mathlib-initiative/mathlib-tactics huggingface dataset:
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 ≤ h

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✝ : 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)‖
x : ℝ
⊢ 0 < (↑x + 1 / ↑n * I).im

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✝ : 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)‖
F : ℝ → ℝ := fun x ↦ ‖1 / ↑h * (1 / 𝕢 h (↑x + 1 / ↑n * I) ^ n * f { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ })‖
⊢ 0 ≤ ↑n ^ e

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: n : ℕ+
x : ↥(↑n).divisorsAntidiagonal
⊢ ↑(divisorsAntidiagonalFactors n x).1 * ↑(divisorsAntidiagonalFactors n x).2 = ↑n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: x : ↥(divisorsAntidiagonal 1)
⊢ divisorsAntidiagonalFactors 1 x = (1, 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: x : ℕ+ × ℕ+
⊢ (↑x.1, ↑x.2) ∈ (↑⟨↑x.1 * ↑x.2, ⋯⟩).divisorsAntidiagonal

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: ⊢ Function.LeftInverse (fun x ↦ ⟨⟨↑x.1 * ↑x.2, ⋯⟩, ⟨(↑x.1, ↑x.2), ⋯⟩⟩) fun x ↦ divisorsAntidiagonalFactors x.fst x.snd

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✝ : 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)‖
F : ℝ → ℝ := fun x ↦ ‖1 / ↑h * (1 / 𝕢 h (↑x + 1 / ↑n * I) ^ n * f { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ })‖
hne : ‖↑n ^ e‖ = ↑n ^ e
x : ℝ
⊢ { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ }.im = ((fun r ↦ r⁻¹) ∘ Nat.cast) n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Summable fun n ↦ ‖↑n ^ k * r ^ n‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ 1 - 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
hc : ↑g 1 0 = 0
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : CompleteSpace 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Summable fun n ↦ ↑n ^ k * r ^ n / (1 - r ^ n)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
⊢ ‖↑z + ↑(↑g 1 1)‖ ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
a : (x : ℕ+) × ↥(↑x).divisorsAntidiagonal
⊢ 0 ≤ ‖↑(↑a.snd).2 ^ k * r ^ ((↑a.snd).1 * (↑a.snd).2)‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
x✝¹ : ↥(↑b).divisorsAntidiagonal
x✝ : x✝¹ ∈ (↑b).divisorsAntidiagonal.attach
⊢ 0 ≤ ‖↑(↑x✝¹).2‖ ^ k

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
x✝¹ : ↥(↑b).divisorsAntidiagonal
x✝ : x✝¹ ∈ (↑b).divisorsAntidiagonal.attach
⊢ 0 ≤ ‖r‖ ^ ((↑x✝¹).1 * (↑x✝¹).2)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Summable fun c ↦ ‖↑↑c ^ (k + 1) * r ^ ↑c‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
⊢ |↑g 1 1| ≤ 1

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
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
⊢ ↑g 0 1 = -1

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
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
⊢ ‖↑z‖ ≤ 1

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✝ : 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)‖
F : ℝ → ℝ := fun x ↦ ‖1 / ↑h * (1 / 𝕢 h (↑x + 1 / ↑n * I) ^ n * f { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ })‖
hne : ‖↑n ^ e‖ = ↑n ^ e
x : ℝ
⊢ F x ≤ 1 / h * (1 / Real.exp (-2 * Real.pi / h)) * (C * ↑n ^ e)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Summable fun c ↦ ↑(↑c.snd).2 ^ k * r ^ ((↑c.snd).1 * (↑c.snd).2)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Summable fun c ↦ ↑↑c.2 ^ k * r ^ (↑c.1 * ↑c.2)

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✝ : 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)‖
F : ℝ → ℝ := fun x ↦ ‖1 / ↑h * (1 / 𝕢 h (↑x + 1 / ↑n * I) ^ n * f { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ })‖
hne : ‖↑n ^ e‖ = ↑n ^ e
this : ∀ (x : ℝ), F x ≤ 1 / h * (1 / Real.exp (-2 * Real.pi / h)) * (C * ↑n ^ e)
⊢ 0 ≤ h

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
this : ∑' (c : ℕ+ × ℕ+), ↑↑c.2 ^ k * r ^ (↑c.1 * ↑c.2) = ∑' (e : ℕ+), ↑((σ k) ↑e) * r ^ ↑e
⊢ ∑' (d : ℕ+) (c : ℕ+), ↑↑c ^ k * r ^ (↑d * ↑c) = ∑' (e : ℕ+), ↑((σ k) ↑e) * r ^ ↑e

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
this : ∑' (c : ℕ+ × ℕ+), ↑↑c.2 ^ k * r ^ (↑c.1 * ↑c.2) = ∑' (e : ℕ+), ↑((σ k) ↑e) * r ^ ↑e
⊢ ∑' (d : ℕ+) (c : ℕ+), ↑↑c ^ k * r ^ (↑d * ↑c) = ∑' (e : ℕ+), ↑((σ k) ↑e) * r ^ ↑e

Tactic:
by rwa [← (summable_prod_mul_pow k hr).tsum_prod]

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
n : ℕ+
i : ℕ
hi : i ∈ (↑n).divisors
⊢ ↑(↑n / i) ^ k * r ^ (i * (↑n / i)) = ↑(↑n / i) ^ k * r ^ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ ∑' (d : ℕ+) (c : ℕ+), ↑↑c ^ k * r ^ (↑d * ↑c) = ∑' (e : ℕ+), ↑((σ k) ↑e) * r ^ ↑e

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k m : ℕ
inst✝ : NeZero m
⊢ 0 ≤ ‖r‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : CompleteSpace 𝕜
inst✝¹ : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k m : ℕ
inst✝ : NeZero m
⊢ ‖r ^ m‖ < 1

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
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
hz' : ‖↑z‖ = 1
⊢ g = T ^ ↑g 0 0 * S

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
this : ∀ (m : ℕ) [NeZero m], ∑' (n : ℕ), (r ^ m) ^ n = (1 - r ^ m)⁻¹
h00 : ∑' (d : ℕ+) (c : ℕ+), ↑↑c ^ k * r ^ (↑d * ↑c) = ∑' (e : ℕ+), ↑((σ k) ↑e) * r ^ ↑e
⊢ Summable (Function.uncurry fun c d ↦ ↑↑c ^ k * r ^ (↑d * ↑c))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
this : ∀ (m : ℕ) [NeZero m], ∑' (n : ℕ), (r ^ m) ^ n = (1 - r ^ m)⁻¹
h00 : ∑' (b : ℕ+) (c : ℕ+), ↑↑b ^ k * r ^ (↑c * ↑b) = ∑' (e : ℕ+), ↑((σ k) ↑e) * r ^ ↑e
b c : ℕ+
⊢ ↑↑b ^ k * (r ^ ↑b) ^ ↑c = ↑↑b ^ k * r ^ (↑c * ↑b)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
⊢ ∑' (n : ℕ+), ↑↑n ^ k * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((σ k) ↑n) * r ^ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
m : ℕ+
⊢ ‖r ^ ↑m‖ < 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ ↑m‖ < 1
m : ℕ+
⊢ ∑' (n : ℕ+), r ^ (↑n * ↑m) = (1 - r ^ ↑m)⁻¹ - 1

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✝ : ModularFormClass F Γ k
f : F
e : ℝ
hF : ⇑f =O[comap UpperHalfPlane.im (𝓝 0)] fun τ ↦ τ.im ^ (-e)
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion Γ.strictWidthInfty ⇑f)) =O[atTop] fun n ↦ ↑n ^ e

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ ↑m‖ < 1
h2 : ∀ (m : ℕ+), ∑' (n : ℕ+), r ^ (↑n * ↑m) = (1 - r ^ ↑m)⁻¹ - 1
⊢ 𝓝 0 = 𝓝 ((1 - 0)⁻¹ - 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ ↑m‖ < 1
h2 : ∀ (m : ℕ+), ∑' (n : ℕ+), r ^ (↑n * ↑m) = (1 - r ^ ↑m)⁻¹ - 1
⊢ 1 - 0 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.TsumDivisorsAntidiagonal

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
⊢ Tendsto (fun m ↦ ∑' (n : ℕ+), r ^ (↑n * ↑m)) atTop (𝓝 0)

Tactic:
by

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

Goal state before:
pp: k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝ : Γ.HasDetOne
f : CuspForm Γ k
⊢ (CuspForm.toModularFormₗ f).IsCuspForm

Tactic:
by

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

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
k : ℤ
F : Type u_1
inst✝⁴ : FunLike F ℍ ℂ
inst✝³ : DiscreteTopology ↥Γ
inst✝² : Γ.HasDetPlusMinusOne
inst✝¹ : Fact (IsCusp ∞ Γ)
inst✝ : ModularFormClass F Γ k
f : F
h : valueAtInfty ⇑f = 0
⊢ IsZeroAtImInfty ⇑f

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 : ℝ
hC : ∀ (τ : ℍ), ‖f τ‖ ≤ C * max 1 (1 / τ.im ^ k)
τ : ℍ
hτ : τ.im ≤ 1
⊢ 0 ≤ (τ.im ^ k)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : T ^ ↑g 0 0 • S • z ∈ 𝒟
hg'✝ : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = 0
hb : ↑g 0 1 = -1
hz' : ‖↑z‖ = 1
hg' : g = T ^ ↑g 0 0 * S
⊢ (S • z).re = -z.re

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 : ℝ
hC : ∀ (τ : ℍ), ‖f τ‖ ≤ C * max 1 (1 / τ.im ^ k)
τ : ℍ
hτ : τ.im ≤ 1
⊢ 0 < τ.im ^ 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
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion Γ.strictWidthInfty ⇑f)) =O[atTop] fun n ↦ ↑n ^ k

Tactic:
by

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

Goal state before:
pp: k : ℤ
f : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k
h : (PowerSeries.coeff 0) (qExpansion 1 ⇑f) = 0
c : OnePoint ℝ
hc : IsCusp c (Matrix.SpecialLinearGroup.mapGL ℝ).range
γ : SL(2, ℤ)
a✝ : (Matrix.SpecialLinearGroup.mapGL ℝ) γ • ∞ = c
⊢ valueAtInfty ⇑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✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
inst✝ : CuspFormClass F Γ k
f : F
C : ℝ
hC : ∀ (τ : ℍ), ‖f τ‖ ≤ C / τ.im ^ (↑k / 2)
τ : ℍ
⊢ 0 ≤ τ.im ^ (-(↑k / 2))

Tactic:
by

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

Goal state before:
pp: k : ℤ
f : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k
h : (PowerSeries.coeff 0) (qExpansion 1 ⇑f) = 0
c : OnePoint ℝ
hc : IsCusp c (Matrix.SpecialLinearGroup.mapGL ℝ).range
⊢ c.IsZeroAt (⇑f) 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
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion Γ.strictWidthInfty ⇑f)) =O[atTop] fun n ↦ ↑n ^ (↑k / 2)

Tactic:
by

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

Goal state before:
pp: k : ℤ
f : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k
⊢ f.IsCuspForm ↔ (PowerSeries.coeff 0) (qExpansion 1 ⇑f) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Interval

Goal state before:
pp: α : Type u_1
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
⊢ Tendsto (fun p ↦ Icc p.1 p.2) (atBot ×ˢ atTop) atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Interval

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : NoBotOrder α
⊢ Tendsto (fun p ↦ Ioc p.1 p.2) (atBot ×ˢ atTop) atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Interval

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : NoTopOrder α
⊢ Tendsto (fun p ↦ Ico p.1 p.2) (atBot ×ˢ atTop) atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Interval

Goal state before:
pp: α : Type u_1
inst✝³ : Preorder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoBotOrder α
inst✝ : NoTopOrder α
⊢ Tendsto (fun p ↦ Ioo p.1 p.2) (atBot ×ˢ atTop) atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: G : Type u_1
inst✝³ : Neg G
inst✝² : Preorder G
inst✝¹ : LocallyFiniteOrder G
inst✝ : atTop.NeBot
⊢ (symmetricIcc G).filter.NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: G : Type u_1
inst✝³ : Neg G
inst✝² : Preorder G
inst✝¹ : LocallyFiniteOrder G
inst✝ : atTop.NeBot
⊢ (symmetricIco G).filter.NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: G : Type u_1
inst✝³ : Neg G
inst✝² : Preorder G
inst✝¹ : LocallyFiniteOrder G
inst✝ : atTop.NeBot
⊢ (symmetricIoc G).filter.NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: G : Type u_1
inst✝³ : Neg G
inst✝² : Preorder G
inst✝¹ : LocallyFiniteOrder G
inst✝ : atTop.NeBot
⊢ (symmetricIoo G).filter.NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: G✝ : Type u_1
inst✝⁹ : Neg G✝
inst✝⁸ : Preorder G✝
inst✝⁷ : LocallyFiniteOrder G✝
inst✝⁶ : atTop.NeBot
G : Type u_2
inst✝⁵ : AddCommGroup G
inst✝⁴ : PartialOrder G
inst✝³ : IsOrderedAddMonoid G
inst✝² : LocallyFiniteOrder G
inst✝¹ : NoTopOrder G
inst✝ : NoBotOrder G
⊢ (symmetricIco G).filter ≤ atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: G✝ : Type u_1
inst✝⁹ : Neg G✝
inst✝⁸ : Preorder G✝
inst✝⁷ : LocallyFiniteOrder G✝
inst✝⁶ : atTop.NeBot
G : Type u_2
inst✝⁵ : AddCommGroup G
inst✝⁴ : PartialOrder G
inst✝³ : IsOrderedAddMonoid G
inst✝² : LocallyFiniteOrder G
inst✝¹ : NoTopOrder G
inst✝ : NoBotOrder G
⊢ (symmetricIoc G).filter ≤ atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: G✝ : Type u_1
inst✝⁹ : Neg G✝
inst✝⁸ : Preorder G✝
inst✝⁷ : LocallyFiniteOrder G✝
inst✝⁶ : atTop.NeBot
G : Type u_2
inst✝⁵ : AddCommGroup G
inst✝⁴ : PartialOrder G
inst✝³ : IsOrderedAddMonoid G
inst✝² : LocallyFiniteOrder G
inst✝¹ : NoTopOrder G
inst✝ : NoBotOrder G
⊢ (symmetricIoo G).filter ≤ atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: ⊢ (symmetricIcc ℤ).filter = Filter.map (fun N ↦ Icc (-↑N) ↑N) atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: s : Set (Finset ℤ)
x✝ : ∃ a, ∀ (b : ℕ), a ≤ b → Icc (-↑b) ↑b ∈ s
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → Icc (-↑b) ↑b ∈ s
b : ℕ
hb : a + 1 ≤ b
⊢ a ≤ b - 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: s : Set (Finset ℤ)
x✝ : ∃ a, ∀ (b : ℕ), a ≤ b → Ioo (-↑b) ↑b ∈ s
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → Ioo (-↑b) ↑b ∈ s
b : ℕ
hb : a - 1 ≤ b
⊢ a ≤ b + 1

Tactic:
by

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

Goal state before:
pp: k : ℤ
f g : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k
hg : (PowerSeries.coeff 0) (qExpansion 1 ⇑g) = 1
⊢ (f - (PowerSeries.coeff 0) (qExpansion 1 ⇑f) • g).IsCuspForm

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: ⊢ symmetricIcc ℤ = symmetricIoo ℤ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs

Goal state before:
pp: m : ℤ
z : ℍ
⊢ 2 ≤ 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs

Goal state before:
pp: m : ℤ
z : ℍ
⊢ Summable fun n ↦ eisSummand 2 ![m, n] z

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs

Goal state before:
pp: z : ℍ
⊢ 2 ≤ 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs

Goal state before:
pp: z : ℍ
⊢ Even 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs

Goal state before:
pp: z : ℍ
⊢ e2Summand 0 z = 2 * riemannZeta 2

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs

Goal state before:
pp: z : ℍ
⊢ Function.Even fun x ↦ e2Summand x z

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs

Goal state before:
pp: ⊢ D2 1 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: α : Type u_1
f : ℤ → α
inst✝² : CommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousMul α
a : α
hf : HasProd f a (symmetricIcc ℤ)
hf2 : Tendsto (fun N ↦ (f ↑N)⁻¹) atTop (𝓝 1)
⊢ HasProd f a (symmetricIco ℤ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f : ℤ → α
a : α
⊢ HasProd f a (symmetricIcc ℤ) ↔ Tendsto (fun N ↦ ∏ n ∈ Icc (-↑N) ↑N, f n) atTop (𝓝 a)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f : ℤ → α
a : α
⊢ HasProd f a (symmetricIco ℤ) ↔ Tendsto (fun N ↦ ∏ n ∈ Ico (-↑N) ↑N, f n) atTop (𝓝 a)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f : ℤ → α
a : α
⊢ HasProd f a (symmetricIoc ℤ) ↔ Tendsto (fun N ↦ ∏ n ∈ Ioc (-↑N) ↑N, f n) atTop (𝓝 a)

Tactic:
by

Goal state after: