goal state after:
z : ℍ
this :
  -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑((σ 1) ↑n) * cexp (2 * ↑π * I * ↑z) ^ ↑n =
    ∑' (m : ℕ), -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑↑n * cexp (2 * ↑π * I * ↑z) ^ ((m + 1) * ↑n)
⊢ Tendsto (fun N ↦ ∑ m ∈ range N, -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑↑n * cexp (2 * ↑π * I * ↑z) ^ ((m + 1) * ↑n)) atTop
    (𝓝 (∑' (m : ℕ), -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑↑n * cexp (2 * ↑π * I * ↑z) ^ ((m + 1) * ↑n)))
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable

Goal state before:
pp: case e'_6
z : ℍ
hS : Summable fun n ↦ ↑((σ 1) (n + 1)) * cexp (2 * ↑π * I * ↑z) ^ (n + 1)
⊢ E2 z - ∑ i ∈ range 1, (if i = 0 then 1 else -24 * ↑((σ 1) i)) • cexp (2 * ↑π * I * ↑z) ^ i =
    ∑' (b : ℕ), -24 * (↑((σ 1) (b + 1)) * cexp (2 * ↑π * I * ↑z) ^ (b + 1))

Tactic:
· rw [E2_eq_tsum_cexp, tsum_pnat_eq_tsum_succ (f := fun n ↦ σ 1 n * 𝕢 z ^ n), tsum_mul_left]
    simp

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion

Goal state before:
pp: k : ℕ
hk2 : Even k
hkn0 : k ≠ 0
m : ℕ := k / 2 - 1
h2 : k = 2 * (m + 1)
⊢ (2 * (m + 1))! = 2 * (m + 1) * (2 * m + 1)!

Tactic:
grind [Nat.factorial_succ]

Goal state after:
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
a b : R
p : ℕ
hp : Odd p
h1 :
  ∀ (i : ℕ),
    (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =
      (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)
s : R := ↑p ^ 2
x : ℕ
hx : x + 1 ∈ range p
this : x.succ - 1 + (p - 1 - x.succ) = p - 2
⊢ a ^ (x + 1 - 1 + (p - 1 - (x + 1))) * ↑p * b * ↑(x + 1) = ↑p * b * a ^ (p - 2) * ↑(x + 1)

Tactic:
rw [this]

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
a b : R
p : ℕ
hp : Odd p
h1 :
  ∀ (i : ℕ),
    (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =
      (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)
s : R := ↑p ^ 2
x : ℕ
hx : x + 1 ∈ range p
this : x.succ - 1 + (p - 1 - x.succ) = p - 2
⊢ a ^ (p - 2) * ↑p * b * ↑(x + 1) = ↑p * b * a ^ (p - 2) * ↑(x + 1)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case refine_2
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hx : ¬↑p ∣ y
k : R
hk : x - y = ↑p * k
⊢ ¬↑p ^ 2 ∣ ↑p * y ^ (p - 1)

Tactic:
rw [pow_two, mul_dvd_mul_iff_left hp.ne_zero]

Goal state after:
case refine_2
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hx : ¬↑p ∣ y
k : R
hk : x - y = ↑p * k
⊢ ¬↑p ∣ y ^ (p - 1)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hxy
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
h_ind : emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a
⊢ ↑p ∣ x ^ p ^ a - y ^ p ^ a

Tactic:
rw [← geom_sum₂_mul]

Goal state after:
case succ.hxy
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
h_ind : emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a
⊢ ↑p ∣ (∑ i ∈ range (p ^ a), x ^ i * y ^ (p ^ a - 1 - i)) * (x - y)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hxy
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ↑p ∣ x ^ p ^ multiplicity p (n + 1) - y ^ p ^ multiplicity p (n + 1)

Tactic:
rw [← geom_sum₂_mul]

Goal state after:
case succ.hxy
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ↑p ∣ (∑ i ∈ range (p ^ multiplicity p (n + 1)), x ^ i * y ^ (p ^ multiplicity p (n + 1) - 1 - i)) * (x - y)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: K : Type u_1
inst✝ : Field K
u : (𝓞 K)ˣ
x : mixedSpace K
h : x ≠ 0
⊢ u • x ≠ 0

Tactic:
obtain ⟨w, h⟩ := exists_normAtPlace_ne_zero_iff.mpr h

Goal state after:
K : Type u_1
inst✝ : Field K
u : (𝓞 K)ˣ
x : mixedSpace K
h✝ : x ≠ 0
w : InfinitePlace K
h : (normAtPlace w) x ≠ 0
⊢ u • x ≠ 0
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
h₁ :
  (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔
    Finite
      (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))
i : Fin (rank K)
⊢ (fun i ↦ (logEmbedding K) (Additive.ofMul (u i))) i ∈ ↑(unitLattice K)

Tactic:
exact ⟨Additive.ofMul (u i), mem_top, rfl⟩

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
h₁ :
  (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔
    Finite
      (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))
⊢ IsMaxRank u ↔ (Subgroup.closure (Set.range u)).FiniteIndex

Tactic:
have h₂ : DiscreteTopology
      (span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i)))) := by
    rw [← SetLike.isDiscrete_iff_discreteTopology]
    refine (inferInstance : DiscreteTopology (unitLattice K)).isDiscrete.mono ?_
    rw [SetLike.coe_subset_coe, Submodule.span_le]
    rintro _ ⟨i, rfl⟩
    exact ⟨Additive.ofMul (u i), mem_top, rfl⟩

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
h₁ :
  (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔
    Finite
      (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))
h₂ : DiscreteTopology ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i))))
⊢ IsMaxRank u ↔ (Subgroup.closure (Set.range u)).FiniteIndex
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
h₁ :
  (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔
    Finite
      (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))
h₂ : DiscreteTopology ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i))))
⊢ IsMaxRank u ↔
    Set.finrank ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)) = rank K

Tactic:
IsMaxRank,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
h₁ :
  (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔
    Finite
      (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))
h₂ : DiscreteTopology ↥(span ℤ (Set.range fun i ↦ (logEmbedding K) (Additive.ofMul (u i))))
⊢ (LinearIndependent ℝ fun i ↦ (logEmbedding K) (Additive.ofMul (u i))) ↔
    Set.finrank ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)) = rank K
Module: Mathlib.NumberTheory.NumberField.Units.Regulator

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : ¬IsMaxRank u
⊢ ¬LinearIndependent ℝ fun i ↦ of (fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))) i

Tactic:
IsMaxRank,

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : ¬LinearIndependent ℝ fun i ↦ (logEmbedding K) (Additive.ofMul (u i))
⊢ ¬LinearIndependent ℝ fun i ↦ of (fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))) i
Module: Mathlib.NumberTheory.NumberField.CMField

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : NumberField K
this : (∃ u, Subgroup.zpowers ((unitsMulComplexConjInv K) u) = ⊤) ↔ (unitsMulComplexConjInv K).range.index = 1
⊢ indexRealUnits K = 2 ↔ ∃ u, Subgroup.zpowers ((unitsMulComplexConjInv K) u) = ⊤

Tactic:
rw [this]

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : CharZero K
inst✝¹ : IsCMField K
inst✝ : NumberField K
this : (∃ u, Subgroup.zpowers ((unitsMulComplexConjInv K) u) = ⊤) ↔ (unitsMulComplexConjInv K).range.index = 1
⊢ indexRealUnits K = 2 ↔ (unitsMulComplexConjInv K).range.index = 1
Module: Mathlib.RingTheory.Unramified.Dedekind

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Module.Finite A B
inst✝² : IsDedekindDomain A
inst✝¹ : IsDomain B
inst✝ : Algebra.FormallyUnramified A B
⊢ ∀ (P : Ideal B), P ≠ ⊥ → ∀ (x : P.IsPrime), IsDiscreteValuationRing (Localization.AtPrime P)

Tactic:
intro q

Goal state after:
A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Module.Finite A B
inst✝² : IsDedekindDomain A
inst✝¹ : IsDomain B
inst✝ : Algebra.FormallyUnramified A B
q : Ideal B
⊢ q ≠ ⊥ → ∀ (x : q.IsPrime), IsDiscreteValuationRing (Localization.AtPrime q)
Module: Mathlib.NumberTheory.NumberField.ExistsRamified

Goal state before:
pp: K : Type u_1
𝒪 : Type u_2
inst✝⁵ : Field K
inst✝⁴ : NumberField K
inst✝³ : CommRing 𝒪
inst✝² : Algebra 𝒪 K
inst✝¹ : IsIntegralClosure 𝒪 ℤ K
inst✝ : IsGalois ℚ K
H : 1 < Module.finrank ℚ K
this✝⁶ : IsDomain 𝒪
this✝⁵ : IsDedekindDomain 𝒪
this✝⁴ : IsFractionRing 𝒪 K
this✝³ : Module.Finite ℤ 𝒪
this✝² : CharZero 𝒪
this✝¹ : MulSemiringAction Gal(K/ℚ) 𝒪 := IsIntegralClosure.MulSemiringAction ℤ ℚ K 𝒪
this✝ : IsGaloisGroup Gal(K/ℚ) ℤ 𝒪
P : Ideal 𝒪
w✝ : P.IsMaximal
hP' : ¬Algebra.IsUnramifiedAt ℤ P
p : ℤ
hp : Ideal.under ℤ P = Ideal.span {p}
hp0 : p ≠ 0
this : Prime p
⊢ ∃ p, Nat.Prime p ∧ ∀ (P : Ideal 𝒪) (x : P.IsPrime), ↑p ∈ P → ¬Algebra.IsUnramifiedAt ℤ P

Tactic:
refine ⟨p.natAbs, Int.prime_iff_natAbs_prime.mp this, fun Q _ hQ ↦ ?_⟩

Goal state after:
K : Type u_1
𝒪 : Type u_2
inst✝⁵ : Field K
inst✝⁴ : NumberField K
inst✝³ : CommRing 𝒪
inst✝² : Algebra 𝒪 K
inst✝¹ : IsIntegralClosure 𝒪 ℤ K
inst✝ : IsGalois ℚ K
H : 1 < Module.finrank ℚ K
this✝⁶ : IsDomain 𝒪
this✝⁵ : IsDedekindDomain 𝒪
this✝⁴ : IsFractionRing 𝒪 K
this✝³ : Module.Finite ℤ 𝒪
this✝² : CharZero 𝒪
this✝¹ : MulSemiringAction Gal(K/ℚ) 𝒪 := IsIntegralClosure.MulSemiringAction ℤ ℚ K 𝒪
this✝ : IsGaloisGroup Gal(K/ℚ) ℤ 𝒪
P : Ideal 𝒪
w✝ : P.IsMaximal
hP' : ¬Algebra.IsUnramifiedAt ℤ P
p : ℤ
hp : Ideal.under ℤ P = Ideal.span {p}
hp0 : p ≠ 0
this : Prime p
Q : Ideal 𝒪
x✝ : Q.IsPrime
hQ : ↑p.natAbs ∈ Q
⊢ ¬Algebra.IsUnramifiedAt ℤ Q
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 | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} ∩
        ↑(ZLattice.comap ℝ (idealLattice K ((FractionalIdeal.mk0 K) J)) ↑↑(toMixed K))) ≃
    { a // mixedEmbedding.norm ↑a ≤ s }

Tactic:
by
  rw [ZLattice.coe_comap]
  refine (((toMixed K).toEquiv.image _).trans (Equiv.subtypeEquivProp ?_)).trans
    (Equiv.subtypeSubtypeEquivSubtypeInter _ (mixedEmbedding.norm · ≤ s)).symm
  ext
  simp_rw [mem_idealSet, Set.mem_image, Set.mem_inter_iff, Set.mem_preimage, SetLike.mem_coe,
    mem_idealLattice, FractionalIdeal.coe_mk0]
  constructor
  · rintro ⟨_, ⟨⟨hx₁, hx₂⟩, _, ⟨x, hx₃, rfl⟩, h⟩, rfl⟩
    exact ⟨⟨hx₁, x, hx₃, h⟩, hx₂⟩
  · rintro ⟨⟨hx₁, ⟨x, hx₂, rfl⟩⟩, hx₃⟩
    exact ⟨(toMixed K).symm (mixedEmbedding K x), ⟨⟨hx₁, hx₃⟩, ⟨(x : K), by simp [hx₂], rfl⟩⟩, rfl⟩

Goal state after:
Module: Mathlib.NumberTheory.SiegelsLemma

Goal state before:
pp: case refine_1
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
v : β → ℤ
hv : 0 ≤ v ∧ v ≤ B'
mulVec_def : A *ᵥ v = fun i ↦ ∑ j, A i j * v j
i : α
⊢ N i ≤ (fun i ↦ ∑ j, A i j * v j) i

Tactic:
simp only [mul_neg]
    gcongr ∑ _ : β, ?_ with j _ -- Get rid of sums
    rw [← mul_comm (v j)] -- Move A i j to the right of the products
    -- We have to distinguish cases: we have now 4 goals
    rcases le_total 0 (A i j) with hsign | hsign

Goal state after:
case h.inl
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
v : β → ℤ
hv : 0 ≤ v ∧ v ≤ B'
mulVec_def : A *ᵥ v = fun i ↦ ∑ j, A i j * v j
i : α
j : β
a✝ : j ∈ univ
hsign : 0 ≤ A i j
⊢ -(↑B * (A i j)⁻) ≤ v j * A i j

case h.inr
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
v : β → ℤ
hv : 0 ≤ v ∧ v ≤ B'
mulVec_def : A *ᵥ v = fun i ↦ ∑ j, A i j * v j
i : α
j : β
a✝ : j ∈ univ
hsign : A i j ≤ 0
⊢ -(↑B * (A i j)⁻) ≤ v j * A i j
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
v : β → ℤ
hv : 0 ≤ v ∧ v ≤ B'
mulVec_def : A *ᵥ v = fun i ↦ ∑ j, A i j * v j
i : α
⊢ (fun i ↦ ∑ j, A i j * v j) i ≤ P i

Tactic:
simp only [mul_neg]
    gcongr ∑ _ : β, ?_ with j _ -- Get rid of sums
    rw [← mul_comm (v j)] -- Move A i j to the right of the products
    -- We have to distinguish cases: we have now 4 goals
    rcases le_total 0 (A i j) with hsign | hsign

Goal state after:
case h.inl
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
v : β → ℤ
hv : 0 ≤ v ∧ v ≤ B'
mulVec_def : A *ᵥ v = fun i ↦ ∑ j, A i j * v j
i : α
j : β
a✝ : j ∈ univ
hsign : 0 ≤ A i j
⊢ v j * A i j ≤ ↑B * (A i j)⁺

case h.inr
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
v : β → ℤ
hv : 0 ≤ v ∧ v ≤ B'
mulVec_def : A *ᵥ v = fun i ↦ ∑ j, A i j * v j
i : α
j : β
a✝ : j ∈ univ
hsign : A i j ≤ 0
⊢ v j * A i j ≤ ↑B * (A i j)⁺
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:
rw [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.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:
rw [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.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 ↦ ∑ x ∈ s, toCompl (f x)

Tactic:
← map_sum

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 ↦ toCompl (∑ x ∈ s, f x)
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: f : AbsoluteValue ℚ ℝ
m n : ℕ
hm : 1 < m
hn : 1 < n
notbdd : ¬∀ (n : ℕ), f ↑n ≤ 1
k : ℕ
hk : k ≠ 0
h_ineq1 : ∀ {m n : ℕ}, 1 < m → 1 < n → f ↑n ≤ ↑m * f ↑m / (f ↑m - 1) * f ↑m ^ logb ↑m ↑n
he : 0 < ↑m * f ↑m / (f ↑m - 1)
⊢ f ↑n ^ k ≤ ((↑m * f ↑m / (f ↑m - 1)) ^ (↑k)⁻¹ * f ↑m ^ logb ↑m ↑n) ^ ↑k

Tactic:
mul_rpow (by positivity) (by positivity),

Goal state after:
f : AbsoluteValue ℚ ℝ
m n : ℕ
hm : 1 < m
hn : 1 < n
notbdd : ¬∀ (n : ℕ), f ↑n ≤ 1
k : ℕ
hk : k ≠ 0
h_ineq1 : ∀ {m n : ℕ}, 1 < m → 1 < n → f ↑n ≤ ↑m * f ↑m / (f ↑m - 1) * f ↑m ^ logb ↑m ↑n
he : 0 < ↑m * f ↑m / (f ↑m - 1)
⊢ f ↑n ^ k ≤ ((↑m * f ↑m / (f ↑m - 1)) ^ (↑k)⁻¹) ^ ↑k * (f ↑m ^ logb ↑m ↑n) ^ ↑k
Module: Mathlib.NumberTheory.Ostrowski

Goal state before:
pp: f : AbsoluteValue ℚ ℝ
notbdd : ¬∀ (n : ℕ), f ↑n ≤ 1
m : ℕ
hm : ¬f ↑m ≤ 1
⊢ f.IsEquiv real

Tactic:
have hfm1 : 1 < f m := lt_of_not_ge hm

Goal state after:
f : AbsoluteValue ℚ ℝ
notbdd : ¬∀ (n : ℕ), f ↑n ≤ 1
m : ℕ
hm : ¬f ↑m ≤ 1
hfm1 : 1 < f ↑m
⊢ f.IsEquiv real
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℕ
⊢ Continuous fun x ↦ Ring.choose x k

Tactic:
by
  simp only [Ring.choose]
  fun_prop

Goal state after:
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n : ℕ
⊢ ∑ x ∈ range (n + 1), ((-1) ^ (n - x) * ↑(n.choose x)) • (mahlerSeries a) (x • 1) =
    ∑ x ∈ range (n + 1), ((-1) ^ (n - x) * ↑(n.choose x)) • ∑ x_1 ∈ range (n + 1), (x • 1).choose x_1 • a x_1

Tactic:
refine Finset.sum_congr rfl fun j hj ↦ ?_

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n j : ℕ
hj : j ∈ range (n + 1)
⊢ ((-1) ^ (n - j) * ↑(n.choose j)) • (mahlerSeries a) (j • 1) =
    ((-1) ^ (n - j) * ↑(n.choose j)) • ∑ x ∈ range (n + 1), (j • 1).choose x • a x
Module: Mathlib.NumberTheory.Padics.MahlerBasis

Goal state before:
pp: case a
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Module ℤ_[p] E
inst✝² : IsBoundedSMul ℤ_[p] E
a✝ : E
n✝ : ℕ
x : ℤ_[p]
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
n : ℕ
⊢ ‖mahlerTerm (Δ_[1]^[n] (⇑f) 0) n‖ =
    ‖({ toFun := fun f ↦ { toFun := fun x ↦ Δ_[1]^[x] (⇑f) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ },
              map_add' := ⋯, map_smul' := ⋯ }
            (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f) 0) b)).toBCF
        n‖

Tactic:
simp [(hasSum_mahler f).tsum_eq]

Goal state after:
Module: Mathlib.NumberTheory.Padics.WithVal

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ : ℚ
⊢ ∀ (q : ℚ), ⋯.extend coe' ↑q = ↑((WithVal.equiv (Rat.padicValuation p)).symm q)

Tactic:
intro q

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
a✝ q : ℚ
⊢ ⋯.extend coe' ↑q = ↑((WithVal.equiv (Rat.padicValuation p)).symm q)
Module: Mathlib.NumberTheory.PowModTotient

Goal state before:
pp: x k n : ℕ
hn : 1 < n
h : x.Coprime n
⊢ x ^ (k + φ n) % n = x ^ k % n

Tactic:
pow_add,

Goal state after:
x k n : ℕ
hn : 1 < n
h : x.Coprime n
⊢ x ^ k * x ^ φ n % n = x ^ k % n
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : d ≠ 0
a : Solution₁ d
ha : a.x = 1
prop : d * a.y ^ 2 = a.x ^ 2 - 1
⊢ a = 1

Tactic:
rw [ha, one_pow, sub_self, mul_eq_zero, or_iff_right h₀, sq_eq_zero_iff] at prop

Goal state after:
d : ℤ
h₀ : d ≠ 0
a : Solution₁ d
ha : a.x = 1
prop : a.y = 0
⊢ a = 1
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite
hm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = 0}.Infinite
q : ℚ
a : ℤ
hq : ↑q.den ^ 2 * a ^ 2 = ↑q.den ^ 2 * d
ha : q.num = ↑q.den * a
⊢ False

Tactic:
mul_right_inj' (pow_pos (Int.natCast_pos.mpr q.pos) 2).ne'

Goal state after:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite
hm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = 0}.Infinite
q : ℚ
a : ℤ
hq : a ^ 2 = d
ha : q.num = ↑q.den * a
⊢ False
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: case inr
d : ℤ
a : Solution₁ d
h : IsFundamental a
H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
n : ℤ
hn : n < 0
m : ℤ := -n - 1
⊢ (a ^ n).y < (a ^ (n + 1)).y

Tactic:
have hm : n = -m - 1 := by simp only [m, neg_sub, sub_neg_eq_add, add_tsub_cancel_left]

Goal state after:
case inr
d : ℤ
a : Solution₁ d
h : IsFundamental a
H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
n : ℤ
hn : n < 0
m : ℤ := -n - 1
hm : n = -m - 1
⊢ (a ^ n).y < (a ^ (n + 1)).y
Module: Mathlib.NumberTheory.RamificationInertia.HilbertTheory

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : Field K
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
p : Ideal A
P : Ideal B
inst✝¹² : P.LiesOver p
inst✝¹¹ : FiniteDimensional K L
inst✝¹⁰ : MulSemiringAction Gal(L/K) B
inst✝⁹ : IsGaloisGroup Gal(L/K) A B
inst✝⁸ : IsDedekindDomain A
inst✝⁷ : IsDedekindDomain B
inst✝⁶ : Module.Finite A B
inst✝⁵ : Module.IsTorsionFree A B
inst✝⁴ : Ring.HasFiniteQuotients A
inst✝³ : P.IsMaximal
D : Type u_5
inst✝² : Field D
inst✝¹ : Algebra D L
inst✝ : IsDecompositionField K L P D
hp : p ≠ ⊥
this✝ : p.IsMaximal
this : Finite (A ⧸ p)
⊢ Module.finrank D L = p.ramificationIdxIn B * p.inertiaDegIn B

Tactic:
← IsGaloisGroup.card_eq_finrank (stabilizer Gal(L/K) P) D L,

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : Field K
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
p : Ideal A
P : Ideal B
inst✝¹² : P.LiesOver p
inst✝¹¹ : FiniteDimensional K L
inst✝¹⁰ : MulSemiringAction Gal(L/K) B
inst✝⁹ : IsGaloisGroup Gal(L/K) A B
inst✝⁸ : IsDedekindDomain A
inst✝⁷ : IsDedekindDomain B
inst✝⁶ : Module.Finite A B
inst✝⁵ : Module.IsTorsionFree A B
inst✝⁴ : Ring.HasFiniteQuotients A
inst✝³ : P.IsMaximal
D : Type u_5
inst✝² : Field D
inst✝¹ : Algebra D L
inst✝ : IsDecompositionField K L P D
hp : p ≠ ⊥
this✝ : p.IsMaximal
this : Finite (A ⧸ p)
⊢ Nat.card ↥(stabilizer Gal(L/K) P) = p.ramificationIdxIn B * p.inertiaDegIn B
Module: Mathlib.NumberTheory.Pell

Goal state before:
pp: d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hay : 0 ≤ a.y
ax : ℕ
hax' : ↑ax = a.x
hax : 0 < ↑ax
⊢ ∃ n, a = a₁ ^ n

Tactic:
induction ax using Nat.strong_induction_on generalizing a with

Goal state after: