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

Goal state before:
pp: case refine_1
r : ℕ
inst✝ : NeZero r
⊢ MapsTo (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)

pp: case refine_2
r : ℕ
inst✝ : NeZero r
⊢ InjOn (divIntMap ↑r) (gammaSet 1 r 0)

pp: case refine_3
r : ℕ
inst✝ : NeZero r
⊢ SurjOn (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)

Tactic:
·

Goal state after:
case refine_1
r : ℕ
inst✝ : NeZero r
⊢ MapsTo (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inr
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = -1
hd : ↑g 1 1 = -1
⊢ ∃ n, ∀ (z : ℍ), g • z = T ^ n • z

Tactic:
·

Goal state after:
case inr
g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : ↑g 0 0 * ↑g 1 1 = 1
ha : ↑g 0 0 = -1
hd : ↑g 1 1 = -1
⊢ ∃ n, ∀ (z : ℍ), g • z = T ^ n • z
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: case refine_2
r : ℕ
inst✝ : NeZero r
⊢ InjOn (divIntMap ↑r) (gammaSet 1 r 0)

pp: case refine_3
r : ℕ
inst✝ : NeZero r
⊢ SurjOn (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)

Tactic:
·

Goal state after:
case refine_2
r : ℕ
inst✝ : NeZero r
⊢ InjOn (divIntMap ↑r) (gammaSet 1 r 0)
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: case refine_3
r : ℕ
inst✝ : NeZero r
⊢ SurjOn (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)

Tactic:
·

Goal state after:
case refine_3
r : ℕ
inst✝ : NeZero r
⊢ SurjOn (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: case neg
E : Type u_1
inst✝ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t
hf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ
F : ℝ
τ : ℍ
g : SL(2, ℤ)
hF𝒟 : ‖f τ‖ ≤ F * (g • τ).im ^ t
hg✝ : g • τ ∈ 𝒟
hg : ¬↑g 1 0 = 0
⊢ τ.im / Complex.normSq (↑(↑g 1 0) * ↑τ + ↑(↑g 1 1)) ≤ max τ.im (1 / τ.im)

Tactic:
·

Goal state after:
case neg
E : Type u_1
inst✝ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t
hf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ
F : ℝ
τ : ℍ
g : SL(2, ℤ)
hF𝒟 : ‖f τ‖ ≤ F * (g • τ).im ^ t
hg✝ : g • τ ∈ 𝒟
hg : ¬↑g 1 0 = 0
⊢ τ.im / Complex.normSq (↑(↑g 1 0) * ↑τ + ↑(↑g 1 1)) ≤ max τ.im (1 / τ.im)
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: case h.left
r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : x ∈ gammaSet 1 1 0
⊢ ↑r * x ∈ gammaSet 1 r 0

pp: case h.right
r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : x ∈ gammaSet 1 1 0
⊢ ↑r * x / ↑r = x

Tactic:
·

Goal state after:
case h.left
r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : x ∈ gammaSet 1 1 0
⊢ ↑r * x ∈ gammaSet 1 r 0
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: case h.right
r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : x ∈ gammaSet 1 1 0
⊢ ↑r * x / ↑r = x

Tactic:
·

Goal state after:
case h.right
r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : x ∈ gammaSet 1 1 0
⊢ ↑r * x / ↑r = x
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: r : ℕ
inst✝ : NeZero r
⊢ BijOn (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)

Tactic:
by
  refine ⟨?_, ?_, ?_⟩
  · intro x hx
    simp only [divIntMap, mem_gammaSet_one] at *
    exact finGcdMap_div _ hx.2
  · intro x hx v hv hv2
    ext i
    exact (Int.ediv_left_inj (gammaSet_div_gcd hx i) (gammaSet_div_gcd hv i)).mp
      (congr_fun hv2 i)
  · intro x hx
    use r • x
    simp only [nsmul_eq_mul, divIntMap, Int.cast_natCast]
    constructor
    · rw [mem_gammaSet_one, Int.isCoprime_iff_gcd_eq_one] at hx
      exact ⟨Subsingleton.eq_zero _, by simp [Int.gcd_mul_left, hx]⟩
    · ext i
      simp_all [NeZero.ne r]

