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

Goal state before:
pp: k : ℤ
hk : 3 ≤ k
z : ℍ
⊢ Summable fun x ↦ ‖eisSummand k x z‖

Tactic:
by

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

Goal state before:
pp: N : ℕ
a : Fin 2 → ZMod N
k : ℤ
hk : 3 ≤ k
z : ℍ
⊢ ‖eisensteinSeries a k z‖ ≤ ∑' (x : Fin 2 → ℤ), ‖eisSummand k x z‖

Tactic:
by

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

Goal state before:
pp: h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
⊢ (fun τ ↦ f τ - valueAtInfty f) =O[atImInfty] fun τ ↦ rexp (-2 * π * τ.im / h)

Tactic:
by

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

Goal state before:
pp: h : ℝ
f : ℍ → ℂ
hf : IsZeroAtImInfty f
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
⊢ f =O[atImInfty] fun τ ↦ rexp (-2 * π * τ.im / h)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝⁴ : Γ.IsArithmetic
F : Type u_2
F' : Type u_3
f : F
f' : F'
inst✝³ : FunLike F ℍ ℂ
inst✝² : FunLike F' ℍ ℂ
inst✝¹ : CuspFormClass F Γ k
inst✝ : ModularFormClass F' Γ k
⊢ Continuous fun τ ↦ ‖petersson k (⇑f) (⇑f') τ‖

Tactic:
by

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

Goal state before:
pp: k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f : F
inst✝ : ModularFormClass F Γ k
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
n : ℕ
t : ℝ
ht : 0 < t
u : ℝ
⊢ 0 < (↑u + ↑t * I).im

Tactic:
by

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

Goal state before:
pp: k : ℤ
F : Type u_1
inst✝⁴ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
f : F
inst✝³ : ModularFormClass F Γ k
inst✝² : Γ.HasDetPlusMinusOne
inst✝¹ : DiscreteTopology ↥Γ
inst✝ : Fact (IsCusp OnePoint.infty Γ)
⊢ ∃ c > 0, (fun τ ↦ f τ - valueAtInfty ⇑f) =O[atImInfty] fun τ ↦ rexp (-c * τ.im)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝⁴ : Γ.IsArithmetic
F : Type u_2
F' : Type u_3
f : F
f' : F'
inst✝³ : FunLike F ℍ ℂ
inst✝² : FunLike F' ℍ ℂ
inst✝¹ : CuspFormClass F Γ k
inst✝ : ModularFormClass F' Γ k
g : SL(2, ℤ)
⊢ (toConjAct (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g))⁻¹ • Γ).IsArithmetic

Tactic:
by

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

Goal state before:
pp: k : ℤ
F : Type u_1
inst✝⁴ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
f : F
inst✝³ : ModularFormClass F Γ k
inst✝² : Γ.HasDetPlusMinusOne
inst✝¹ : DiscreteTopology ↥Γ
inst✝ : Fact (IsCusp OnePoint.infty Γ)
hf : IsZeroAtImInfty ⇑f
⊢ ∃ c > 0, ⇑f =O[atImInfty] fun τ ↦ rexp (-c * τ.im)

Tactic:
by

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

Goal state before:
pp: k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f : F
inst✝ : CuspFormClass F Γ k
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
⊢ (PowerSeries.coeff 0) (qExpansion h ⇑f) = 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝⁴ : Γ.IsArithmetic
F : Type u_2
F' : Type u_3
f : F
f' : F'
inst✝³ : FunLike F ℍ ℂ
inst✝² : FunLike F' ℍ ℂ
inst✝¹ : CuspFormClass F Γ k
inst✝ : ModularFormClass F' Γ k
⊢ ∃ C, ∀ (τ : ℍ), ‖petersson k (⇑f) (⇑f') τ‖ ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝⁴ : Γ.IsArithmetic
F : Type u_2
F' : Type u_3
f : F
f' : F'
inst✝³ : FunLike F ℍ ℂ
inst✝² : FunLike F' ℍ ℂ
inst✝¹ : ModularFormClass F Γ k
inst✝ : CuspFormClass F' Γ k
⊢ ∃ C, ∀ (τ : ℍ), ‖petersson k (⇑f) (⇑f') τ‖ ≤ C

Tactic:
by

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
a : Fin 2 → ZMod N
k : ℤ
hk : 3 ≤ k
A : SL(2, ℤ)
z : ℍ
hz : 2 ≤ z.im
n : ℤ
hn : ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im
⊢ 2 < ↑k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
inst✝ : CuspFormClass F Γ k
f : F
C : ℝ
τ : ℍ
hC : ‖petersson k (⇑f) (⇑f) τ‖ ≤ C
⊢ 0 ≤ ‖f τ‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
inst✝ : CuspFormClass F Γ k
f : F
C : ℝ
τ : ℍ
hC : ‖petersson k (⇑f) (⇑f) τ‖ ≤ C
⊢ 0 ≤ √C / τ.im ^ (↑k / 2)

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℂ → ℂ
hfcts : ContinuousAt (Periodic.cuspFunction h f) 0
hgcts : ContinuousAt (Periodic.cuspFunction h g) 0
⊢ Periodic.cuspFunction h (f * g) 0 = Periodic.cuspFunction h f 0 * Periodic.cuspFunction h g 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
⊢ ∃ g, g • z ∈ 𝒟

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℍ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
⊢ (PowerSeries.coeff 0) (qExpansion h (f * g)) =
    (PowerSeries.coeff 0) (qExpansion h f) * (PowerSeries.coeff 0) (qExpansion h g)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : 3 * c' ^ 2 ≤ 4
⊢ c' ^ 2 ≤ 1 ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : 3 * c ^ 2 ≤ 4
⊢ |↑g 1 0| ≤ 1

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℍ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
⊢ cuspFunction h (f * g) = cuspFunction h f * cuspFunction h g

Tactic:
by

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

Goal state before:
pp: h : ℝ
f : ℍ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
a : ℂ
⊢ cuspFunction h (a • f) = a • cuspFunction h f

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℍ → ℂ
hf : AnalyticAt ℂ (cuspFunction h f) 0
hg : AnalyticAt ℂ (cuspFunction h g) 0
n✝ i : ℕ
hi : i ∈ Finset.range (n✝ + 1)
⊢ i ≤ n✝

Tactic:
by

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
a : Fin 2 → ZMod N
k : ℤ
hk : 3 ≤ k
A : SL(2, ℤ)
⊢ IsBoundedAtImInfty (⇑(eisensteinSeriesSIF a k) ∣[k] A)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
this : c ≠ 0 → 9 * c ^ 4 ≤ 16
⊢ 3 * c ^ 2 ≤ 4

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
inst✝ : CuspFormClass F Γ k
f : F
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C / τ.im ^ (↑k / 2)

Tactic:
by

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

Goal state before:
pp: k : ℤ
hk : 3 ≤ k
N : ℕ
a : Fin 2 → ZMod N
⊢ 2 < ↑k

Tactic:
by

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

Goal state before:
pp: k : ℤ
hk : 3 ≤ k
N : ℕ
a : Fin 2 → ZMod N
hk' : 2 < ↑k
p_sum : Summable fun x ↦ ‖↑x‖ ^ (-k)
K : Set ℍ
hK : IsCompact K
A B : ℝ
hB : 0 < B
HABK : K ⊆ verticalStrip A B
p : ↑(gammaSet N 1 a)
z : ℍ
hz : z ∈ verticalStrip A B
⊢ 0 ≤ ↑k

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℍ → ℂ
hf : AnalyticAt ℂ (cuspFunction h f) 0
hg : AnalyticAt ℂ (cuspFunction h g) 0
⊢ qExpansion h (f * g) = qExpansion h f * qExpansion h g

Tactic:
by

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

Goal state before:
pp: k : ℤ
hk : 3 ≤ k
N : ℕ
a : Fin 2 → ZMod N
⊢ TendstoLocallyUniformly (fun s x ↦ ∑ x_1 ∈ s, eisSummand k (↑x_1) x) (fun x ↦ eisensteinSeries a k x) Filter.atTop

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
⊢ g • z ∈ 𝒟

Tactic:
by

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

Goal state before:
pp: h : ℝ
f : ℍ → ℂ
hf : AnalyticAt ℂ (cuspFunction h f) 0
a : ℂ
⊢ qExpansion h (a • f) = a • qExpansion h f

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
⊢ z ∈ 𝒟

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
f : F
C : ℝ
τ : ℍ
k : ℕ
inst✝ : ModularFormClass F Γ ↑k
hC : ‖petersson (↑k) (⇑f) (⇑f) τ‖ ≤ C * max τ.im (1 / τ.im) ^ ↑k
⊢ 0 ≤ ‖petersson (↑k) (⇑f) (⇑f) τ‖ / max τ.im (1 / τ.im) ^ ↑k

Tactic:
by

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

Goal state before:
pp: h : ℝ
f : ℍ → ℂ
hf : AnalyticAt ℂ (cuspFunction h f) 0
⊢ qExpansion h (-f) = -qExpansion h f

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
⊢ 3 * 3 * c ^ 4 ≤ 4 * (g • z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℍ → ℂ
hf : AnalyticAt ℂ (cuspFunction h f) 0
hg : AnalyticAt ℂ (cuspFunction h g) 0
⊢ qExpansion h (f + g) = qExpansion h f + qExpansion h g

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
f : F
C : ℝ
τ : ℍ
k : ℕ
inst✝ : ModularFormClass F Γ ↑k
hC : ‖petersson (↑k) (⇑f) (⇑f) τ‖ ≤ C * max τ.im (1 / τ.im) ^ ↑k
⊢ 0 < max τ.im (1 / τ.im) ^ ↑k

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℍ → ℂ
hf : AnalyticAt ℂ (cuspFunction h f) 0
hg : AnalyticAt ℂ (cuspFunction h g) 0
⊢ AnalyticAt ℂ (cuspFunction h (-g)) 0

Tactic:
by

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

Goal state before:
pp: h : ℝ
f g : ℍ → ℂ
hf : AnalyticAt ℂ (cuspFunction h f) 0
hg : AnalyticAt ℂ (cuspFunction h g) 0
⊢ qExpansion h (f - g) = qExpansion h f - qExpansion h g

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
h₁ : 3 * 3 * c ^ 4 ≤ 4 * (g • z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4
h₂ : (c * z.im) ^ 4 / normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z) ^ 2 ≤ 1
⊢ 9 * c ^ 4 ≤ c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16

Tactic:
by

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

Goal state before:
pp: h : ℝ
this : cuspFunction h 0 = 0
⊢ qExpansion h 0 = 0

Tactic:
by

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

Goal state before:
pp: h : ℝ
⊢ qExpansion h 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
h₁ : 3 * 3 * c ^ 4 ≤ 4 * (g • z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4
h₂ : (c * z.im) ^ 4 / normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z) ^ 2 ≤ 1
⊢ c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16 =
    c ^ 4 * z.im ^ 4 / normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z) ^ 2 * 16

Tactic:
by

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

Goal state before:
pp: k : ℤ
N : ℕ
hk : 3 ≤ k
a : Fin 2 → ZMod N
⊢ TendstoLocallyUniformlyOn (fun s ↦ (fun z ↦ ∑ x ∈ s, eisSummand k (↑x) z) ∘ ↑ofComplex)
    (⇑(eisensteinSeriesSIF a k) ∘ ↑ofComplex) Filter.atTop {z | 0 < z.im}

Tactic:
by

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

Goal state before:
pp: k : ℤ
a : Fin 2 → ℤ
⊢ DifferentiableOn ℂ (fun z ↦ (↑(a 0) * z + ↑(a 1)) ^ (-k)) {z | 0 < z.im}

Tactic:
by

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

Goal state before:
pp: h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
⊢ qExpansion h f = 0 ↔ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
h₁ : 3 * 3 * c ^ 4 ≤ 4 * (g • z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4
h₂ : (c * z.im) ^ 4 / normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z) ^ 2 ≤ 1
⊢ c ^ 4 * z.im ^ 4 / normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z) ^ 2 * 16 ≤ 16

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
⊢ |↑g 1 0| ≤ 1

Tactic:
by

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

Goal state before:
pp: h : ℝ
m : ℕ
⊢ cuspFunction h 1 = 1

Tactic:
by

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

Goal state before:
pp: h : ℝ
m : ℕ
h1 : cuspFunction h 1 = 1
⊢ iteratedDeriv m 1 0 = if m = 0 then 1 else 0

Tactic:
by

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

Goal state before:
pp: h : ℝ
⊢ qExpansion h 1 = 1

Tactic:
by

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

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
k : ℤ
f : ModularForm Γ k
⊢ qExpansion h ⇑f = 0 ↔ f = 0

Tactic:
by

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

Goal state before:
pp: k : ℤ
a : Fin 2 → ℤ
⊢ DifferentiableOn ℂ (eisSummand k a ∘ ↑ofComplex) {z | 0 < z.im}

Tactic:
by

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

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
inst✝ : Γ.HasDetPlusMinusOne
⊢ qExpansion h ⇑1 = 1

Tactic:
by

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

Goal state before:
pp: k : ℤ
N : ℕ
hk : 3 ≤ k
a : Fin 2 → ZMod N
τ : ℍ
this : DifferentiableAt ℂ (⇑(eisensteinSeriesSIF a k) ∘ ↑ofComplex) ↑τ
⊢ MDiffAt ⇑(eisensteinSeriesSIF a k) τ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
f : F
C : ℝ
τ : ℍ
k : ℕ
inst✝ : ModularFormClass F Γ ↑k
hC : ‖f τ‖ ^ 2 ≤ C * (max τ.im (1 / τ.im) ^ ↑k / ‖↑τ.im ^ ↑k‖)
hC' : 0 ≤ C
h : 0 < ‖↑τ.im ^ ↑k‖
⊢ 0 ≤ ‖f τ‖

Tactic:
by

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

Goal state before:
pp: k : ℤ
N : ℕ
hk : 3 ≤ k
a : Fin 2 → ZMod N
⊢ MDiff ⇑(eisensteinSeriesSIF a k)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
this :
  ∀ {g : SL(2, ℤ)} {z : ℍ},
    z ∈ 𝒟 →
      g • z ∈ 𝒟 →
        ↑g 1 0 = 0 →
          0 ≤ ↑g 1 1 → (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1
hd : ¬0 ≤ ↑g 1 1
⊢ ↑(-g) 1 0 = 0

Tactic:
by

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

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
inst✝ : Γ.HasDetPlusMinusOne
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
f : ModularForm Γ k
n : ℕ
⊢ qExpansion h ⇑(f.pow n) = qExpansion h ⇑f ^ n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
f : F
C : ℝ
τ : ℍ
k : ℕ
inst✝ : ModularFormClass F Γ ↑k
hC : ‖f τ‖ ^ 2 ≤ C * (max τ.im (1 / τ.im) ^ ↑k / ‖↑τ.im ^ ↑k‖)
hC' : 0 ≤ C
h : 0 < ‖↑τ.im ^ ↑k‖
⊢ 0 ≤ √C * max 1 (1 / τ.im ^ ↑k)

Tactic:
by

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

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝ : Γ.HasDetPlusMinusOne
hΓ : ∃ h ∈ Γ.strictPeriods, 0 < h
a b : ℤ
f : ModularForm Γ a
g : ModularForm Γ b
hf : f ≠ 0
hg : g ≠ 0
⊢ f.mul g ≠ 0

Tactic:
by

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

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
inst✝ : Γ.HasDetPlusMinusOne
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
a b : ℤ
f : ModularForm Γ a
g : ModularForm Γ b
⊢ qExpansion h ⇑(((DirectSum.of (ModularForm Γ) a) f * (DirectSum.of (ModularForm Γ) b) g) (a + b)) =
    qExpansion h ⇑f * qExpansion h ⇑g

Tactic:
by

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

Goal state before:
pp: k : ℤ
N : ℕ
inst✝ : NeZero N
hk : 3 ≤ k
a : Fin 2 → ZMod N
c : OnePoint ℝ
hc : IsCusp c (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) Γ(N))
⊢ c.IsBoundedAt (⇑(eisensteinSeriesSIF a k)) k

Tactic:
by

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

Goal state before:
pp: ⊢ 3 ≤ 4

Tactic:
by

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

Goal state before:
pp: ⊢ 3 ≤ 6

Tactic:
by

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

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
inst✝ : Γ.HasDetPlusMinusOne
hh : 0 < h
hΓ : h ∈ Γ.strictPeriods
f : ModularForm Γ k
n : ℕ
⊢ qExpansion h ⇑(((DirectSum.of (ModularForm Γ) k) f ^ n) (↑n * k)) = qExpansion h ⇑f ^ n

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
c : ℕ → ℂ
hh : 0 < h
hfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0
hf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)
⊢ HasFPowerSeriesOnBall (cuspFunction h ⇑f) (qExpansionFormalMultilinearSeries h f) 0 1

Tactic:
have h1 := (hasFPowerSeriesOnBall_cuspFunction hh hfanalytic hf).hasFPowerSeriesAt

Goal state after:
F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
c : ℕ → ℂ
hh : 0 < h
hfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0
hf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)
h1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0
⊢ HasFPowerSeriesOnBall (cuspFunction h ⇑f) (qExpansionFormalMultilinearSeries h f) 0 1
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
c : ℕ → ℂ
hh : 0 < h
hfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0
hf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)
h1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0
⊢ HasFPowerSeriesAt (cuspFunction h ⇑f) (qExpansionFormalMultilinearSeries h f) 0

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
c : ℕ → ℂ
hh : 0 < h
hfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0
hf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)
⊢ HasFPowerSeriesOnBall (cuspFunction h ⇑f) (qExpansionFormalMultilinearSeries h f) 0 1

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
c : ℕ → ℂ
hh : 0 < h
hfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0
hf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)
m : ℕ
⊢ c m = (PowerSeries.coeff m) (qExpansion h ⇑f)

Tactic:
have h1 := (hasFPowerSeriesOnBall_cuspFunction hh hfanalytic hf).hasFPowerSeriesAt

Goal state after:
F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
c : ℕ → ℂ
hh : 0 < h
hfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0
hf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)
m : ℕ
h1 : HasFPowerSeriesAt (cuspFunction h ⇑f) (FormalMultilinearSeries.ofScalars ℂ c) 0
⊢ c m = (PowerSeries.coeff m) (qExpansion h ⇑f)
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
c : ℕ → ℂ
hh : 0 < h
hfanalytic : AnalyticAt ℂ (cuspFunction h ⇑f) 0
hf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)
m : ℕ
⊢ c m = (PowerSeries.coeff m) (qExpansion h ⇑f)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd : 0 ≤ ↑g 1 1
⊢ ↑g 1 1 = 1 ∨ ↑g 1 1 = -1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd : 0 ≤ ↑g 1 1
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
⊢ ↑g 1 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
⊢ ↑g 0 0 = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
f : F
C : ℝ
τ : ℍ
k : ℕ
inst✝ : ModularFormClass F Γ ↑k
hC : ‖f τ‖ ^ 2 ≤ C * (max τ.im (1 / τ.im) ^ ↑k / ‖↑τ.im ^ ↑k‖)
hC' : 0 ≤ C
h✝ : 0 < ‖↑τ.im ^ ↑k‖
t : NNReal
h : τ.im = ↑t
ht : 0 < t
⊢ 0 ≤ ‖↑(t ^ k)‖₊

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
hk : 0 ≤ k
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
inst✝ : ModularFormClass F Γ k
f : F
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C * max 1 (1 / τ.im ^ k)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
⊢ g = T ^ b

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
hgz : g = T ^ b
⊢ (g • z).re = ↑b + z.re

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
hgz : g = T ^ b
hre : (g • z).re = ↑b + z.re
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

Tactic:
have := (abs_sub_abs_le_abs_add ..).trans (hre ▸ hg.2)

Goal state after:
g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1
hd : ↑g 1 1 = 1
ha : ↑g 0 0 = 1
b : ℤ := ↑g 0 1
hgz : g = T ^ b
hre : (g • z).re = ↑b + z.re
this : |↑b| - |z.re| ≤ 1 / 2
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
inst✝ : ModularFormClass F Γ k
f : F
e : ℝ
hF : ⇑f =O[comap UpperHalfPlane.im (𝓝 0)] fun τ ↦ τ.im ^ (-e)
h : ℝ := Γ.strictWidthInfty
hh : 0 < h
this : NeZero h
hΓ : h ∈ Γ.strictPeriods
C : ℝ
Cpos : C > 0
hC : ∀ᶠ (b : ℝ) in 𝓝 0, ∀ (a : ℍ), a.im = b → ‖f a‖ ≤ C * ‖a.im ^ (-e)‖
n : ℕ
hn : 0 < n
hn' : ∀ (a : ℍ), a.im = ((fun r ↦ r⁻¹) ∘ Nat.cast) n → ‖f a‖ ≤ C * ‖a.im ^ (-e)‖
⊢ 0 < 1 / ↑n

Tactic:
by

Goal state after: