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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
x : ℝ
hx : x ∈ {1}ᶜ
x✝ : x ∈ Ioi 0
⊢ ↑x ^ (s - 1) • (Ioo 0 1).indicator (fun t ↦ P.f₀ - (P.ε * ↑(t ^ (-P.k))) • P.g₀) x =
    ↑x ^ (s - 1) • (Ioo 0 1).indicator (fun t ↦ P.f₀ - (P.ε * ↑(t ^ (-P.k))) • P.g₀) x

Tactic:
]

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

Goal state before:
pp: k : ℕ
a : ℝ
ha : a ≤ 1
t : ℝ
n : ℕ
this : ↑(Int.negSucc n) + a = -(↑n + (1 - a))
⊢ (↑n + (1 - a)) ^ k * rexp (-π * (↑n + (1 - a)) ^ 2 * t) = (↑n + (1 - a)) ^ k * rexp (-π * (↑n + (1 - a)) ^ 2 * t)

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)
h_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i ↦ if p i = 0 then 0 else a i
i : ι
⊢ (if p i = 0 then 0 else a i) = 0 ∨ 0 < p i

Tactic:
split_ifs with h <;> try tauto

Goal state after:
case neg
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)
h_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i ↦ if p i = 0 then 0 else a i
i : ι
h : ¬p i = 0
⊢ a i = 0 ∨ 0 < p i
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable

Goal state before:
pp: z τ : ℂ
hτ : τ.im ≤ 0
⊢ ¬0 < τ.im

Tactic:
]

Goal state after:
z τ : ℂ
hτ : τ.im ≤ 0
⊢ ¬0 < τ.im
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds

Goal state before:
pp: case ofNat
k : ℕ
a : ℝ
ha : a ∈ Icc 0 1
t : ℝ
ht : 0 < t
a✝ : ℕ
⊢ f_nat k a t a✝ = Int.rec (f_nat k a t) (f_nat k (1 - a) t) (Int.ofNat a✝)

Tactic:
]

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

Goal state before:
pp: case negSucc
k : ℕ
a : ℝ
ha : a ∈ Icc 0 1
t : ℝ
ht : 0 < t
a✝ : ℕ
⊢ f_nat k (1 - a) t a✝ = Int.rec (f_nat k a t) (f_nat k (1 - a) t) (Int.negSucc a✝)

Tactic:
]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
x : ℝ
x✝ : x ∈ Ioc 0 1
hx : 0 < x
right✝ : x ≤ 1
⊢ ↑x ^ (s - 1) • P.f₀ - P.ε • ↑x ^ (s - ↑P.k - 1) • P.g₀ = ↑x ^ (s - 1) • P.f₀ - P.ε • ↑x ^ (s - ↑P.k - 1) • P.g₀

Tactic:
]

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

Goal state before:
pp: a : ℝ
ha : a ∈ Ico 0 1
p : ℝ
hp : 0 < p
hp' : (fun t ↦ F_nat 0 a t - if a = 0 then 1 else 0) =O[atTop] fun t ↦ rexp (-p * t)
q : ℝ
hq : 0 < q
hq' : (fun t ↦ F_nat 0 (1 - a) t) =O[atTop] fun t ↦ rexp (-q * t)
t : ℝ
ht : 0 < t
⊢ (F_nat 0 a t + F_nat 0 (1 - a) t - if a = 0 then 1 else 0) =
    (F_nat 0 a t - if a = 0 then 1 else 0) + F_nat 0 (1 - a) t

Tactic:
]

Goal state after:
a : ℝ
ha : a ∈ Ico 0 1
p : ℝ
hp : 0 < p
hp' : (fun t ↦ F_nat 0 a t - if a = 0 then 1 else 0) =O[atTop] fun t ↦ rexp (-p * t)
q : ℝ
hq : 0 < q
hq' : (fun t ↦ F_nat 0 (1 - a) t) =O[atTop] fun t ↦ rexp (-q * t)
t : ℝ
ht : 0 < t
⊢ (F_nat 0 a t + F_nat 0 (1 - a) t - if a = 0 then 1 else 0) =
    (F_nat 0 a t - if a = 0 then 1 else 0) + F_nat 0 (1 - a) t
Module: Mathlib.NumberTheory.LSeries.AbstractFuncEq

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ (∫ (x : ℝ) in Ioc 0 1, ↑x ^ (s - 1)) • P.f₀ - P.ε • (∫ (x : ℝ) in Ioc 0 1, ↑x ^ (s - ↑P.k - 1)) • P.g₀ =
    (∫ (x : ℝ) in Ioc 0 1, ↑x ^ (s - 1)) • P.f₀ - P.ε • (∫ (x : ℝ) in Ioc 0 1, ↑x ^ (s - ↑P.k - 1)) • P.g₀

pp: case hf
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ Integrable (fun x ↦ ↑x ^ (s - 1) • P.f₀) (volume.restrict (Ioc 0 1))

pp: case hg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ Integrable (fun x ↦ P.ε • ↑x ^ (s - ↑P.k - 1) • P.g₀) (volume.restrict (Ioc 0 1))

Tactic:
]

Goal state after:
case hf
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ Integrable (fun x ↦ ↑x ^ (s - 1) • P.f₀) (volume.restrict (Ioc 0 1))

case hg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ Integrable (fun x ↦ P.ε • ↑x ^ (s - ↑P.k - 1) • P.g₀) (volume.restrict (Ioc 0 1))
Module: Mathlib.NumberTheory.LSeries.AbstractFuncEq

Goal state before:
pp: case hf
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ IntervalIntegrable (fun x ↦ ↑x ^ (s - 1)) volume 0 1

Tactic:
]

Goal state after:
case hf
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ IntervalIntegrable (fun x ↦ ↑x ^ (s - 1)) volume 0 1
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: case neg
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), 0 ≤ p i
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re
hs' : s ≠ 0
a' : ι → ℂ := fun i ↦ if p i = 0 then 0 else a i
hp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i
this✝ : ∀ (i : ι) (t : ℝ), (if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) = a' i * ↑(rexp (-π * p i * t))
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a' i * ↑(rexp (-π * p i * t))) (F t)
i : ι
h✝ : ¬p i = 0
this : 0 ≤ p i
⊢ ‖a i‖ / p i ^ s.re ≤ ‖a i‖ / p i ^ s.re

Tactic:
]

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

Goal state before:
pp: case hg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ IntervalIntegrable (fun x ↦ ↑x ^ (s - ↑P.k - 1)) volume 0 1

Tactic:
]

Goal state after:
case hg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
P : WeakFEPair E
inst✝ : CompleteSpace E
s : ℂ
hs : P.k < s.re
h_re1 : -1 < (s - 1).re
h_re2 : -1 < (s - ↑P.k - 1).re
⊢ IntervalIntegrable (fun x ↦ ↑x ^ (s - ↑P.k - 1)) volume 0 1
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
⊢ 0 < s.re / 2

Tactic:
]

Goal state after:
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
⊢ 0 < s.re / 2
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a : UnitAddCircle
x ξ : ℝ
this : cexp (-↑π * I * (I * ↑x + 2 * (↑ξ * I * ↑x))) = ↑(rexp (π * (x + 2 * ξ * x)))
⊢ rexp (-π * (ξ + 1) ^ 2 * x + π * (x + 2 * ξ * x)) * (jacobiTheta₂ (↑ξ * I * ↑x) (I * ↑x)).re =
    rexp (-π * ξ ^ 2 * x) * (jacobiTheta₂ (↑ξ * I * ↑x) (I * ↑x)).re

Tactic:
]

Goal state after:
a : UnitAddCircle
x ξ : ℝ
this : cexp (-↑π * I * (I * ↑x + 2 * (↑ξ * I * ↑x))) = ↑(rexp (π * (x + 2 * ξ * x)))
⊢ rexp (-π * (ξ + 1) ^ 2 * x + π * (x + 2 * ξ * x)) * (jacobiTheta₂ (↑ξ * I * ↑x) (I * ↑x)).re =
    rexp (-π * ξ ^ 2 * x) * (jacobiTheta₂ (↑ξ * I * ↑x) (I * ↑x)).re
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable

Goal state before:
pp: z τ : ℂ
hτ : 0 < τ.im
eval_fst_CLM : (ℂ × ℂ →L[ℂ] ℂ) →L[ℂ] ℂ := { toFun := fun f ↦ f (1, 0), map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }
step2 : ∀ (n : ℤ), (jacobiTheta₂_term_fderiv n z τ) (1, 0) = jacobiTheta₂'_term n z τ
step1 : HasSum (fun n ↦ jacobiTheta₂'_term n z τ) ((jacobiTheta₂_fderiv z τ) (1, 0))
step3 : HasDerivAt (fun x ↦ jacobiTheta₂ x τ) (∑' (b : ℤ), jacobiTheta₂'_term b z τ) z
⊢ HasDerivAt (fun x ↦ jacobiTheta₂ x τ) (jacobiTheta₂' z τ) z

Tactic:
]

Goal state after:
z τ : ℂ
hτ : 0 < τ.im
eval_fst_CLM : (ℂ × ℂ →L[ℂ] ℂ) →L[ℂ] ℂ := { toFun := fun f ↦ f (1, 0), map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }
step2 : ∀ (n : ℤ), (jacobiTheta₂_term_fderiv n z τ) (1, 0) = jacobiTheta₂'_term n z τ
step1 : HasSum (fun n ↦ jacobiTheta₂'_term n z τ) ((jacobiTheta₂_fderiv z τ) (1, 0))
step3 : HasDerivAt (fun x ↦ jacobiTheta₂ x τ) (∑' (b : ℤ), jacobiTheta₂'_term b z τ) z
⊢ HasDerivAt (fun x ↦ jacobiTheta₂ x τ) (jacobiTheta₂' z τ) z
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: case e'_5
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑π ^ (-s / 2) * Complex.Gamma (s / 2) * a i = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) * a i

Tactic:
]

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

Goal state before:
pp: case inl
a : UnitAddCircle
x : ℝ
hx : x ≤ 0
⊢ 0 = 0

pp: case inl.hx
a : UnitAddCircle
x : ℝ
hx : x ≤ 0
⊢ 1 / x ≤ 0

Tactic:
]

Goal state after:
case inl.hx
a : UnitAddCircle
x : ℝ
hx : x ≤ 0
⊢ 1 / x ≤ 0
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: case e'_6
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑|r i| ^ s = ↑|r i| ^ (↑2 * (s / 2))

pp: case e'_6.hlt
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ -π < ↑2 * (↑|r i|).arg

pp: case e'_6.hle
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑2 * (↑|r i|).arg ≤ π

Tactic:
]

Goal state after:
case e'_6
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑|r i| ^ s = ↑|r i| ^ (↑2 * (s / 2))

case e'_6.hlt
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ -π < ↑2 * (↑|r i|).arg

case e'_6.hle
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑2 * (↑|r i|).arg ≤ π
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case inr.H
x : ℝ
hx : 0 < x
a : ℝ
⊢ cexp (-↑π * ↑a ^ 2 * ↑x) *
      (1 / (-I * (I * ↑x)) ^ (1 / 2) * cexp (-↑π * I * (↑a * I * ↑x) ^ 2 / (I * ↑x)) *
        jacobiTheta₂ (↑a * I * ↑x / (I * ↑x)) (-1 / (I * ↑x))) =
    ↑(1 / x ^ (1 / 2)) * jacobiTheta₂ (↑a) (I * ↑(1 / x))

Tactic:
]

Goal state after:
case inr.H
x : ℝ
hx : 0 < x
a : ℝ
⊢ cexp (-↑π * ↑a ^ 2 * ↑x) *
      (1 / (-I * (I * ↑x)) ^ (1 / 2) * cexp (-↑π * I * (↑a * I * ↑x) ^ 2 / (I * ↑x)) *
        jacobiTheta₂ (↑a * I * ↑x / (I * ↑x)) (-1 / (I * ↑x))) =
    ↑(1 / x ^ (1 / 2)) * jacobiTheta₂ (↑a) (I * ↑(1 / x))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: x : ℝ
hx : 0 < x
a : ℝ
⊢ I / ↑x = I / ↑x

Tactic:
]

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

Goal state before:
pp: x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
⊢ ↑a * I * ↑x = ↑a * (I * ↑x)

Tactic:
]

Goal state after:
x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
⊢ ↑a * I * ↑x = ↑a * (I * ↑x)
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: case e'_6.hlt
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ -π < ↑2 * 0

Tactic:
]

Goal state after:
case e'_6.hlt
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ -π < ↑2 * 0
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: case e'_6.hle
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑2 * 0 ≤ π

Tactic:
]

Goal state after:
case e'_6.hle
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ ↑2 * 0 ≤ π
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
h2 : ↑a * I * ↑x / (I * ↑x) = ↑a
⊢ 1 / ↑x ^ (1 / 2) = 1 / ↑x ^ (1 / 2)

Tactic:
]

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

Goal state before:
pp: case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s ↦ (s - ↑P.k) • P.Λ₀ s) (𝓝 ↑P.k) (𝓝 ((↑P.k - ↑P.k) • P.Λ₀ ↑P.k))

Tactic:
]

Goal state after:
case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s ↦ (s - ↑P.k) • P.Λ₀ s) (𝓝 ↑P.k) (𝓝 ((↑P.k - ↑P.k) • P.Λ₀ ↑P.k))
Module: Mathlib.NumberTheory.LSeries.AbstractFuncEq

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s ↦ (s - ↑P.k) • (1 / s) • P.f₀) (𝓝 ↑P.k) (𝓝 ((↑P.k - ↑P.k) • (1 / ↑P.k) • P.f₀))

Tactic:
]

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s ↦ (s - ↑P.k) • (1 / s) • P.f₀) (𝓝 ↑P.k) (𝓝 ((↑P.k - ↑P.k) • (1 / ↑P.k) • P.f₀))
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)
i : ι
⊢ |r i| ^ s.re = |r i| ^ s.re

Tactic:
]

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

Goal state before:
pp: x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
h2 : ↑a * I * ↑x / (I * ↑x) = ↑a
h3 : 1 / (-I * (I * ↑x)) ^ (1 / 2) = 1 / ↑(x ^ (1 / 2))
⊢ -↑π * I * (↑a ^ 2 * -1 * ↑x ^ 2) = -(-↑π * ↑a ^ 2 * ↑x) * (I * ↑x)

Tactic:
]

Goal state after:
x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
h2 : ↑a * I * ↑x / (I * ↑x) = ↑a
h3 : 1 / (-I * (I * ↑x)) ^ (1 / 2) = 1 / ↑(x ^ (1 / 2))
⊢ -↑π * I * (↑a ^ 2 * -1 * ↑x ^ 2) = -(-↑π * ↑a ^ 2 * ↑x) * (I * ↑x)
Module: Mathlib.NumberTheory.LSeries.MellinEqDirichlet

Goal state before:
pp: ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(r i) * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs₁ : s ≠ 0
⊢ 0 < s.re + 1

Tactic:
]

Goal state after:
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(r i) * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re
hs₁ : s ≠ 0
⊢ 0 < s.re + 1
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable

Goal state before:
pp: z τ : ℂ
n : ℤ
this : cexp (↑π * I * ↑n ^ 2 * 2) = 1
⊢ cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ) = cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable

Goal state before:
pp: z τ : ℂ
n : ℤ
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ 𝓝 0 = 𝓝 0

Tactic:
]

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

Goal state before:
pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s ↦ s • (P.ε / (↑P.k - s)) • P.g₀) (𝓝[≠] 0) (𝓝 (0 • (P.ε / (↑P.k - 0)) • P.g₀))

Tactic:
]

Goal state after:
case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s ↦ s • (P.ε / (↑P.k - s)) • P.g₀) (𝓝[≠] 0) (𝓝 (0 • (P.ε / (↑P.k - 0)) • P.g₀))
Module: Mathlib.NumberTheory.LSeries.AbstractFuncEq

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
hP : IsStrongFEPair P
s : ℂ
t : ℝ
ht₁ : t ≠ 1
ht₀ : 0 < t
ht : t < 1
⊢ ↑t ^ (s - 1) • (((Ioo 0 1).indicator fun x ↦ P.f x) + (Ioi 1).indicator fun x ↦ P.f x) t = ↑t ^ (s - 1) • P.f t

Tactic:
]

Goal state after:
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
hP : IsStrongFEPair P
s : ℂ
t : ℝ
ht₁ : t ≠ 1
ht₀ : 0 < t
ht : t < 1
⊢ ↑t ^ (s - 1) • (((Ioo 0 1).indicator fun x ↦ P.f x) + (Ioi 1).indicator fun x ↦ P.f x) t = ↑t ^ (s - 1) • P.f t
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: case inr.H
x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
h2 : ↑a * I * ↑x / (I * ↑x) = ↑a
h3 : 1 / (-I * (I * ↑x)) ^ (1 / 2) = 1 / ↑(x ^ (1 / 2))
h4 : -↑π * I * (↑a * I * ↑x) ^ 2 / (I * ↑x) = -(-↑π * ↑a ^ 2 * ↑x)
⊢ 1 / ↑(x ^ (1 / 2)) * jacobiTheta₂ (↑a) (-1 / (I * ↑x)) = 1 / ↑(x ^ (1 / 2)) * jacobiTheta₂ (↑a) (-1 / (I * ↑x))

Tactic:
]

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

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
hP : IsStrongFEPair P
s : ℂ
t : ℝ
ht₁ : t ≠ 1
ht₀ : 0 < t
ht : t < 1
⊢ ↑t ^ (s - 1) • P.f t = ↑t ^ (s - 1) • P.f t

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable

Goal state before:
pp: z τ : ℂ
⊢ ∑' (a : ℤ), (starRingEnd ℂ) (jacobiTheta₂_term a (-z) τ) = jacobiTheta₂ ((starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)

Tactic:
]

Goal state after:
z τ : ℂ
⊢ ∑' (a : ℤ), (starRingEnd ℂ) (jacobiTheta₂_term a (-z) τ) = jacobiTheta₂ ((starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a t : ℝ
ht : 0 < t
⊢ HasSum (fun x ↦ ↑(rexp (-π * (↑x + a) ^ 2 * t))) (cexp (-↑π * ↑a ^ 2 * ↑t) * jacobiTheta₂ (↑a * I * ↑t) (I * ↑t))

Tactic:
]

Goal state after:
a t : ℝ
ht : 0 < t
⊢ HasSum (fun x ↦ ↑(rexp (-π * (↑x + a) ^ 2 * t))) (cexp (-↑π * ↑a ^ 2 * ↑t) * jacobiTheta₂ (↑a * I * ↑t) (I * ↑t))
Module: Mathlib.NumberTheory.LSeries.AbstractFuncEq

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
hP : IsStrongFEPair P
s : ℂ
t : ℝ
ht₁ : t ≠ 1
ht₀ : 0 < t
ht : ¬t < 1
⊢ ↑t ^ (s - 1) • P.f t = ↑t ^ (s - 1) • P.f t

Tactic:
]

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

Goal state before:
pp: a t : ℝ
ht : 0 < t
n : ℤ
⊢ cexp (-(↑π * (↑n + ↑a) ^ 2 * ↑t)) =
    cexp (-(↑π * ↑a ^ 2 * ↑t) + (2 * ↑π * I * ↑n * (↑a * I * ↑t) + ↑π * I * ↑n ^ 2 * (I * ↑t)))

Tactic:
]

Goal state after:
a t : ℝ
ht : 0 < t
n : ℤ
⊢ cexp (-(↑π * (↑n + ↑a) ^ 2 * ↑t)) =
    cexp (-(↑π * ↑a ^ 2 * ↑t) + (2 * ↑π * I * ↑n * (↑a * I * ↑t) + ↑π * I * ↑n ^ 2 * (I * ↑t)))
Module: Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable

Goal state before:
pp: z τ : ℂ
n : ℤ
this : cexp (↑π * I * ↑n ^ 2 * 2) = 1
⊢ 2 * ↑π * I * ↑n * (cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ)) =
    2 * ↑π * I * ↑n * (cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ))

Tactic:
]

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

Goal state before:
pp: z τ J J' : ℂ
⊢ cexp (↑π * I * (z + 1) ^ 2 * τ + -↑π * I * (τ + 2 * (z * τ))) * (J' - 2 * ↑π * I * J + (z * J + J) * (2 * ↑π * I)) =
    cexp (↑π * I * z ^ 2 * τ) * (J' + z * J * (2 * ↑π * I))

Tactic:
]

Goal state after:
z τ J J' : ℂ
⊢ cexp (↑π * I * (z + 1) ^ 2 * τ + -↑π * I * (τ + 2 * (z * τ))) * (J' - 2 * ↑π * I * J + (z * J + J) * (2 * ↑π * I)) =
    cexp (↑π * I * z ^ 2 * τ) * (J' + z * J * (2 * ↑π * I))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a t : ℝ
ht : 0 < t
⊢ HasSum (fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t))) (jacobiTheta₂ (↑a) (I * ↑t))

Tactic:
]

Goal state after:
a t : ℝ
ht : 0 < t
⊢ HasSum (fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t))) (jacobiTheta₂ (↑a) (I * ↑t))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaEven

Goal state before:
pp: a t : ℝ
ht : 0 < t
n : ℤ
⊢ cexp (2 * ↑π * I * ↑a * ↑n + -(↑π * ↑n ^ 2 * ↑t)) = cexp (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2 * (I * ↑t))

Tactic:
]

Goal state after:
a t : ℝ
ht : 0 < t
n : ℤ
⊢ cexp (2 * ↑π * I * ↑a * ↑n + -(↑π * ↑n ^ 2 * ↑t)) = cexp (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2 * (I * ↑t))
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: case inl
z : ℂ
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: z τ : ℂ
hτ : τ ≠ 0
⊢ -(2 * ↑π) = -(2 * ↑π)

Tactic:
]

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

Goal state before:
pp: a t : ℝ
ht : 0 < t
⊢ HasSum (fun x ↦ ↑(2 * Real.cos (2 * π * a * (↑x + 1)) * rexp (-π * (↑x + 1) ^ 2 * t))) (↑(cosKernel (↑a) t) - 1)

Tactic:
]

Goal state after:
a t : ℝ
ht : 0 < t
⊢ HasSum (fun x ↦ ↑(2 * Real.cos (2 * π * a * (↑x + 1)) * rexp (-π * (↑x + 1) ^ 2 * t))) (↑(cosKernel (↑a) t) - 1)
Module: Mathlib.NumberTheory.LSeries.HurwitzZetaOdd

Goal state before:
pp: z τ : ℂ
hτ : τ ≠ 0
aux1 : -2 * ↑π / (2 * ↑π * I) = I
⊢ 1 / τ = 1 / τ

Tactic:
]

Goal state after: