mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)
x : ℍ
hx : x ∈ interior 𝒟
ξ : ℍ
hξ : ξ ∈ 𝒟
⊢ ξ ∈ (fun τ ↦ ‖↑τ‖) ⁻¹' Set.Ici 1

Tactic:
simpa using

Goal state after:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)
x : ℍ
hx : x ∈ interior 𝒟
ξ : ℍ
hξ : ξ ∈ 𝒟
⊢ 1 ≤ ‖↑ξ‖
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)
x : ℍ
hx : x ∈ interior 𝒟
ξ : ℍ
hξ : ξ ∈ 𝒟
⊢ ξ ∈ UpperHalfPlane.re ⁻¹' Set.Icc (-(1 / 2)) (1 / 2)

Tactic:
simpa using

Goal state after:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)
x : ℍ
hx : x ∈ interior 𝒟
ξ : ℍ
hξ : ξ ∈ 𝒟
⊢ -2⁻¹ ≤ ξ.re ∧ ξ.re ≤ 2⁻¹
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: y : ℝ
z : ℂ
hz : 0 < z.im
h : { coe := z, coe_im_pos := hz } ∈ truncatedFundamentalDomain y
⊢ 1 ≤ ‖↑{ coe := z, coe_im_pos := hz }‖

Tactic:
simpa using

Goal state after:
y : ℝ
z : ℂ
hz : 0 < z.im
h : { coe := z, coe_im_pos := hz } ∈ truncatedFundamentalDomain y
⊢ 1 ≤ ‖z‖
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: y : ℝ
z : ℂ
hz : 0 ≤ z.im
h1 : z.im ≤ y
h2 : |z.re| ≤ 1 / 2
h3 : 0 = z.im
⊢ ‖z‖ < 1

Tactic:
simpa using

Goal state after:
y : ℝ
z : ℂ
hz : 0 ≤ z.im
h1 : z.im ≤ y
h2 : |z.re| ≤ 1 / 2
h3 : 0 = z.im
⊢ |z.re| < 1
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion

Goal state before:
pp: k : ℕ
hk : 3 ≤ k
hk2 : Even k
⊢ (PowerSeries.coeff 0) (qExpansion 1 ⇑(E hk)) = 1

Tactic:
simpa using

Goal state after:
k : ℕ
hk : 3 ≤ k
hk2 : Even k
⊢ PowerSeries.constantCoeff (qExpansion 1 ⇑(E hk)) = 1
Module: Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing

Goal state before:
pp: z : ℍ
g : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 12
hg : CuspForm.toModularFormₗ g = E₄CubeSubE₆SqForm
c : ℂ
hc : c • CuspForm.discriminant = g
hgE : ⇑g = ⇑E₄CubeSubE₆SqForm
hcΔ : c • ⇑CuspForm.discriminant = ⇑g
hgΔ : qExpansion 1 ⇑E₄CubeSubE₆SqForm = c • qExpansion 1 ⇑CuspForm.discriminant
⊢ c = 1728

Tactic:
simpa using

Goal state after:
z : ℍ
g : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 12
hg : CuspForm.toModularFormₗ g = E₄CubeSubE₆SqForm
c : ℂ
hc : c • CuspForm.discriminant = g
hgE : ⇑g = ⇑E₄CubeSubE₆SqForm
hcΔ : c • ⇑CuspForm.discriminant = ⇑g
hgΔ : qExpansion 1 ⇑E₄CubeSubE₆SqForm = c • qExpansion 1 ⇑CuspForm.discriminant
⊢ c = 1728
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inl
τ : ℍ
h : 1 / 2 ≤ τ.im
⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im ∧ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1

Tactic:
exact ⟨1, (one_smul SL(2, ℤ) τ).symm ▸ h, by
      simp only [map_one, denom_one, norm_one, le_refl]⟩

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing

Goal state before:
pp: ⊢ (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (modularForm CuspForm.discriminant) =
    (1 / 1728) •
      ((DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (E₄.pow 3) -
        (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (E₆.pow 2))

Tactic:
rw [← map_sub (DirectSum.of (ModularForm 𝒮ℒ) 12), ← DirectSum.of_smul]

Goal state after:
⊢ (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (modularForm CuspForm.discriminant) =
    (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) ((1 / 1728) • (E₄.pow 3 - E₆.pow 2))
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inr
τ : ℍ
h : τ.im ≤ 1 / 2
γ : SL(2, ℤ)
hγ : 1 / 2 ≤ (γ • τ).im
h1 : τ.im * ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ^ 2 ≤ τ.im
⊢ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1

Tactic:
simpa using

Goal state after:
case inr
τ : ℍ
h : τ.im ≤ 1 / 2
γ : SL(2, ℤ)
hγ : 1 / 2 ≤ (γ • τ).im
h1 : τ.im * ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ^ 2 ≤ τ.im
⊢ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝² : FunLike F ℍ ℂ
k : ℤ
inst✝¹ : SlashInvariantFormClass F 𝒢 k
inst✝ : 𝒢.IsFiniteRelIndex ℋ
h : GL (Fin 2) ℝ
hh : h ∈ ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
⊢ (let this := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ);
      ∑ q, quotientFunc f q) ∣[k]
      h =
    let this := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ);
    ∑ q, quotientFunc f q

Tactic:
simpa using

Goal state after:
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝² : FunLike F ℍ ℂ
k : ℤ
inst✝¹ : SlashInvariantFormClass F 𝒢 k
inst✝ : 𝒢.IsFiniteRelIndex ℋ
h : GL (Fin 2) ℝ
hh : h ∈ ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
⊢ ∑ x, quotientFunc f (⟨h, hh⟩⁻¹ • x) = ∑ q, quotientFunc f q
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : SlashInvariantFormClass F 𝒢 k
inst✝¹ : 𝒢.IsFiniteRelIndex ℋ
inst✝ : ℋ.HasDetPlusMinusOne
h : GL (Fin 2) ℝ
hh : h ∈ ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
⊢ (let this := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ);
      ∏ q, quotientFunc f q) ∣[k * ↑(Nat.card (↥ℋ ⧸ 𝒢.subgroupOf ℋ))]
      h =
    let this := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ);
    ∏ q, quotientFunc f q

Tactic:
simpa using

Goal state after:
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : SlashInvariantFormClass F 𝒢 k
inst✝¹ : 𝒢.IsFiniteRelIndex ℋ
inst✝ : ℋ.HasDetPlusMinusOne
h : GL (Fin 2) ℝ
hh : h ∈ ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
⊢ ∏ x, quotientFunc f (⟨h, hh⟩⁻¹ • x) = ∏ q, quotientFunc f q
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝² : FunLike F ℍ ℂ
k : ℤ
inst✝¹ : 𝒢.IsFiniteRelIndex ℋ
inst✝ : ModularFormClass F 𝒢 k
γ : GL (Fin 2) ℝ
h : IsCusp (γ • OnePoint.infty) ℋ
x✝¹ : ↥ℋ
r : GL (Fin 2) ℝ
hr : r ∈ ℋ
x✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ
⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toConjAct (↑⟨r, hr⟩)⁻¹⁻¹ • 𝒢)

Tactic:
simpa using

Goal state after:
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝² : FunLike F ℍ ℂ
k : ℤ
inst✝¹ : 𝒢.IsFiniteRelIndex ℋ
inst✝ : ModularFormClass F 𝒢 k
γ : GL (Fin 2) ℝ
h : IsCusp (γ • OnePoint.infty) ℋ
x✝¹ : ↥ℋ
r : GL (Fin 2) ℝ
hr : r ∈ ℋ
x✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ
⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toConjAct r • 𝒢)
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝² : FunLike F ℍ ℂ
k : ℤ
inst✝¹ : 𝒢.IsFiniteRelIndex ℋ
inst✝ : CuspFormClass F 𝒢 k
γ : GL (Fin 2) ℝ
h : IsCusp (γ • OnePoint.infty) ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
x✝¹ : ↥ℋ
r : GL (Fin 2) ℝ
hr : r ∈ ℋ
x✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ
⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toConjAct (↑⟨r, hr⟩)⁻¹⁻¹ • 𝒢)

Tactic:
simpa using

Goal state after:
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝² : FunLike F ℍ ℂ
k : ℤ
inst✝¹ : 𝒢.IsFiniteRelIndex ℋ
inst✝ : CuspFormClass F 𝒢 k
γ : GL (Fin 2) ℝ
h : IsCusp (γ • OnePoint.infty) ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
x✝¹ : ↥ℋ
r : GL (Fin 2) ℝ
hr : r ∈ ℋ
x✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ
⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toConjAct r • 𝒢)
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
γ : GL (Fin 2) ℝ
h : IsCusp (γ • OnePoint.infty) ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
x✝¹ : ↥ℋ
r : GL (Fin 2) ℝ
hr : r ∈ ℋ
x✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ
⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toConjAct (↑⟨r, hr⟩)⁻¹⁻¹ • 𝒢)

Tactic:
simpa using

Goal state after:
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
γ : GL (Fin 2) ℝ
h : IsCusp (γ • OnePoint.infty) ℋ
this : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)
x✝¹ : ↥ℋ
r : GL (Fin 2) ℝ
hr : r ∈ ℋ
x✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ
⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toConjAct r • 𝒢)
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
hf : ∃ i ∈ Finset.univ, quotientFunc f i = 0
⊢ ⇑f = 0

Tactic:
simpa using

Goal state after:
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
hf : ∃ i ∈ Finset.univ, quotientFunc f i = 0
⊢ ⇑f = 0
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: case refine_2
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
hf : ⇑f = 0
τ : ℍ
⊢ (ModularForm.norm ℋ f) τ = 0 τ

Tactic:
simpa using

Goal state after:
case refine_2
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
hf : ⇑f = 0
τ : ℍ
⊢ ∃ a ∈ ℋ, (⇑f ∣[k] a⁻¹) τ = 0
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
hf : ⇑f = 0
τ : ℍ
⊢ 1 ∈ ℋ ∧ (⇑f ∣[k] 1⁻¹) τ = 0

Tactic:
simpa using

Goal state after:
𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)
F : Type u_1
f : F
inst✝³ : FunLike F ℍ ℂ
k : ℤ
inst✝² : 𝒢.IsFiniteRelIndex ℋ
inst✝¹ : ℋ.HasDetPlusMinusOne
inst✝ : ModularFormClass F 𝒢 k
hf : ⇑f = 0
τ : ℍ
⊢ f τ = 0
Module: Mathlib.NumberTheory.ModularForms.NormTrace

Goal state before:
pp: 𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : 𝒢.IsArithmetic
inst✝ : 𝒢.HasDetOne
f : ModularForm 𝒢 0
this : ModularFormClass
  (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range
    (0 *
      ↑(Nat.card
          (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearGroup.mapGL ℝ).range))))
  (Matrix.SpecialLinearGroup.mapGL ℝ).range 0 :=
  Eq.mpr
    (id
      (congrArg
        (fun _a ↦
          ModularFormClass (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range _a)
            (Matrix.SpecialLinearGroup.mapGL ℝ).range 0)
        (zero_mul
          ↑(Nat.card
              (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearGroup.mapGL ℝ).range)))))
    inferInstance
c : ℂ
hc : ⇑(ModularForm.norm (Matrix.SpecialLinearGroup.mapGL ℝ).range (f - const (f I))) = Function.const ℍ c
⊢ (ModularForm.norm (Matrix.SpecialLinearGroup.mapGL ℝ).range (f - const (f I))) I = 0

Tactic:
simpa using

Goal state after:
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : 𝒢.IsArithmetic
inst✝ : 𝒢.HasDetOne
f : ModularForm 𝒢 0
this : ModularFormClass
  (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range
    (0 *
      ↑(Nat.card
          (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearGroup.mapGL ℝ).range))))
  (Matrix.SpecialLinearGroup.mapGL ℝ).range 0 :=
  Eq.mpr
    (id
      (congrArg
        (fun _a ↦
          ModularFormClass (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range _a)
            (Matrix.SpecialLinearGroup.mapGL ℝ).range 0)
        (zero_mul
          ↑(Nat.card
              (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearGroup.mapGL ℝ).range)))))
    inferInstance
c : ℂ
hc : ⇑(ModularForm.norm (Matrix.SpecialLinearGroup.mapGL ℝ).range (f - const (f I))) = Function.const ℍ c
⊢ ∃ a, ((⇑f - Function.const ℍ (f I)) ∣[0] ((Matrix.SpecialLinearGroup.mapGL ℝ) a)⁻¹) I = 0
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: R : Type u_1
n : ℕ
inst✝ : CommRing R
x y p : R
h : p ∣ x - y
⊢ p ∣ y - x

Tactic:
simpa using

Goal state after:
R : Type u_1
n : ℕ
inst✝ : CommRing R
x y p : R
h : p ∣ x - y
⊢ p ∣ y - x
Module: Mathlib.NumberTheory.NumberField.InfiniteAdeleRing

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K∞
v : InfinitePlace K
hv : ¬IsUnit (x v)
⊢ ‖x v‖ ^ v.mult = 0

Tactic:
simpa using

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K∞
v : InfinitePlace K
hv : ¬IsUnit (x v)
⊢ x v = 0
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
r : ℝ
hpos : 0 < s.re
hs : r + 1 < s.re
hΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s
hcoeff : (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ r
hh : 0 < h Γ
hΓ : h Γ ∈ Γ.strictPeriods
⊢ (fun x ↦ ‖(PowerSeries.coeff x) (qExpansion (h Γ) ⇑f)‖) =O[atTop] fun x ↦ ↑x ^ r

Tactic:
simpa using

Goal state after:
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
r : ℝ
hpos : 0 < s.re
hs : r + 1 < s.re
hΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s
hcoeff : (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ r
hh : 0 < h Γ
hΓ : h Γ ∈ Γ.strictPeriods
⊢ (fun x ↦ (PowerSeries.coeff x) (qExpansion (h Γ) ⇑f)) =O[atTop] fun x ↦ ↑x ^ r
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case refine_2
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : ↑k + 1 < s.re
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ ↑k

Tactic:
simpa using

Goal state after:
case refine_2
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : ↑k + 1 < s.re
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ k
Module: Mathlib.NumberTheory.NumberField.Completion.InfinitePlace

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
L : Type u_2
inst✝² : Field L
inst✝¹ : Algebra K L
w : InfinitePlace L
v : InfinitePlace K
inst✝ : w.LiesOver v
x : WithAbs ↑v
⊢ ‖(algebraMap (WithAbs ↑v) (WithAbs ↑w)) x‖ = ‖x‖

Tactic:
simpa using

Goal state after:
K : Type u_1
inst✝³ : Field K
L : Type u_2
inst✝² : Field L
inst✝¹ : Algebra K L
w : InfinitePlace L
v : InfinitePlace K
inst✝ : w.LiesOver v
x : WithAbs ↑v
⊢ ↑w ((algebraMap (WithAbs ↑v) (WithAbs ↑w)) x).ofAbs = ↑v x.ofAbs
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case refine_2
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : CuspFormClass F Γ k
hk : 0 < k
hs : ↑k / 2 + 1 < s.re
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ (↑k / 2)

Tactic:
simpa using

Goal state after:
case refine_2
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : CuspFormClass F Γ k
hk : 0 < k
hs : ↑k / 2 + 1 < s.re
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ (↑k / 2)
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: case refine_2
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
QU : DFunLike.coe ⁻¹' U ⊆ DFunLike.coe ⁻¹' Q
⊢ DFunLike.coe ⁻¹' U ∈ 𝓝 x

Tactic:
simpa using

Goal state after:
case refine_2
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
QU : DFunLike.coe ⁻¹' U ⊆ DFunLike.coe ⁻¹' Q
⊢ DFunLike.coe ⁻¹' U ∈ comap DFunLike.coe (𝓝 ⇑x)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: m : ℤ
eq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))
⊢ 8 ∣ (2 * m + 1) ^ 2 - 1

Tactic:
simpa using

Goal state after:
m : ℤ
eq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))
⊢ 8 ∣ 4 * (m * (m + 1))
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: m : ℕ
eq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))
⊢ 8 ∣ (2 * m + 1) ^ 2 - 1

Tactic:
simpa using

Goal state after:
m : ℕ
eq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))
⊢ 8 ∣ 4 * (m * (m + 1))
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
⊢ Odd y

Tactic:
simpa using

Goal state after:
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
⊢ Odd y
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
⊢ Odd y

Tactic:
simpa using

Goal state after:
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
⊢ Odd y
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hxy
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 (n + 1)

Tactic:
simpa using

Goal state after:
case succ.hxy
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 (n + 1)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hx
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)

Tactic:
simpa using

Goal state after:
case succ.hx
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x n : ℕ
h1x : 1 < x
hx : ¬2 ∣ x
hn : n ≠ 0
hneven : Even n
⊢ padicValNat 2 (x ^ n - 1) + 1 = padicValNat 2 (x + 1) + padicValNat 2 (x - 1) + padicValNat 2 n

Tactic:
simpa using

Goal state after:
x n : ℕ
h1x : 1 < x
hx : ¬2 ∣ x
hn : n ≠ 0
hneven : Even n
⊢ padicValNat 2 (x ^ n - 1) + 1 = padicValNat 2 (x + 1) + padicValNat 2 (x - 1) + padicValNat 2 n
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
𝓕✝ 𝓖 : Filter ι
S : ι → Type u_3
inst✝⁴ : (i : ι) → SetLike (S i) (R i)
B : (i : ι) → S i
T : Set ι
𝓕 : Filter ι
inst✝³ : (i : ι) → TopologicalSpace (R i)
hBopen : Fact (∀ (i : ι), IsOpen[inst✝³ i] ↑(B i))
inst✝² : (i : ι) → Group (R i)
inst✝¹ : ∀ (i : ι), SubgroupClass (S i) (R i)
inst✝ : ∀ (i : ι), IsTopologicalGroup (R i)
hAcompact : ∀ (i : ι), CompactSpace ↥(B i)
i : ι
x : R i
⊢ x ∈ x • ↑(B i)

Tactic:
simpa using

Goal state after:
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
𝓕✝ 𝓖 : Filter ι
S : ι → Type u_3
inst✝⁴ : (i : ι) → SetLike (S i) (R i)
B : (i : ι) → S i
T : Set ι
𝓕 : Filter ι
inst✝³ : (i : ι) → TopologicalSpace (R i)
hBopen : Fact (∀ (i : ι), IsOpen[inst✝³ i] ↑(B i))
inst✝² : (i : ι) → Group (R i)
inst✝¹ : ∀ (i : ι), SubgroupClass (S i) (R i)
inst✝ : ∀ (i : ι), IsTopologicalGroup (R i)
hAcompact : ∀ (i : ι), CompactSpace ↥(B i)
i : ι
x : R i
⊢ x ∈ x • ↑(B i)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
h : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w
a : mixedSpace K
ha : a ∈ A
⊢ normAtComplexPlaces a ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0

Tactic:
simpa using

Goal state after:
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
h : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w
a : mixedSpace K
ha : a ∈ A
⊢ ∀ (i : InfinitePlace K), i.IsComplex → 0 ≤ normAtComplexPlaces a i
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : mixedSpace K
hx : x ∈ fundamentalCone K
hy : ∀ (w : InfinitePlace K), (normAtPlace w) y = (normAtPlace w) x
⊢ y ∉ {x | mixedEmbedding.norm x = 0}

Tactic:
simpa using

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x y : mixedSpace K
hx : x ∈ fundamentalCone K
hy : ∀ (w : InfinitePlace K), (normAtPlace w) y = (normAtPlace w) x
⊢ ¬mixedEmbedding.norm x = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord

Goal state before:
pp: case refine_2
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
h : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w
x : realSpace K
x✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
hx₁ : normAtComplexPlaces (mixedSpaceOfRealSpace x) ∈ normAtComplexPlaces '' A
hx₂ : x ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
w : InfinitePlace K
hw : w.IsComplex
⊢ 0 ≤ x w

Tactic:
simpa using

Goal state after:
case refine_2
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A
h : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w
x : realSpace K
x✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
hx₁ : normAtComplexPlaces (mixedSpaceOfRealSpace x) ∈ normAtComplexPlaces '' A
hx₂ : x ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.univ else Set.Ici 0
w : InfinitePlace K
hw : w.IsComplex
⊢ 0 ≤ x w
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
⊢ regOfFamily u ≠ 0 → IsMaxRank u

Tactic:
simpa using

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
⊢ ¬regOfFamily u = 0 → IsMaxRank u