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

Goal state before:
pp: case pos
r : ℕ
v : Fin 2 → ℤ
hv : v ∈ gammaSet 1 r 0
hr : r = 0
⊢ v = r • divIntMap (↑r) v

pp: case neg
r : ℕ
v : Fin 2 → ℤ
hv : v ∈ gammaSet 1 r 0
hr : ¬r = 0
⊢ v = r • divIntMap (↑r) v

Tactic:
·

Goal state after:
case pos
r : ℕ
v : Fin 2 → ℤ
hv : v ∈ gammaSet 1 r 0
hr : r = 0
⊢ v = r • divIntMap (↑r) v
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: case neg
r : ℕ
v : Fin 2 → ℤ
hv : v ∈ gammaSet 1 r 0
hr : ¬r = 0
⊢ v = r • divIntMap (↑r) v

Tactic:
·

Goal state after:
case neg
r : ℕ
v : Fin 2 → ℤ
hv : v ∈ gammaSet 1 r 0
hr : ¬r = 0
⊢ v = r • divIntMap (↑r) v
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case left
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ 1 ≤ normSq ↑(g • z)

pp: case right
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ |(g • z).re| ≤ 1 / 2

Tactic:
·

Goal state after:
case left
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ 1 ≤ normSq ↑(g • z)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case right
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ |(g • z).re| ≤ 1 / 2

Tactic:
·

Goal state after:
case right
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ |(g • z).re| ≤ 1 / 2
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case right.left
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ -(1 / 2) ≤ (g • z).re

pp: case right.right
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ (g • z).re ≤ 1 / 2

Tactic:
·

Goal state after:
case right.left
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ -(1 / 2) ≤ (g • z).re
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case right.right
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ (g • z).re ≤ 1 / 2

Tactic:
·

Goal state after:
case right.right
z : ℍ
g₀ : SL(2, ℤ)
hg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im
g : SL(2, ℤ)
hg : ↑g 1 = ↑g₀ 1
hg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
⊢ (g • z).re ≤ 1 / 2
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
hk' : 2 < ↑k
⊢ Summable fun i ↦ r { coe := { re := ↑N, im := 2 }, coe_im_pos := ⋯ } ^ (-k) * ‖i‖ ^ (-k)

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
hk' : 2 < ↑k
⊢ ∀ (i : Fin 2 → ℤ),
    ‖eisSummand k i (ModularGroup.T ^ (↑N * n) • z)‖ ≤
      r { coe := { re := ↑N, im := 2 }, coe_im_pos := ⋯ } ^ (-k) * ‖i‖ ^ (-k)

Tactic:
·

Goal state after:
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
hk' : 2 < ↑k
⊢ Summable fun i ↦ r { coe := { re := ↑N, im := 2 }, coe_im_pos := ⋯ } ^ (-k) * ‖i‖ ^ (-k)
Module: Mathlib.NumberTheory.ModularForms.QExpansion

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

pp: case neg
h : ℝ
f g : ℍ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
z : ℂ
H : ¬z = 0
⊢ cuspFunction h (f * g) z = (cuspFunction h f * cuspFunction h g) z

Tactic:
·

Goal state after:
case pos
h : ℝ
f g : ℍ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
z : ℂ
H : z = 0
⊢ cuspFunction h (f * g) z = (cuspFunction h f * cuspFunction h g) z
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
hk' : 2 < ↑k
⊢ ∀ (i : Fin 2 → ℤ),
    ‖eisSummand k i (ModularGroup.T ^ (↑N * n) • z)‖ ≤
      r { coe := { re := ↑N, im := 2 }, coe_im_pos := ⋯ } ^ (-k) * ‖i‖ ^ (-k)

Tactic:
·

Goal state after:
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
hk' : 2 < ↑k
⊢ ∀ (i : Fin 2 → ℤ),
    ‖eisSummand k i (ModularGroup.T ^ (↑N * n) • z)‖ ≤
      r { coe := { re := ↑N, im := 2 }, coe_im_pos := ⋯ } ^ (-k) * ‖i‖ ^ (-k)
Module: Mathlib.NumberTheory.ModularForms.QExpansion

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

Tactic:
·

Goal state after:
case neg
h : ℝ
f g : ℍ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
z : ℂ
H : ¬z = 0
⊢ cuspFunction h (f * g) z = (cuspFunction h f * cuspFunction h g) z
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inl
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
hc : c = 0
⊢ 3 * c ^ 2 ≤ 4

pp: case inr
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
hc : c ≠ 0
⊢ 3 * c ^ 2 ≤ 4

Tactic:
·

Goal state after:
case inl
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
hc : c = 0
⊢ 3 * c ^ 2 ≤ 4
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inr
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
hc : c ≠ 0
⊢ 3 * c ^ 2 ≤ 4

Tactic:
·

Goal state after:
case inr
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
hc : c ≠ 0
⊢ 3 * c ^ 2 ≤ 4
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
⊢ ‖f τ‖ ^ 2 ≤ ‖(starRingEnd ℂ) (f τ) * f τ * ↑τ.im ^ k‖ / τ.im ^ (↑k / 2 * ↑2)

Tactic:
simp [abs_of_pos τ.im_pos, field]

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 n z ↦ ∑ x ∈ n, eisSummand k (↑x) z) (⇑(eisensteinSeriesSIF a k)) Filter.atTop ⊤

pp: k : ℤ
N : ℕ
hk : 3 ≤ k
a : Fin 2 → ZMod N
⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph UpperHalfPlane.coe isOpenEmbedding_coe).symm)
    (IsOpenEmbedding.toOpenPartialHomeomorph UpperHalfPlane.coe isOpenEmbedding_coe).target ⊤

Tactic:
·

Goal state after:
k : ℤ
N : ℕ
hk : 3 ≤ k
a : Fin 2 → ZMod N
⊢ TendstoLocallyUniformlyOn (fun n z ↦ ∑ x ∈ n, eisSummand k (↑x) z) (⇑(eisensteinSeriesSIF a k)) Filter.atTop ⊤
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence

Goal state before:
pp: k : ℤ
N : ℕ
hk : 3 ≤ k
a : Fin 2 → ZMod N
⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph UpperHalfPlane.coe isOpenEmbedding_coe).symm)
    (IsOpenEmbedding.toOpenPartialHomeomorph UpperHalfPlane.coe isOpenEmbedding_coe).target ⊤

Tactic:
·

Goal state after:
k : ℤ
N : ℕ
hk : 3 ≤ k
a : Fin 2 → ZMod N
⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph UpperHalfPlane.coe isOpenEmbedding_coe).symm)
    (IsOpenEmbedding.toOpenPartialHomeomorph UpperHalfPlane.coe isOpenEmbedding_coe).target ⊤
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case mp
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

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

Tactic:
·

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

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

Tactic:
·

Goal state after:
case mpr
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
⊢ f = 0 → qExpansion h f = 0
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.MDifferentiable

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

pp: case inr
k : ℤ
⊢ DifferentiableOn ℂ (fun z ↦ (↑(0 0) * z + ↑(0 1)) ^ (-k)) {z | 0 < z.im}

Tactic:
·

Goal state after:
case inl
k : ℤ
a : Fin 2 → ℤ
ha : a ≠ 0
⊢ DifferentiableOn ℂ (fun z ↦ (↑(a 0) * z + ↑(a 1)) ^ (-k)) {z | 0 < z.im}
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.MDifferentiable

Goal state before:
pp: case inl.hf
k : ℤ
a : Fin 2 → ℤ
ha : a ≠ 0
⊢ DifferentiableOn ℂ (fun x ↦ ↑(a 0) * x + ↑(a 1)) {z | 0 < z.im}

pp: case h
k : ℤ
a : Fin 2 → ℤ
ha : a ≠ 0
⊢ (∀ x ∈ {z | 0 < z.im}, ↑(a 0) * x + ↑(a 1) ≠ 0) ∨ 0 ≤ -k

Tactic:
·

Goal state after:
case inl.hf
k : ℤ
a : Fin 2 → ℤ
ha : a ≠ 0
⊢ DifferentiableOn ℂ (fun x ↦ ↑(a 0) * x + ↑(a 1)) {z | 0 < z.im}
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.MDifferentiable

Goal state before:
pp: case h
k : ℤ
a : Fin 2 → ℤ
ha : a ≠ 0
⊢ (∀ x ∈ {z | 0 < z.im}, ↑(a 0) * x + ↑(a 1) ≠ 0) ∨ 0 ≤ -k

Tactic:
·

Goal state after:
case h
k : ℤ
a : Fin 2 → ℤ
ha : a ≠ 0
⊢ (∀ x ∈ {z | 0 < z.im}, ↑(a 0) * x + ↑(a 1) ≠ 0) ∨ 0 ≤ -k
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.MDifferentiable

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

Tactic:
·

Goal state after:
case inr
k : ℤ
⊢ DifferentiableOn ℂ (fun z ↦ (↑(0 0) * z + ↑(0 1)) ^ (-k)) {z | 0 < z.im}
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case inl
h : ℝ
m : ℕ
⊢ cuspFunction h 1 0 = 1 0

pp: case inr
h : ℝ
m : ℕ
q : ℂ
hq : q ≠ 0
⊢ cuspFunction h 1 q = 1 q

Tactic:
·

Goal state after:
case inl
h : ℝ
m : ℕ
⊢ cuspFunction h 1 0 = 1 0
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case inr
h : ℝ
m : ℕ
q : ℂ
hq : q ≠ 0
⊢ cuspFunction h 1 q = 1 q

Tactic:
·

Goal state after:
case inr
h : ℝ
m : ℕ
q : ℂ
hq : q ≠ 0
⊢ cuspFunction h 1 q = 1 q
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inr
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 = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

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 = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

Tactic:
·

Goal state after:
case inr
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 = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inr
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
⊢ 0 ≤ ↑(-g) 1 1

pp: case inr
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
this : (-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 = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

Tactic:
·

Goal state after:
case inr
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
⊢ 0 ≤ ↑(-g) 1 1
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case inl
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 ^ 0
hre : (g • z).re = ↑b + z.re
hb : b = 0
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

pp: case inr.inl
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 ^ 1
hre : (g • z).re = ↑b + z.re
hb : b = 1
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

pp: case inr.inr
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 ^ (-1)
hre : (g • z).re = ↑b + z.re
hb : b = -1
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

Tactic:
·

Goal state after:
case inl
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 ^ 0
hre : (g • z).re = ↑b + z.re
hb : b = 0
⊢ (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.Modular

Goal state before:
pp: case inr.inl
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 ^ 1
hre : (g • z).re = ↑b + z.re
hb : b = 1
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

pp: case inr.inr
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 ^ (-1)
hre : (g • z).re = ↑b + z.re
hb : b = -1
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

Tactic:
·

Goal state after:
case inr.inl
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 ^ 1
hre : (g • z).re = ↑b + z.re
hb : b = 1
⊢ (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.Modular

Goal state before:
pp: case inr.inr
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 ^ (-1)
hre : (g • z).re = ↑b + z.re
hb : b = -1
⊢ (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1

Tactic:
·

Goal state after:
case inr.inr
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 ^ (-1)
hre : (g • z).re = ↑b + z.re
hb : b = -1
⊢ (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: case a
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 := ⋯
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 := ⋯
hne : ‖↑n ^ e‖ = ↑n ^ e
x : ℝ
⊢ -2 * Real.pi / h ≤ ↑n * (-2 * Real.pi * (↑x + 1 / ↑n * I).im / h)

pp: case h₂
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 := ⋯
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 := ⋯
hne : ‖↑n ^ e‖ = ↑n ^ e
x : ℝ
⊢ ‖f { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ }‖ ≤ C * ↑n ^ e

Tactic:
·

Goal state after:
case a
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 := ⋯
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 := ⋯
hne : ‖↑n ^ e‖ = ↑n ^ e
x : ℝ
⊢ -2 * Real.pi / h ≤ ↑n * (-2 * Real.pi * (↑x + 1 / ↑n * I).im / h)
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: case h₂
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 := ⋯
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 := ⋯
hne : ‖↑n ^ e‖ = ↑n ^ e
x : ℝ
⊢ ‖f { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ }‖ ≤ C * ↑n ^ e

Tactic:
·

Goal state after:
case h₂
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 := ⋯
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 := ⋯
hne : ‖↑n ^ e‖ = ↑n ^ e
x : ℝ
⊢ ‖f { coe := ↑x + 1 / ↑n * I, coe_im_pos := ⋯ }‖ ≤ C * ↑n ^ e