mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: case refine_1
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
⊢ IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ioc 1 T) volume

pp: case refine_2
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
t : ℝ
ht : t ∈ Set.Ioc 1 T
⊢ ‖S t - l * ↑t‖ * t ^ (-s - 1) ≤ ‖S t - l * ↑t‖ * t ^ (-1 - 1)

Tactic:
· exact h₁.mono_set <| Set.Ioc_subset_Ioi_self.trans Set.Ioi_subset_Ici_self

Goal state after:
case refine_2
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
t : ℝ
ht : t ∈ Set.Ioc 1 T
⊢ ‖S t - l * ↑t‖ * t ^ (-s - 1) ≤ ‖S t - l * ↑t‖ * t ^ (-1 - 1)
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: K : Type u_1
inst✝² : NontriviallyNormedField K
inst✝¹ : IsUltrametricDist K
inst✝ : IsDiscreteValuationRing ↥𝒪[K]
ϖ : ↥𝒪[K]
h : Irreducible ϖ
⊢ ↑𝓂[K] = Metric.closedBall 0 ‖ϖ‖

Tactic:
simp [h.maximalIdeal_eq_setOfPred_le_v_coe, Set.ext_iff, ← NNReal.coe_le_coe]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: case mp.mk
K : Type u_1
Γ₀ : Type u_2
inst✝⁴ : Field K
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : Valued K Γ₀
inst✝¹ : v.RankOne
inst✝ : IsDiscreteValuationRing ↥𝒪[K]
H : TotallyBounded Set.univ
p : ↥𝒪[K]
hp : Irreducible p
t : Set ↥𝒪[K]
ht : t.Finite
a✝ : IsLocalRing.ResidueField ↥𝒪[K]
x y : ↥𝒪[K]
hy : y ∈ t
hy' : dist x y < ‖p‖
⊢ y - x ∈ IsLocalRing.maximalIdeal ↥𝒪[K]

Tactic:
hp.maximalIdeal_eq,

Goal state after:
case mp.mk
K : Type u_1
Γ₀ : Type u_2
inst✝⁴ : Field K
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : Valued K Γ₀
inst✝¹ : v.RankOne
inst✝ : IsDiscreteValuationRing ↥𝒪[K]
H : TotallyBounded Set.univ
p : ↥𝒪[K]
hp : Irreducible p
t : Set ↥𝒪[K]
ht : t.Finite
a✝ : IsLocalRing.ResidueField ↥𝒪[K]
x y : ↥𝒪[K]
hy : y ∈ t
hy' : dist x y < ‖p‖
⊢ y - x ∈ Ideal.span {p}
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ ‖(↑s - 1) * ↑s * ∫ (t : ℝ) in Set.Ioi 1, (S t - l * ↑t) * ↑t ^ (-↑s - 1)‖ ≤
    (s - 1) * s * ∫ (t : ℝ) in Set.Ioi 1, ‖(S t - l * ↑t) * ↑t ^ (-↑s - 1)‖

Tactic:
by
      rw [norm_mul, show ((s : ℂ) - 1) * s = ((s - 1) * s : ℝ) by simp, norm_real,
        Real.norm_of_nonneg hs']
      exact mul_le_mul_of_nonneg_left (norm_integral_le_integral_norm _) hs'

Goal state after:
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: case hfs
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ioc 1 T) volume

pp: case hft
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ioi T) volume

Tactic:
· exact h₁.mono_set <| Set.Ioc_subset_Ioi_self.trans Set.Ioi_subset_Ici_self

Goal state after:
case hft
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ioi T) volume
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: case refine_2
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
h₁ : ∀ {C ε : ℝ}, Tendsto (fun s ↦ (s - 1) * s * C + s * ε) (𝓝[>] 1) (𝓝 ε)
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝[>] 1) fun s ↦ ‖(↑s - 1) * LSeries f ↑s - ↑s * l‖
ε : ℝ
hε : 0 < ε
C : ℝ
hC₁ : C ≥ 0
hC₂ : (fun s ↦ ‖(↑s - 1) * LSeries f ↑s - ↑s * l‖) ≤ᶠ[𝓝[>] 1] fun s ↦ (s - 1) * s * C + s * ε
⊢ limsup (fun x ↦ ‖(↑x - 1) * LSeries f ↑x - ↑x * l‖) (𝓝[>] 1) ≤ ε

Tactic:
refine le_of_le_of_eq (limsup_le_limsup hC₂ ?_ h₁.isBoundedUnder_le) h₁.limsup_eq

