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

Goal state before:
pp: case neg.inr
R : Type u_1
inst✝ : Ring R
𝒢 : Subgroup (GL (Fin 2) R)
h✝ : 𝒢.strictPeriods < 𝒢.periods
u b : R
hu_mem : upperRightHom u ∈ 𝒢 ∨ -upperRightHom u ∈ 𝒢
hu_notMem : upperRightHom u ∉ 𝒢
h : -upperRightHom b ∈ 𝒢
⊢ upperRightHom b * upperRightHom u ∈ 𝒢 ∨ upperRightHom b ∈ 𝒢

Tactic:
·

Goal state after:
case neg.inr
R : Type u_1
inst✝ : Ring R
𝒢 : Subgroup (GL (Fin 2) R)
h✝ : 𝒢.strictPeriods < 𝒢.periods
u b : R
hu_mem : upperRightHom u ∈ 𝒢 ∨ -upperRightHom u ∈ 𝒢
hu_notMem : upperRightHom u ∉ 𝒢
h : -upperRightHom b ∈ 𝒢
⊢ upperRightHom b * upperRightHom u ∈ 𝒢 ∨ upperRightHom b ∈ 𝒢
Module: Mathlib.NumberTheory.ModularForms.Identities

Goal state before:
pp: case mp
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ Γ, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ s
⊢ f ∣[k] γ = f

pp: case mpr
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ f ∣[k] γ = f

Tactic:
·

Goal state after:
case mp
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ Γ, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ s
⊢ f ∣[k] γ = f
Module: Mathlib.NumberTheory.ModularForms.Identities

Goal state before:
pp: case mpr
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ f ∣[k] γ = f

Tactic:
·

Goal state after:
case mpr
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ f ∣[k] γ = f
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case left
R : Type u_1
inst✝ : Ring R
𝒢 : Subgroup (GL (Fin 2) R)
⊢ 𝒢.strictPeriods.relIndex 𝒢.periods ≠ 0

pp: case right
R : Type u_1
inst✝ : Ring R
𝒢 : Subgroup (GL (Fin 2) R)
⊢ 𝒢.periods.relIndex 𝒢.strictPeriods ≠ 0

Tactic:
·

Goal state after:
case left
R : Type u_1
inst✝ : Ring R
𝒢 : Subgroup (GL (Fin 2) R)
⊢ 𝒢.strictPeriods.relIndex 𝒢.periods ≠ 0
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case right
R : Type u_1
inst✝ : Ring R
𝒢 : Subgroup (GL (Fin 2) R)
⊢ 𝒢.periods.relIndex 𝒢.strictPeriods ≠ 0

Tactic:
·

Goal state after:
case right
R : Type u_1
inst✝ : Ring R
𝒢 : Subgroup (GL (Fin 2) R)
⊢ 𝒢.periods.relIndex 𝒢.strictPeriods ≠ 0
Module: Mathlib.NumberTheory.ModularForms.Identities

Goal state before:
pp: case mpr.mul
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ ∀ (x y : GL (Fin 2) ℝ),
    x ∈ Subgroup.closure s → y ∈ Subgroup.closure s → f ∣[k] x = f → f ∣[k] y = f → f ∣[k] (x * y) = f

pp: case mpr.inv
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ ∀ x ∈ Subgroup.closure s, f ∣[k] x = f → f ∣[k] x⁻¹ = f

pp: case mpr.hx
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ γ ∈ Subgroup.closure s

Tactic:
·

Goal state after:
case mpr.mul
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ ∀ (x y : GL (Fin 2) ℝ),
    x ∈ Subgroup.closure s → y ∈ Subgroup.closure s → f ∣[k] x = f → f ∣[k] y = f → f ∣[k] (x * y) = f
Module: Mathlib.NumberTheory.ModularForms.Identities

Goal state before:
pp: case mpr.inv
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ ∀ x ∈ Subgroup.closure s, f ∣[k] x = f → f ∣[k] x⁻¹ = f

pp: case mpr.hx
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ γ ∈ Subgroup.closure s

Tactic:
·

Goal state after:
case mpr.inv
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ ∀ x ∈ Subgroup.closure s, f ∣[k] x = f → f ∣[k] x⁻¹ = f
Module: Mathlib.NumberTheory.ModularForms.Identities

Goal state before:
pp: case mpr.hx
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ γ ∈ Subgroup.closure s

Tactic:
·

Goal state after:
case mpr.hx
f : ℍ → ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
s : Set (GL (Fin 2) ℝ)
hΓ : Γ = Subgroup.closure s
k : ℤ
h : ∀ γ ∈ s, f ∣[k] γ = f
γ : GL (Fin 2) ℝ
hγ : γ ∈ Γ
⊢ γ ∈ Subgroup.closure s
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: z : ℍ
this : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
⊢ Tendsto (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) cofinite atTop

Tactic:
have f_def : ⇑f = fun p : Fin 2 → ℝ => (p 0 : ℂ) * ↑z + p 1 := by
    ext1
    dsimp only [π₀, π₁, f, LinearMap.coe_proj, real_smul, LinearMap.coe_smulRight,
      LinearMap.add_apply]
    rw [mul_one]

Goal state after:
z : ℍ
this : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
⊢ Tendsto (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) cofinite atTop
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case pos
𝒢 : Subgroup (GL (Fin 2) ℝ)
h : DiscreteTopology ↥𝒢.strictPeriods
⊢ 𝒢.strictWidthInfty ∈ 𝒢.strictPeriods

pp: case neg
𝒢 : Subgroup (GL (Fin 2) ℝ)
h : ¬DiscreteTopology ↥𝒢.strictPeriods
⊢ 𝒢.strictWidthInfty ∈ 𝒢.strictPeriods

Tactic:
·

Goal state after:
case pos
𝒢 : Subgroup (GL (Fin 2) ℝ)
h : DiscreteTopology ↥𝒢.strictPeriods
⊢ 𝒢.strictWidthInfty ∈ 𝒢.strictPeriods
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case neg
𝒢 : Subgroup (GL (Fin 2) ℝ)
h : ¬DiscreteTopology ↥𝒢.strictPeriods
⊢ 𝒢.strictWidthInfty ∈ 𝒢.strictPeriods

Tactic:
·

Goal state after:
case neg
𝒢 : Subgroup (GL (Fin 2) ℝ)
h : ¬DiscreteTopology ↥𝒢.strictPeriods
⊢ 𝒢.strictWidthInfty ∈ 𝒢.strictPeriods
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mp
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
⊢ 0 < 𝒢.strictWidthInfty → IsCusp ∞ 𝒢

pp: case mpr
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
⊢ IsCusp ∞ 𝒢 → 0 < 𝒢.strictWidthInfty

Tactic:
·

Goal state after:
case mp
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
⊢ 0 < 𝒢.strictWidthInfty → IsCusp ∞ 𝒢
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mp.refine_1
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
h : 0 < 𝒢.strictWidthInfty
⊢ (Additive.toMul (upperRightHom.toAddMonoidHom 𝒢.strictWidthInfty)).IsParabolic

pp: case mp.refine_2
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
h : 0 < 𝒢.strictWidthInfty
⊢ Additive.toMul (upperRightHom.toAddMonoidHom 𝒢.strictWidthInfty) • ∞ = ∞

Tactic:
·

Goal state after:
case mp.refine_1
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
h : 0 < 𝒢.strictWidthInfty
⊢ (Additive.toMul (upperRightHom.toAddMonoidHom 𝒢.strictWidthInfty)).IsParabolic
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mp.refine_2
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
h : 0 < 𝒢.strictWidthInfty
⊢ Additive.toMul (upperRightHom.toAddMonoidHom 𝒢.strictWidthInfty) • ∞ = ∞

Tactic:
·

Goal state after:
case mp.refine_2
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
h : 0 < 𝒢.strictWidthInfty
⊢ Additive.toMul (upperRightHom.toAddMonoidHom 𝒢.strictWidthInfty) • ∞ = ∞
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mpr
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
⊢ IsCusp ∞ 𝒢 → 0 < 𝒢.strictWidthInfty

Tactic:
·

Goal state after:
case mpr
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
⊢ IsCusp ∞ 𝒢 → 0 < 𝒢.strictWidthInfty
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mpr.inl
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
x : ℝ
hx : x ≠ 0
hgg : upperRightHom x ∈ 𝒢
hgi : ↑(upperRightHom x) 1 0 = 0
⊢ ∃ a, ¬a = 0 ∧ upperRightHom a ∈ 𝒢

pp: case mpr.inr
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
x : ℝ
hx : x ≠ 0
hgg : -upperRightHom x ∈ 𝒢
hgi : ↑(-upperRightHom x) 1 0 = 0
⊢ ∃ a, ¬a = 0 ∧ upperRightHom a ∈ 𝒢

Tactic:
·

Goal state after:
case mpr.inl
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
x : ℝ
hx : x ≠ 0
hgg : upperRightHom x ∈ 𝒢
hgi : ↑(upperRightHom x) 1 0 = 0
⊢ ∃ a, ¬a = 0 ∧ upperRightHom a ∈ 𝒢
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mpr.inr
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
x : ℝ
hx : x ≠ 0
hgg : -upperRightHom x ∈ 𝒢
hgi : ↑(-upperRightHom x) 1 0 = 0
⊢ ∃ a, ¬a = 0 ∧ upperRightHom a ∈ 𝒢

Tactic:
·

Goal state after:
case mpr.inr
𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝¹ : DiscreteTopology ↥𝒢.strictPeriods
inst✝ : 𝒢.HasDetPlusMinusOne
x : ℝ
hx : x ≠ 0
hgg : -upperRightHom x ∈ 𝒢
hgi : ↑(-upperRightHom x) 1 0 = 0
⊢ ∃ a, ¬a = 0 ∧ upperRightHom a ∈ 𝒢
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case «0»
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
g : ℂ →ₗ[ℝ] Fin 2 → ℝ := LinearMap.pi ![imLm, imLm ∘ₗ (↑z • ↑↑conjAe)]
c : Fin 2 → ℝ
hz : (↑z).im ≠ 0
⊢ (↑z).im⁻¹ * g (f c) ((fun i ↦ i) ⟨0, ⋯⟩) = c ((fun i ↦ i) ⟨0, ⋯⟩)

pp: case «1»
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
g : ℂ →ₗ[ℝ] Fin 2 → ℝ := LinearMap.pi ![imLm, imLm ∘ₗ (↑z • ↑↑conjAe)]
c : Fin 2 → ℝ
hz : (↑z).im ≠ 0
⊢ (↑z).im⁻¹ * g (f c) ((fun i ↦ i) ⟨1, ⋯⟩) = c ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
·

Goal state after:
case «0»
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
g : ℂ →ₗ[ℝ] Fin 2 → ℝ := LinearMap.pi ![imLm, imLm ∘ₗ (↑z • ↑↑conjAe)]
c : Fin 2 → ℝ
hz : (↑z).im ≠ 0
⊢ (↑z).im⁻¹ * g (f c) ((fun i ↦ i) ⟨0, ⋯⟩) = c ((fun i ↦ i) ⟨0, ⋯⟩)
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mp
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
⊢ (∃ x_1, (↑(↑x_1 0 0) = 1 ∧ ↑(↑x_1 0 1) = 0 ∧ ↑(↑x_1 1 0) = 0 ∧ ↑(↑x_1 1 1) = 1) ∧ (mapGL ℝ) x_1 = upperRightHom x) →
    x ∈ AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)

pp: case mpr
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
⊢ x ∈ AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N) →
    ∃ x_1, (↑(↑x_1 0 0) = 1 ∧ ↑(↑x_1 0 1) = 0 ∧ ↑(↑x_1 1 0) = 0 ∧ ↑(↑x_1 1 1) = 1) ∧ (mapGL ℝ) x_1 = upperRightHom x

Tactic:
·

Goal state after:
case mp
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
⊢ (∃ x_1, (↑(↑x_1 0 0) = 1 ∧ ↑(↑x_1 0 1) = 0 ∧ ↑(↑x_1 1 0) = 0 ∧ ↑(↑x_1 1 1) = 1) ∧ (mapGL ℝ) x_1 = upperRightHom x) →
    x ∈ AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case «1»
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
g : ℂ →ₗ[ℝ] Fin 2 → ℝ := LinearMap.pi ![imLm, imLm ∘ₗ (↑z • ↑↑conjAe)]
c : Fin 2 → ℝ
hz : (↑z).im ≠ 0
⊢ (↑z).im⁻¹ * g (f c) ((fun i ↦ i) ⟨1, ⋯⟩) = c ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
·

Goal state after:
case «1»
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
g : ℂ →ₗ[ℝ] Fin 2 → ℝ := LinearMap.pi ![imLm, imLm ∘ₗ (↑z • ↑↑conjAe)]
c : Fin 2 → ℝ
hz : (↑z).im ≠ 0
⊢ (↑z).im⁻¹ * g (f c) ((fun i ↦ i) ⟨1, ⋯⟩) = c ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: case mpr
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
⊢ x ∈ AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N) →
    ∃ x_1, (↑(↑x_1 0 0) = 1 ∧ ↑(↑x_1 0 1) = 0 ∧ ↑(↑x_1 1 0) = 0 ∧ ↑(↑x_1 1 1) = 1) ∧ (mapGL ℝ) x_1 = upperRightHom x

Tactic:
·

Goal state after:
case mpr
N : ℕ
x : ℝ
this : AddSubgroup.zmultiples ↑N = AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N)
⊢ x ∈ AddSubgroup.map (Int.castAddHom ℝ) (AddSubgroup.zmultiples ↑N) →
    ∃ x_1, (↑(↑x_1 0 0) = 1 ∧ ↑(↑x_1 0 1) = 0 ∧ ↑(↑x_1 1 0) = 0 ∧ ↑(↑x_1 1 1) = 1) ∧ (mapGL ℝ) x_1 = upperRightHom x
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case inl
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
hq : ‖0‖ < 1
⊢ DifferentiableAt ℂ (cuspFunction h f) 0

pp: case inr
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
q : ℂ
hq : ‖q‖ < 1
hq' : q ≠ 0
⊢ DifferentiableAt ℂ (cuspFunction h f) q

Tactic:
·

Goal state after:
case inl
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
hq : ‖0‖ < 1
⊢ DifferentiableAt ℂ (cuspFunction h f) 0
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case inr
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
q : ℂ
hq : ‖q‖ < 1
hq' : q ≠ 0
⊢ DifferentiableAt ℂ (cuspFunction h f) q

Tactic:
·

Goal state after:
case inr
h : ℝ
f : ℍ → ℂ
hh : 0 < h
hfper : Periodic (f ∘ ↑ofComplex) ↑h
hfhol : MDiff f
hfbdd : IsBoundedAtImInfty f
q : ℂ
hq : ‖q‖ < 1
hq' : q ≠ 0
⊢ DifferentiableAt ℂ (cuspFunction h f) q
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case pos
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f : F
inst✝ : SlashInvariantFormClass F Γ k
hΓ : h ∈ Γ.strictPeriods
w : ℂ
hw : 0 < w.im
⊢ (⇑f ∘ ↑ofComplex) (w + ↑h) = (⇑f ∘ ↑ofComplex) w

pp: case neg
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f : F
inst✝ : SlashInvariantFormClass F Γ k
hΓ : h ∈ Γ.strictPeriods
w : ℂ
hw : w.im ≤ 0
⊢ (⇑f ∘ ↑ofComplex) (w + ↑h) = (⇑f ∘ ↑ofComplex) w

Tactic:
·

Goal state after:
case pos
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f : F
inst✝ : SlashInvariantFormClass F Γ k
hΓ : h ∈ Γ.strictPeriods
w : ℂ
hw : 0 < w.im
⊢ (⇑f ∘ ↑ofComplex) (w + ↑h) = (⇑f ∘ ↑ofComplex) w
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case e'_4
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
hf : f.ker = ⊥
hf' : IsClosedEmbedding ⇑f
⊢ cofinite = Filter.coprodᵢ fun x ↦ cofinite

pp: case e'_5
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
hf : f.ker = ⊥
hf' : IsClosedEmbedding ⇑f
⊢ cocompact (Fin 2 → ℝ) = Filter.coprodᵢ fun x ↦ cocompact ℝ

Tactic:
·

Goal state after:
case e'_4
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
hf : f.ker = ⊥
hf' : IsClosedEmbedding ⇑f
⊢ cofinite = Filter.coprodᵢ fun x ↦ cofinite
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case e'_5
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
hf : f.ker = ⊥
hf' : IsClosedEmbedding ⇑f
⊢ cocompact (Fin 2 → ℝ) = Filter.coprodᵢ fun x ↦ cocompact ℝ

Tactic:
·

Goal state after:
case e'_5
z : ℍ
this✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule
π₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0
π₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1
f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1
f_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)
this : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normSq ∘ ⇑f ∘ fun p ↦ Int.cast ∘ p
hf : f.ker = ⊥
hf' : IsClosedEmbedding ⇑f
⊢ cocompact (Fin 2 → ℝ) = Filter.coprodᵢ fun x ↦ cocompact ℝ
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: case neg
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f : F
inst✝ : SlashInvariantFormClass F Γ k
hΓ : h ∈ Γ.strictPeriods
w : ℂ
hw : w.im ≤ 0
⊢ (⇑f ∘ ↑ofComplex) (w + ↑h) = (⇑f ∘ ↑ofComplex) w

Tactic:
·

Goal state after:
case neg
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
Γ : Subgroup (GL (Fin 2) ℝ)
h : ℝ
f : F
inst✝ : SlashInvariantFormClass F Γ k
hΓ : h ∈ Γ.strictPeriods
w : ℂ
hw : w.im ≤ 0
⊢ (⇑f ∘ ↑ofComplex) (w + ↑h) = (⇑f ∘ ↑ofComplex) w
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case e'_3.«0».«0»
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun g ↦ (↑g).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix : Tendsto (fun m ↦ m.map Int.cast) cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime (↑g 1 0) (↑g 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun t ↦ of ![![t, -↑1], Int.cast ∘ ↑g 1]
hmB : Continuous mB
hf₂ : IsClosedEmbedding ⇑(lcRow0Extend hcd)
⊢ (mB ∘ fun g_1 ↦ (lcRow0 (↑g 1)) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) ↑g_1)) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨0, ⋯⟩)
      ((fun i ↦ i) ⟨0, ⋯⟩) =
    (⇑(lcRow0Extend hcd) ∘ f₁ ∘ fun a ↦ ↑a) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

pp: case e'_3.«0».«1»
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun g ↦ (↑g).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix : Tendsto (fun m ↦ m.map Int.cast) cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime (↑g 1 0) (↑g 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun t ↦ of ![![t, -↑1], Int.cast ∘ ↑g 1]
hmB : Continuous mB
hf₂ : IsClosedEmbedding ⇑(lcRow0Extend hcd)
⊢ (mB ∘ fun g_1 ↦ (lcRow0 (↑g 1)) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) ↑g_1)) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨0, ⋯⟩)
      ((fun i ↦ i) ⟨1, ⋯⟩) =
    (⇑(lcRow0Extend hcd) ∘ f₁ ∘ fun a ↦ ↑a) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

pp: case e'_3.«1»
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun g ↦ (↑g).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix : Tendsto (fun m ↦ m.map Int.cast) cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime (↑g 1 0) (↑g 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun t ↦ of ![![t, -↑1], Int.cast ∘ ↑g 1]
hmB : Continuous mB
hf₂ : IsClosedEmbedding ⇑(lcRow0Extend hcd)
j : Fin 2
⊢ (mB ∘ fun g_1 ↦ (lcRow0 (↑g 1)) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) ↑g_1)) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨1, ⋯⟩) j =
    (⇑(lcRow0Extend hcd) ∘ f₁ ∘ fun a ↦ ↑a) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨1, ⋯⟩) j

Tactic:
·

Goal state after:
case e'_3.«0».«0»
f₁ : SL(2, ℤ) → Matrix (Fin 2) (Fin 2) ℝ := fun g ↦ (↑g).map Int.cast
cocompact_ℝ_to_cofinite_ℤ_matrix : Tendsto (fun m ↦ m.map Int.cast) cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
hf₁ : Tendsto f₁ cofinite (cocompact (Matrix (Fin 2) (Fin 2) ℝ))
g : SL(2, ℤ)
hcd : IsCoprime (↑g 1 0) (↑g 1 1)
mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun t ↦ of ![![t, -↑1], Int.cast ∘ ↑g 1]
hmB : Continuous mB
hf₂ : IsClosedEmbedding ⇑(lcRow0Extend hcd)
⊢ (mB ∘ fun g_1 ↦ (lcRow0 (↑g 1)) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) ↑g_1)) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨0, ⋯⟩)
      ((fun i ↦ i) ⟨0, ⋯⟩) =
    (⇑(lcRow0Extend hcd) ∘ f₁ ∘ fun a ↦ ↑a) ⟨g, ⋯⟩ ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)