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

Goal state before:
pp: 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
⊢ 0 < (g • τ).im ^ t

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
⊢ (T • z).im = z.im

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
⊢ (T⁻¹ • z).re = z.re - 1

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
k : ℤ
inst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k
hk : k ≤ 0
f : F
z : ℍ
⊢ 𝕢 1 ↑z ∈ Metric.ball 0 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
⊢ (T⁻¹ • z).im = 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
n : ℕ
R : ℝ
hR : 0 < R
hR' : R < 1
⊢ (PowerSeries.coeff n) (qExpansion h f) =
    (2 * ↑π * Complex.I)⁻¹ * ∮ (z : ℂ) in C(0, R), cuspFunction h f z / z ^ (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
hc : ↑g 1 0 = 0
had : (↑g).det = 1
⊢ ↑g 0 0 * ↑g 1 1 = 1

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
k : ℤ
inst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k
hk : k ≤ 0
f : F
⊢ ⇑f = Function.const ℍ (UpperHalfPlane.cuspFunction 1 (⇑f) 0)

Tactic:
by

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

Goal state before:
pp: k : ℤ
F : Type u_1
inst✝ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f✝ : F
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
n : ℕ
t : ℝ
ht : 0 < t
u : ℝ
⊢ 0 < (↑u + ↑t * ↑I).im

Tactic:
by

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

Goal state before:
pp: 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‖ / (g • τ).im ^ t ≤ F
hg : g • τ ∈ 𝒟
⊢ 0 ≤ ‖f τ‖ / (g • τ).im ^ t

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
n : ℕ
t : ℝ
ht : 0 < t
R : ℝ := rexp (-2 * π * t / h)
hR0 : 0 < R
⊢ 0 < 2 * π * t

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
n : ℕ
t : ℝ
ht : 0 < t
R : ℝ := rexp (-2 * π * t / h)
hR0 : 0 < R
⊢ -2 * π * t / h < 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: 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
this : g = T ^ ↑g 0 1
⊢ ∀ (z : ℍ), g • z = T ^ ↑g 0 1 • z

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
k : ℤ
inst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k
hk : k < 0
f : F
⊢ ⇑f = 0

Tactic:
by

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

Goal state before:
pp: ⊢ const 1 ≠ 0

Tactic:
by

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

Goal state before:
pp: ⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 0) = 1

Tactic:
by

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

Goal state before:
pp: k : ℤ
hk : k < 0
⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k) = 0

Tactic:
by

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

Goal state before:
pp: 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
hg' : ↑g 1 1 = 0
⊢ 0 = 1

Tactic:
by

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

Goal state before:
pp: N r : ℕ
⊢ Pairwise (Disjoint on gammaSet N r)

Tactic:
by

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

Goal state before:
pp: r : ℕ
a : Fin 2 → ZMod 1
⊢ gammaSet 1 r a = {v | (v 0).gcd (v 1) = r}

Tactic:
by

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

Goal state before:
pp: r : ℕ
v : Fin 2 → ℤ
⊢ v ∈ gammaSet 1 r 0 ↔ (v 0).gcd (v 1) = r

Tactic:
by

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

Goal state before:
pp: r : ℕ
inst✝ : NeZero r
v : Fin 2 → ℤ
hv : finGcdMap v = r
⊢ IsCoprime ((v / ↑r) 0) ((v / ↑r) 1)

Tactic:
by

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

Goal state before:
pp: r : ℕ
a : ℤ
v : Fin 2 → ℤ
hv : finGcdMap v = r
⊢ finGcdMap (a • v) = a.natAbs * r

Tactic:
by

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

Goal state before:
pp: v : Fin 2 → ℤ
⊢ v ∈ gammaSet 1 1 0 ↔ IsCoprime (v 0) (v 1)

Tactic:
by

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

Goal state before:
pp: r : ℕ
v : Fin 2 → ℤ
hv : v ∈ gammaSet 1 r 0
i : Fin 2
⊢ ↑r ∣ v i

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: 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
this : g = -T ^ (-↑g 0 1)
⊢ ∀ (z : ℍ), g • z = T ^ (-↑g 0 1) • z

Tactic:
by

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

Goal state before:
pp: case refine_3
r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : x ∈ gammaSet 1 1 0
⊢ x ∈ divIntMap ↑r '' gammaSet 1 r 0

Tactic:
use r • x

Goal state after:
case h
r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : x ∈ gammaSet 1 1 0
⊢ r • x ∈ gammaSet 1 r 0 ∧ divIntMap (↑r) (r • x) = x
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs

Goal state before:
pp: r : ℕ
inst✝ : NeZero r
x : Fin 2 → ℤ
hx : (x 0).gcd (x 1) = 1
⊢ ((↑r * x) 0).gcd ((↑r * x) 1) = r

Tactic:
by

Goal state after:
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

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

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

Tactic:
by

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

Goal state before:
pp: N r : ℕ
a : Fin 2 → ZMod N
⊢ (Fin 2 → ℤ) ≃ (r : ℕ) × ↑(gammaSet 1 r 0)

Tactic:
by

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

Goal state before:
pp: 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
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C * max τ.im (1 / τ.im) ^ t

Tactic:
by

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

Goal state before:
pp: r : ℕ
inst✝ : NeZero r
v : Fin 2 → ℤ
hab : finGcdMap v = r
A : SL(2, ℤ)
⊢ v = r • (v / ↑r)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
hc : ↑g 1 0 = 0
⊢ ∃ n, ∀ (z : ℍ), g • z = T ^ n • z

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
hc : ↑g 1 0 = 1
hg : 1 = (↑g).det
⊢ ↑g 0 1 = ↑g 0 0 * ↑g 1 1 - 1

Tactic:
by

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

Goal state before:
pp: r : ℕ
inst✝ : NeZero r
v : Fin 2 → ℤ
hab : finGcdMap v = r
A : SL(2, ℤ)
⊢ finGcdMap (v ᵥ* ↑A) = r

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
τ : ℍ
g h : SL(2, ℤ)
hgh : g ≈ h
⊢ ∃ j ∈ Γ, h = g * j

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
τ : ℍ
g h : SL(2, ℤ)
hgh : g ≈ h
⊢ f (g⁻¹ • τ) = f (h⁻¹ • τ)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
hc : ↑g 1 0 = 1
⊢ g = T ^ ↑g 0 0 * S * T ^ ↑g 1 1

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
n : ℕ
t : ℝ
ht : 0 < t
R : ℝ := rexp (-2 * π * t / h)
hR0 : 0 < R
hR1 : R < 1
u : ℝ
⊢ 0 < (↑u + ↑t * ↑I).im

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
f' : ℍ → SL(2, ℤ) ⧸ Γ → E := fun τ ↦ Quotient.lift (fun g ↦ f (g⁻¹ • τ)) ⋯
γ : SL(2, ℤ) ⧸ Γ
g : SL(2, ℤ)
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ f' x ↑g

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
h : 1 < normSq ↑z
⊢ normSq ↑(S • z) < 1

Tactic:
by

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

Goal state before:
pp: N r : ℕ
a : Fin 2 → ZMod N
inst✝ : NeZero r
v : Fin 2 → ℤ
hv : v ∈ gammaSet N r a
γ : SL(2, ℤ)
⊢ v ᵥ* ↑γ ∈ gammaSet N r (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ))

Tactic:
by

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

Goal state before:
pp: case e_f
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
n : ℕ
t : ℝ
ht : 0 < t
R : ℝ := rexp (-2 * π * t / h)
hR0 : 0 < R
hR1 : R < 1
u : ℝ
⊢ (2 * ↑π * Complex.I)⁻¹ * ↑(2 * π / h) *
      deriv (circleMap 0 R) (u * (2 * π / h)) •
        (cuspFunction h f (circleMap 0 R (u * (2 * π / h))) / circleMap 0 R (u * (2 * π / h)) ^ (n + 1)) =
    1 / ↑h * (1 / 𝕢 h (↑u + ↑t * ↑I) ^ n * f { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ })

Tactic:
let τ : ℍ := ⟨u + t * I, by simpa using ht⟩

Goal state after:
case e_f
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
n : ℕ
t : ℝ
ht : 0 < t
R : ℝ := rexp (-2 * π * t / h)
hR0 : 0 < R
hR1 : R < 1
u : ℝ
τ : ℍ := { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ }
⊢ (2 * ↑π * Complex.I)⁻¹ * ↑(2 * π / h) *
      deriv (circleMap 0 R) (u * (2 * π / h)) •
        (cuspFunction h f (circleMap 0 R (u * (2 * π / h))) / circleMap 0 R (u * (2 * π / h)) ^ (n + 1)) =
    1 / ↑h * (1 / 𝕢 h (↑u + ↑t * ↑I) ^ n * f { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ })
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
h : normSq ↑z < 1
⊢ z.im < z.im / normSq ↑z

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
h : normSq ↑z < 1
⊢ z.im < (S • z).im

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: ⊢ ρ ∈ 𝒟

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: ⊢ I ∈ 𝒟

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
h : z ∈ 𝒟ᵒ
⊢ |2 * z.re| < 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
h : z ∈ 𝒟ᵒ
⊢ 1 < z.re * z.re + z.im * z.im

Tactic:
by

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

Goal state before:
pp: N r : ℕ
a : Fin 2 → ZMod N
inst✝ : NeZero r
γ : SL(2, ℤ)
v : ↑(gammaSet N r (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)))
⊢ ↑v ᵥ* ↑γ⁻¹ ∈ gammaSet N r a

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
n : ℕ
t : ℝ
ht : 0 < t
R : ℝ := rexp (-2 * π * t / h)
hR0 : 0 < R
hR1 : R < 1
u : ℝ
τ : ℍ := { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ }
⊢ circleMap 0 R (u * (2 * π / h)) = 𝕢 h ↑τ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
f' : ℍ → SL(2, ℤ) ⧸ Γ → E := fun τ ↦ Quotient.lift (fun g ↦ f (g⁻¹ • τ)) ⋯
hf'_cont : ∀ (γ : SL(2, ℤ) ⧸ Γ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ f' x γ
τ : ℍ
g : SL(2, ℤ)
γ : SL(2, ℤ) ⧸ Γ
⊢ f' (g • τ) (g • γ) = f' τ γ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
h : z ∈ 𝒟ᵒ
⊢ 3 < 4 * z.im ^ 2

Tactic:
by

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

Goal state before:
pp: N r : ℕ
a : Fin 2 → ZMod N
inst✝ : NeZero r
γ : SL(2, ℤ)
v : ↑(gammaSet N r a)
⊢ (fun v ↦ ⟨↑v ᵥ* ↑γ⁻¹, ⋯⟩) ((fun v ↦ ⟨↑v ᵥ* ↑γ, ⋯⟩) v) = v

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: τ : ℍ
h : τ ∈ 𝒟
⊢ 1 ≤ τ.re * τ.re + τ.im * τ.im

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: τ : ℍ
h : τ ∈ 𝒟
⊢ 3 ≤ 4 * τ.im ^ 2

Tactic:
by

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

Goal state before:
pp: N r : ℕ
a : Fin 2 → ZMod N
inst✝ : NeZero r
γ : SL(2, ℤ)
v : ↑(gammaSet N r (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)))
⊢ (fun v ↦ ⟨↑v ᵥ* ↑γ, ⋯⟩) ((fun v ↦ ⟨↑v ᵥ* ↑γ⁻¹, ⋯⟩) v) = v

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
hz : z ∈ 𝒟ᵒ
n : ℤ
hz₁ : 1 < z.re * z.re + z.im * z.im
hzn : 0 ≤ ↑n * (2 * z.re) + ↑n * ↑n
⊢ 1 < (z.re + ↑n) * (z.re + ↑n) + z.im * z.im

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
hz : z ∈ 𝒟ᵒ
n : ℤ
⊢ 1 < normSq ↑(T ^ n • z)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
n : ℤ
hz : z ∈ 𝒟ᵒ
hg : T ^ n • z ∈ 𝒟ᵒ
this : |↑n| < 1
⊢ n = 0

Tactic:
by

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

Goal state before:
pp: k : ℤ
i : Fin 2 → ℤ
A : SL(2, ℤ)
z : ℍ
a b c d u v : ℂ
hc : c * ↑z + d ≠ 0
⊢ ↑z * c + d ≠ 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
f' : ℍ → SL(2, ℤ) ⧸ Γ → E := fun τ ↦ Quotient.lift (fun g ↦ f (g⁻¹ • τ)) ⋯
hf'_cont : ∀ (γ : SL(2, ℤ) ⧸ Γ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ f' x γ
hf'_inv : ∀ (τ : ℍ) (g : SL(2, ℤ)) (γ : SL(2, ℤ) ⧸ Γ), f' (g • τ) (g • γ) = f' τ γ
hf'_infty : ∀ (γ : SL(2, ℤ) ⧸ Γ), (fun x ↦ f' x γ) =O[atImInfty] fun z ↦ z.im ^ t
this : Fintype (SL(2, ℤ) ⧸ Γ)
⊢ Continuous fun a ↦ ∑ c, ‖f' a c‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
n : ℤ
hz : z ∈ 𝒟ᵒ
hg : T ^ n • z ∈ 𝒟ᵒ
⊢ n = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
f' : ℍ → SL(2, ℤ) ⧸ Γ → E := fun τ ↦ Quotient.lift (fun g ↦ f (g⁻¹ • τ)) ⋯
hf'_cont : ∀ (γ : SL(2, ℤ) ⧸ Γ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ f' x γ
hf'_inv : ∀ (τ : ℍ) (g : SL(2, ℤ)) (γ : SL(2, ℤ) ⧸ Γ), f' (g • τ) (g • γ) = f' τ γ
hf'_infty : ∀ (γ : SL(2, ℤ) ⧸ Γ), (fun x ↦ f' x γ) =O[atImInfty] fun z ↦ z.im ^ t
this : Fintype (SL(2, ℤ) ⧸ Γ)
g : SL(2, ℤ)
τ : ℍ
⊢ ∀ (x : SL(2, ℤ) ⧸ Γ), ‖f' τ x‖ = ‖f' (g • τ) ((MulAction.toPerm g) x)‖

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: 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|
⊢ (g • z).im = (g₀ • z).im

Tactic:
by

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

Goal state before:
pp: k : ℤ
i : Fin 2 → ℤ
A : SL(2, ℤ)
z : ℍ
a b c d u v : ℂ
hc : c * ↑z + d ≠ 0
⊢ (u * ((a * ↑z + b) / (c * ↑z + d)) + v) ^ (-k) = (c * ↑z + d) ^ k * ((u * a + v * c) * ↑z + (u * b + v * d)) ^ (-k)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: 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|
⊢ ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
t : ℝ
ht : 0 ≤ t
hf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t
Γ : Subgroup SL(2, ℤ)
inst✝ : Γ.FiniteIndex
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C * max τ.im (1 / τ.im) ^ t

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf_infinity : IsBoundedAtImInfty f
hf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ
⊢ f =O[atImInfty] fun z ↦ z.im ^ 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf_infinity : IsBoundedAtImInfty f
hf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf_infinity : ∀ (g : SL(2, ℤ)), IsBoundedAtImInfty fun τ ↦ f (g • τ)
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝ : Γ.IsArithmetic
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
⊢ ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ 0

Tactic:
by

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

Goal state before:
pp: k : ℤ
i : Fin 2 → ℤ
A : SL(2, ℤ)
z : ℍ
⊢ eisSummand k i (A • z) =
    denom (SpecialLinearGroup.toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) A)) ↑z ^ k * eisSummand k (i ᵥ* ↑A) z

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
f : ℍ → E
hf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf_infinity : ∀ (g : SL(2, ℤ)), IsBoundedAtImInfty fun τ ↦ f (g • τ)
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝ : Γ.IsArithmetic
hf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ
⊢ ∃ C, ∀ (τ : ℍ), ‖f τ‖ ≤ C

Tactic:
by

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

Goal state before:
pp: N : ℕ
a : Fin 2 → ZMod N
k : ℤ
γ : SL(2, ℤ)
z : ℍ
⊢ eisensteinSeries a k (γ • z) *
      (denom (SpecialLinearGroup.toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑z ^ k)⁻¹ =
    eisensteinSeries (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)) k z

Tactic:
mul_inv_eq_iff_eq_mul₀ (zpow_ne_zero _ <| denom_ne_zero _ z),

Goal state after:
N : ℕ
a : Fin 2 → ZMod N
k : ℤ
γ : SL(2, ℤ)
z : ℍ
⊢ eisensteinSeries a k (γ • z) =
    eisensteinSeries (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)) k z *
      denom (SpecialLinearGroup.toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑z ^ k
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: k : ℤ
hk : 0 ≤ k
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✝ : ModularFormClass F' Γ k
⊢ ∃ C, ∀ (τ : ℍ), ‖petersson k (⇑f) (⇑f') τ‖ ≤ C * max τ.im (1 / τ.im) ^ k

Tactic:
by

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

Goal state before:
pp: N : ℕ
a : Fin 2 → ZMod N
k : ℤ
γ : SL(2, ℤ)
⊢ eisensteinSeries a k ∣[k] γ = eisensteinSeries (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)) k

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
n : ℕ
t : ℝ
ht : 0 < t
⊢ (PowerSeries.coeff n) (qExpansion h f) =
    1 / ↑h * ∫ (u : ℝ) in 0..h, 1 / 𝕢 h (↑u + ↑t * ↑I) ^ n * f { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ }

Tactic:
by

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

Goal state before:
pp: N r : ℕ
a : Fin 2 → ZMod N
inst✝ : NeZero r
k : ℤ
A : GL (Fin 2) ℝ
hA : A ∈ Subgroup.map (SpecialLinearGroup.mapGL ℝ) Γ(N)
⊢ eisensteinSeries a k ∣[k] A = eisensteinSeries a k

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
this : (f ∘ ↑ofComplex) =O[I∞] (1 ∘ ↑ofComplex)
z : ℂ
hz : z ∈ Complex.im ⁻¹' Ioi 0
⊢ DifferentiableAt ℂ (f ∘ ↑ofComplex) z

Tactic:
by

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

Goal state before:
pp: k : ℤ
hk : 3 ≤ k
z : ℍ
⊢ 2 < ↑k

Tactic:
by

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

Goal state before:
pp: k : ℤ
hk : 3 ≤ k
z : ℍ
hk' : 2 < ↑k
b : Fin 2 → ℤ
⊢ ‖eisSummand k b z‖ ≤ r z ^ (-↑k) * ‖b‖ ^ (-↑k)

Tactic:
simp only [eisSummand, norm_zpow]

Goal state after:
k : ℤ
hk : 3 ≤ k
z : ℍ
hk' : 2 < ↑k
b : Fin 2 → ℤ
⊢ ‖↑(b 0) * ↑z + ↑(b 1)‖ ^ (-k) ≤ r z ^ (-↑k) * ‖b‖ ^ (-↑k)