Goal state after:
case refine_2
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
h₁ : ∀ {C ε : ℝ}, Tendsto (fun s ↦ (s - 1) * s * C + s * ε) (𝓝[>] 1) (𝓝 ε)
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝[>] 1) fun s ↦ ‖(↑s - 1) * LSeries f ↑s - ↑s * l‖
ε : ℝ
hε : 0 < ε
C : ℝ
hC₁ : C ≥ 0
hC₂ : (fun s ↦ ‖(↑s - 1) * LSeries f ↑s - ↑s * l‖) ≤ᶠ[𝓝[>] 1] fun s ↦ (s - 1) * s * C + s * ε
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝[>] 1) fun x ↦ ‖(↑x - 1) * LSeries f ↑x - ↑x * l‖
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: q : ℕ
inst✝ : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
this : ω ^ ((q + 1) / 4) * (ω ^ ((q + 1) / 4) * ωb ^ ((q + 1) / 4)) = -ωb ^ ((q + 1) / 4)
div4 : (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4
⊢ ω ^ ((q + 1) / 4) + ωb ^ ((q + 1) / 4) = 0

Tactic:
← mul_pow,

Goal state after:
q : ℕ
inst✝ : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
this : ω ^ ((q + 1) / 4) * (ω * ωb) ^ ((q + 1) / 4) = -ωb ^ ((q + 1) / 4)
div4 : (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4
⊢ ω ^ ((q + 1) / 4) + ωb ^ ((q + 1) / 4) = 0
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: p' : ℕ
w : 3 ≤ p' + 2
hp : Nat.Prime (mersenne (p' + 2))
this : Fact (Nat.Prime (mersenne (p' + 2)))
hp' : p' = p' + 2 - 2
⊢ LucasLehmerTest (p' + 2)

Tactic:
dsimp [LucasLehmerTest, lucasLehmerResidue]

Goal state after:
p' : ℕ
w : 3 ≤ p' + 2
hp : Nat.Prime (mersenne (p' + 2))
this : Fact (Nat.Prime (mersenne (p' + 2)))
hp' : p' = p' + 2 - 2
⊢ sZMod (p' + 2) (p' + 2 - 2) = 0
Module: Mathlib.NumberTheory.ModularForms.CongruenceSubgroups

Goal state before:
pp: case mp
N : ℕ
A : SL(2, ℤ)
ha : A ∈ Subgroup.map ((Gamma0 N).subtype.comp (Gamma1' N).subtype) ⊤
⊢ ↑(↑A 0 0) = 1 ∧ ↑(↑A 1 1) = 1 ∧ ↑(↑A 1 0) = 0

Tactic:
Subgroup.mem_map

Goal state after:
case mp
N : ℕ
A : SL(2, ℤ)
ha : ∃ x ∈ ⊤, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A
⊢ ↑(↑A 0 0) = 1 ∧ ↑(↑A 1 1) = 1 ∧ ↑(↑A 1 0) = 0
Module: Mathlib.NumberTheory.ModularForms.CongruenceSubgroups

Goal state before:
pp: case mpr
N : ℕ
A : SL(2, ℤ)
ha : ↑(↑A 0 0) = 1 ∧ ↑(↑A 1 1) = 1 ∧ ↑(↑A 1 0) = 0
⊢ A ∈ Subgroup.map ((Gamma0 N).subtype.comp (Gamma1' N).subtype) ⊤

Tactic:
Subgroup.mem_map

Goal state after:
case mpr
N : ℕ
A : SL(2, ℤ)
ha : ↑(↑A 0 0) = 1 ∧ ↑(↑A 1 1) = 1 ∧ ↑(↑A 1 0) = 0
⊢ ∃ x ∈ ⊤, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A
Module: Mathlib.NumberTheory.ModularForms.Cusps

Goal state before:
pp: 𝒢 : Subgroup (GL (Fin 2) ℝ)
inst✝ : 𝒢.IsArithmetic
a b : SL(2, ℤ)
hab : a ≈ b
⊢ ⟦⟨(mapGL ℝ) a⁻¹ • ∞, ⋯⟩⟧ = ⟦⟨(mapGL ℝ) b⁻¹ • ∞, ⋯⟩⟧

Tactic:
by
      rw [← Quotient.eq_iff_equiv, Quotient.eq, QuotientGroup.leftRel_apply] at hab
      refine Quotient.eq.mpr ⟨⟨_, hab⟩, ?_⟩
      simp [mul_smul]

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✝ : SlashInvariantFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k
hk : k ≤ 0
f : F
τ : ℍ
γ : SL(2, ℤ)
hγ : 1 / 2 ≤ (γ • τ).im
hdenom : ‖denom (Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1
⊢ ‖f τ‖ ≤ ‖f (γ • τ)‖

Tactic:
by
    have : SlashInvariantFormClass F Γ(1) k := Gamma_one_coe_eq_SL ▸ ‹_›
    simpa only [slash_action_eqn_SL'' _ (mem_Gamma_one γ), norm_mul, norm_zpow]
      using le_mul_of_one_le_left (norm_nonneg _) <|
        one_le_zpow_of_nonpos₀ (norm_pos_iff.2 (denom_ne_zero _ _)) hdenom hk

Goal state after:
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:
ext i
    simp_all [Pi.smul_apply, divIntMap, ← Int.mul_ediv_assoc _ (gammaSet_div_gcd hv i)]

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
τ : ℍ := { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ }
this : circleMap 0 R (u * (2 * π / h)) = 𝕢 h ↑τ
⊢ (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:
simp_rw [deriv_circleMap, this, show ↑I = Complex.I by rfl, show u + t * Complex.I = τ by rfl,
    show ⟨↑τ, τ.2⟩ = τ by rfl, eq_cuspFunction _ hh.ne' hfper, smul_eq_mul, pow_succ]

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 := ⋯ }
this : circleMap 0 R (u * (2 * π / h)) = 𝕢 h ↑τ
⊢ (2 * ↑π * Complex.I)⁻¹ * ↑(2 * π / h) * (𝕢 h ↑τ * Complex.I * (f τ / (𝕢 h ↑τ ^ n * 𝕢 h ↑τ))) =
    1 / ↑h * (1 / 𝕢 h ↑τ ^ n * f τ)
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
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
hg' : (g • z).re < -(1 / 2)
⊢ |(g • z).re + 1| < |(g • z).re|

Tactic:
cases abs_cases ((g • z).re + 1) <;> cases abs_cases (g • z).re

Goal state after:
case right.left.inl.inl
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
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
hg' : (g • z).re < -(1 / 2)
h✝¹ : |(g • z).re + 1| = (g • z).re + 1 ∧ 0 ≤ (g • z).re + 1
h✝ : |(g • z).re| = (g • z).re ∧ 0 ≤ (g • z).re
⊢ |(g • z).re + 1| < |(g • z).re|

case right.left.inl.inr
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
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
hg' : (g • z).re < -(1 / 2)
h✝¹ : |(g • z).re + 1| = (g • z).re + 1 ∧ 0 ≤ (g • z).re + 1
h✝ : |(g • z).re| = -(g • z).re ∧ (g • z).re < 0
⊢ |(g • z).re + 1| < |(g • z).re|

case right.left.inr.inl
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
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
hg' : (g • z).re < -(1 / 2)
h✝¹ : |(g • z).re + 1| = -((g • z).re + 1) ∧ (g • z).re + 1 < 0
h✝ : |(g • z).re| = (g • z).re ∧ 0 ≤ (g • z).re
⊢ |(g • z).re + 1| < |(g • z).re|

case right.left.inr.inr
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
hg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im
hg' : (g • z).re < -(1 / 2)
h✝¹ : |(g • z).re + 1| = -((g • z).re + 1) ∧ (g • z).re + 1 < 0
h✝ : |(g • z).re| = -(g • z).re ∧ (g • z).re < 0
⊢ |(g • z).re + 1| < |(g • z).re|
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
    obtain ⟨A, (hA : A ∈ Γ(N)), rfl⟩ := hA
    simp [SpecialLinearGroup.mapGL, ← SL_slash, eisensteinSeries_slash_apply, Gamma_mem'.mp hA]

Goal state after:
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:
right

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 := ⋯
hgz : g = T ^ (-1)
hre : (g • z).re = ↑b + z.re
hb : b = -1
⊢ (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨ g = 1 ∨ g = -1
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case hre
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 = 1
hgeq : g = T ^ ↑g 0 0 * S * T
hnorm : normSq ↑z = 1
hre : z.re = -1 / 2
⊢ z.re = ρ.re

Tactic:
simp [hre, ρ]

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 = 1
hgeq : g = T ^ ↑g 0 0 * S * T
hnorm : normSq ↑z = 1
hre : z.re = -1 / 2
hρ : z = ρ
⊢ T ^ ↑g 0 0 * S * T = S * T ∨ T ^ ↑g 0 0 * S * T = T * S * T

Tactic:
have hSTρ : (S * T) • ρ = ρ := by
    rw [mul_smul, ← SL_neg_smul S, ← S_inv, inv_smul_eq_iff, eq_comm, UpperHalfPlane.ext_iff,
      modular_S_smul, modular_T_smul, UpperHalfPlane.coe_mk, coe_vadd,
      ← mul_one (_ : ℂ)⁻¹, inv_mul_eq_iff_eq_mul₀ (neg_ne_zero.mpr ρ.ne_zero)]
    grind [ρ_sq, ofReal_one]

Goal state after:
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 = 1
hgeq : g = T ^ ↑g 0 0 * S * T
hnorm : normSq ↑z = 1
hre : z.re = -1 / 2
hρ : z = ρ
hSTρ : (S * T) • ρ = ρ
⊢ T ^ ↑g 0 0 * S * T = S * T ∨ T ^ ↑g 0 0 * S * T = T * S * T
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : T ^ ↑g 0 0 • (S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this✝¹ : ↑g 0 1 = -↑g 0 0 - 1
hgeq : g = T ^ ↑g 0 0 * S * T⁻¹
hnorm : normSq ↑z + (-2 * z.re + 1) ≤ 1
this✝ : normSq (↑z - 1) = normSq ↑z + (-2 * z.re + 1)
h : normSq ↑z = 1
h' : z.re = 1 / 2
hρ : z = 1 +ᵥ ρ
this : S • ρ = T • ρ
⊢ (S * T⁻¹) • (1 +ᵥ ρ) = 1 +ᵥ ρ

Tactic:
rw [mul_smul, ← SL_neg_smul S, ← S_inv, inv_smul_eq_iff, ← zpow_neg_one,
    modular_T_zpow_smul, Int.cast_neg, Int.cast_one, neg_vadd_vadd,
    ← inv_smul_eq_iff, S_inv, SL_neg_smul, this, modular_T_smul]

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 * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this✝¹ : ↑g 0 1 = -↑g 0 0 - 1
hgeq : g = T ^ ↑g 0 0 * S * T⁻¹
hnorm : normSq ↑z + (-2 * z.re + 1) ≤ 1
this✝ : normSq (↑z - 1) = normSq ↑z + (-2 * z.re + 1)
h : normSq ↑z = 1
h' : z.re = 1 / 2
hρ : z = 1 +ᵥ ρ
this : S • ρ = T • ρ
⊢ g = S * T⁻¹ ∨ g = T⁻¹ * S * T⁻¹

Tactic:
have : (S * T⁻¹) • ((1 : ℝ) +ᵥ ρ) = (1 : ℝ) +ᵥ ρ := by
    rw [mul_smul, ← SL_neg_smul S, ← S_inv, inv_smul_eq_iff, ← zpow_neg_one,
    modular_T_zpow_smul, Int.cast_neg, Int.cast_one, neg_vadd_vadd,
    ← inv_smul_eq_iff, S_inv, SL_neg_smul, this, modular_T_smul]

Goal state after:
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : T ^ ↑g 0 0 • (S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟
hg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1
hc : ↑g 1 0 = 1
hd : ↑g 1 1 = -1
this✝² : ↑g 0 1 = -↑g 0 0 - 1
hgeq : g = T ^ ↑g 0 0 * S * T⁻¹
hnorm : normSq ↑z + (-2 * z.re + 1) ≤ 1
this✝¹ : normSq (↑z - 1) = normSq ↑z + (-2 * z.re + 1)
h : normSq ↑z = 1
h' : z.re = 1 / 2
hρ : z = 1 +ᵥ ρ
this✝ : S • ρ = T • ρ
this : (S * T⁻¹) • (1 +ᵥ ρ) = 1 +ᵥ ρ
⊢ g = S * T⁻¹ ∨ g = T⁻¹ * S * T⁻¹
Module: Mathlib.NumberTheory.ModularForms.Derivative

Goal state before:
pp: F : ℍ → ℂ
n : ℕ
hF : MDiff F
z : ℍ
hFz : DifferentiableAt ℂ (F ∘ ↑ofComplex) ↑z
⊢ (2 * ↑π * I)⁻¹ * deriv ((F ^ n) ∘ ↑ofComplex) ↑z = ↑n z * F z ^ (n - 1) * ((2 * ↑π * I)⁻¹ * deriv (F ∘ ↑ofComplex) ↑z)

Tactic:
show (F ^ n) ∘ ofComplex = (F ∘ ofComplex) ^ n from rfl,

Goal state after:
F : ℍ → ℂ
n : ℕ
hF : MDiff F
z : ℍ
hFz : DifferentiableAt ℂ (F ∘ ↑ofComplex) ↑z
⊢ (2 * ↑π * I)⁻¹ * deriv (F ∘ ↑ofComplex ^ n) ↑z = ↑n z * F z ^ (n - 1) * ((2 * ↑π * I)⁻¹ * deriv (F ∘ ↑ofComplex) ↑z)
Module: Mathlib.NumberTheory.ModularForms.Derivative

Goal state before:
pp: k : ℤ
F : ℍ → ℂ
hF : MDiff F
g : GL (Fin 2) ℝ
hg : 0 < (↑g).det
hdet : ↑(Matrix.GeneralLinearGroup.det g) = (↑g).det
hdetℂ : ↑(↑g).det ≠ 0
hσ : ∀ (x : ℂ), (σ g) x = x
z : ℍ
hz : denom g ↑z ≠ 0
h_smul : HasDerivAt (fun w ↦ ↑(g • ↑ofComplex w)) (↑(↑g).det / denom g ↑z ^ 2) ↑z
h_F : HasDerivAt (F ∘ ↑ofComplex) (deriv (F ∘ ↑ofComplex) ↑(g • ↑ofComplex ↑z)) ↑(g • ↑ofComplex ↑z)
⊢ (2 * ↑π * I)⁻¹ * deriv (F ∣[k] g ∘ ↑ofComplex) ↑z =
    (↑(↑g).det)⁻¹ *
        ((σ g) ((2 * ↑π * I)⁻¹ * deriv (F ∘ ↑ofComplex) ↑(g • z)) *
            ↑|↑(Matrix.GeneralLinearGroup.det g)| ^ (k + 2 - 1) *
          denom g ↑z ^ (-(k + 2))) -
      ↑k * (2 * ↑π * I)⁻¹ * (↑(↑g 1 0) / denom g ↑z) *
        ((σ g) (F (g • z)) * ↑|↑(Matrix.GeneralLinearGroup.det g)| ^ (k - 1) * denom g ↑z ^ (-k))

Tactic:
have h_denom : HasDerivAt (fun w ↦ (denom g w) ^ (-k))
      (-k * (g 1 0 : ℂ) * (denom g z) ^ (-k - 1)) ↑z := by
    simpa using hasDerivAt_denom_zpow g (-k) z

Goal state after:
k : ℤ
F : ℍ → ℂ
hF : MDiff F
g : GL (Fin 2) ℝ
hg : 0 < (↑g).det
hdet : ↑(Matrix.GeneralLinearGroup.det g) = (↑g).det
hdetℂ : ↑(↑g).det ≠ 0
hσ : ∀ (x : ℂ), (σ g) x = x
z : ℍ
hz : denom g ↑z ≠ 0
h_smul : HasDerivAt (fun w ↦ ↑(g • ↑ofComplex w)) (↑(↑g).det / denom g ↑z ^ 2) ↑z
h_F : HasDerivAt (F ∘ ↑ofComplex) (deriv (F ∘ ↑ofComplex) ↑(g • ↑ofComplex ↑z)) ↑(g • ↑ofComplex ↑z)
h_denom : HasDerivAt (fun w ↦ denom g w ^ (-k)) (-↑k * ↑(↑g 1 0) * denom g ↑z ^ (-k - 1)) ↑z
⊢ (2 * ↑π * I)⁻¹ * deriv (F ∣[k] g ∘ ↑ofComplex) ↑z =
    (↑(↑g).det)⁻¹ *
        ((σ g) ((2 * ↑π * I)⁻¹ * deriv (F ∘ ↑ofComplex) ↑(g • z)) *
            ↑|↑(Matrix.GeneralLinearGroup.det g)| ^ (k + 2 - 1) *
          denom g ↑z ^ (-(k + 2))) -
      ↑k * (2 * ↑π * I)⁻¹ * (↑(↑g 1 0) / denom g ↑z) *
        ((σ g) (F (g • z)) * ↑|↑(Matrix.GeneralLinearGroup.det g)| ^ (k - 1) * denom g ↑z ^ (-k))
Module: Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula

Goal state before:
pp: case n
k : ℕ
hk : 3 ≤ k
hk2 : Even k
h : E hk ∈ cuspFormSubmodule (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑k
hE : (PowerSeries.coeff 0) (qExpansion 1 ⇑(E hk)) = 1
⊢ False

Tactic:
exact one_ne_zero <| hE.symm.trans <| (isCuspForm_iff_coeffZero_eq_zero _).mp h

Goal state after: