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

Goal state before:
pp: f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ LSeries (fun {n} ↦ if ?m.225 then ?m.227 else f n) s =
    s * ∫ (t : ℝ) in Set.Ioi 1, (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-(s + 1))

pp: f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ {n : ℕ} → Prop

pp: f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ {n : ℕ} → Decidable ?m.225

pp: f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ {n : ℕ} → ℂ

pp: f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ ∀ {n : ℕ}, n ≠ 0 → ¬?m.225

Tactic:
LSeries_eq_mul_integral_aux (if_pos rfl) hr hs hS

Goal state after:
f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ s * ∫ (t : ℝ) in Set.Ioi 1, (∑ k ∈ Icc 1 ⌊t⌋₊, if k = 0 then 0 else f k) * ↑t ^ (-(s + 1)) =
    s * ∫ (t : ℝ) in Set.Ioi 1, (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-(s + 1))

f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ (fun n ↦ ∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) =O[atTop] fun n ↦ ↑n ^ r

f : ℕ → ℂ
r : ℝ
hr : 0 ≤ r
s : ℂ
hs : r < s.re
hS : LSeriesSummable (fun n ↦ if n = 0 then 0 else f n) s
hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ ↑n ^ r
this : ∀ (n : ℕ), (∑ k ∈ Icc 1 n, if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k
⊢ ∀ {n : ℕ}, n ≠ 0 → ¬n = 0
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: a : ℤ
b : ℕ
h : a.gcd ↑b = 1
⊢ J(a ^ 2 | b) = 1

Tactic:
by rw [pow_left, sq_one h]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: a : ℤ
R : Type u_1
inst✝ : Semiring R
χ : R →* ℤ
hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p
b : ℕ
hb : Odd b
⊢ (List.pmap (fun p pp ↦ legendreSym p a) b.primeFactorsList ⋯).prod =
    (List.map (⇑χ ∘ Nat.cast) b.primeFactorsList).prod

Tactic:
← List.pmap_eq_map
    fun _ => prime_of_mem_primeFactorsList

Goal state after:
a : ℤ
R : Type u_1
inst✝ : Semiring R
χ : R →* ℤ
hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p
b : ℕ
hb : Odd b
⊢ (List.pmap (fun p pp ↦ legendreSym p a) b.primeFactorsList ⋯).prod =
    (List.pmap (fun a x ↦ (⇑χ ∘ Nat.cast) a) b.primeFactorsList ⋯).prod
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: m₁ m₂ n : ℕ
⊢ J(χ₄ (↑m₁ * ↑m₂) | n) = J(χ₄ ↑m₁ | n) * J(χ₄ ↑m₂ | n)

Tactic:
map_mul,

Goal state after:
m₁ m₂ n : ℕ
⊢ J(χ₄ ↑m₁ * χ₄ ↑m₂ | n) = J(χ₄ ↑m₁ | n) * J(χ₄ ↑m₂ | n)
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
this : ∀ (z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ), (Set.Icc z 1).Finite
x y : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
hxy : x < y
h : x ≤ 1
pf✝² : MonoidHomClass (K →*₀ Γ₀) K Γ₀
pf✝¹ : MonoidWithZeroHomClass (Valuation K Γ₀) K Γ₀
pf✝ : MonoidHomClass (K →*₀ Γ₀) K Γ₀
⊢ (Set.Icc x y).Finite

Tactic:
rcases le_total y 1 with hy | hy

Goal state after:
case inl
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
this : ∀ (z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ), (Set.Icc z 1).Finite
x y : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
hxy : x < y
h : x ≤ 1
pf✝² : MonoidHomClass (K →*₀ Γ₀) K Γ₀
pf✝¹ : MonoidWithZeroHomClass (Valuation K Γ₀) K Γ₀
pf✝ : MonoidHomClass (K →*₀ Γ₀) K Γ₀
hy : y ≤ 1
⊢ (Set.Icc x y).Finite

case inr
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
this : ∀ (z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ), (Set.Icc z 1).Finite
x y : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
hxy : x < y
h : x ≤ 1
pf✝² : MonoidHomClass (K →*₀ Γ₀) K Γ₀
pf✝¹ : MonoidWithZeroHomClass (Valuation K Γ₀) K Γ₀
pf✝ : MonoidHomClass (K →*₀ Γ₀) K Γ₀
hy : 1 ≤ y
⊢ (Set.Icc x y).Finite
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
⊢ ∀ (a b : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ), (Set.Icc a b).Finite

Tactic:
suffices ∀ z : (MonoidHom.mrange (MonoidWithZeroHom.ofClass (Valued.v (R := K))))ˣ,
      (Set.Icc z 1).Finite by
    rintro x y
    rcases lt_trichotomy y x with hxy | rfl | hxy
    · rw [Set.Icc_eq_empty_of_lt]
      · exact Set.finite_empty
      · simp [hxy]
    · simp
    wlog! h : x ≤ 1 generalizing x y
    · specialize this y⁻¹ x⁻¹ (inv_lt_inv' hxy) (inv_le_one_of_one_le (h.trans hxy).le)
      refine (this.inv).subset ?_
      rw [Set.inv_Icc]
      intro
      simp +contextual
    generalize_proofs _ _ _ _ hxu hyu
    rcases le_total y 1 with hy | hy
    · exact (this x).subset (Set.Icc_subset_Icc_right hy)
    · have H : (Set.Icc y⁻¹ 1).Finite := this _
      refine ((this x).union H.inv).subset (le_of_eq ?_)
      rw [Set.inv_Icc, inv_one, Set.Icc_union_Icc_eq_Icc] <;>
      simp [h, hy]

Goal state after:
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
⊢ ∀ (z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ), (Set.Icc z 1).Finite
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
a c : A
⊢ x * (a * c) = a * (x * c)

Tactic:
rw [← mul_assoc, ← hx1, mul_assoc]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
b a : A
⊢ x * (a * b) = x * a * b

Tactic:
rw [← mul_assoc, ← hx1, mul_assoc]

Goal state after:
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: q : ℕ
⊢ ωb * ω = 1

Tactic:
rw [mul_comm, ω_mul_ωb]

Goal state after:
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: q : ℕ
inst✝ : Fact (Nat.Prime q)
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
w✝ : ℕ
h✝ : q = 2 * w✝ + 1
⊢ 2 ^ (q / 2) * 2 = 2

Tactic:
have leg := legendreSym.eq_pow q 2

Goal state after:
q : ℕ
inst✝ : Fact (Nat.Prime q)
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
w✝ : ℕ
h✝ : q = 2 * w✝ + 1
leg : ↑(legendreSym q 2) = ↑2 ^ (q / 2)
⊢ 2 ^ (q / 2) * 2 = 2
Module: Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete

Goal state before:
pp: case mpr
G : Type u_1
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : TopologicalSpace G
inst✝¹ : OrderTopology G
inst✝ : MulArchimedean G
H : Subgroup G
a✝ : Nontrivial G
this : Dense ↑H ∨ ∃ a, H = closure {a}
⊢ DiscreteTopology ↥H → ∃ g, zpowers g = H

Tactic:
simp only [← Subgroup.zpowers_eq_closure, Eq.comm (a := H)] at this

Goal state after:
case mpr
G : Type u_1
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : TopologicalSpace G
inst✝¹ : OrderTopology G
inst✝ : MulArchimedean G
H : Subgroup G
a✝ : Nontrivial G
this : Dense ↑H ∨ ∃ a, zpowers a = H
⊢ DiscreteTopology ↥H → ∃ g, zpowers g = H
Module: Mathlib.Topology.Compactification.OnePoint.ProjectiveLine

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : DecidableEq K
c : K
g : GL (Fin 2) K
h : ¬↑g 1 0 * c + ↑g 1 1 = 0
⊢ ↑g 1 0 * c ^ 2 + (↑g 1 1 - ↑g 0 0) * c - ↑g 0 1 = 0 ↔ ↑((↑g 0 0 * c + ↑g 0 1) / (↑g 1 0 * c + ↑g 1 1)) = ↑c

Tactic:
rw [coe_eq_coe, div_eq_iff h]
    grind

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

Goal state before:
pp: k : ℤ
A B : GL (Fin 2) ℝ
f : ℍ → ℂ
τ : ℍ
⊢ (σ A) ((σ B) (f (A • B • τ))) * ↑|↑(Matrix.GeneralLinearGroup.det (A * B))| ^ (k - 1) *
      ((σ B) (denom A ↑(B • τ)) ^ (-k) * denom B ↑τ ^ (-k)) =
    (σ A) ((σ B) (f (A • B • τ))) *
        (↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1) * ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1)) *
      ((σ B) (denom A ↑(B • τ) ^ (-k)) * denom B ↑τ ^ (-k))

Tactic:
map_mul,

Goal state after:
k : ℤ
A B : GL (Fin 2) ℝ
f : ℍ → ℂ
τ : ℍ
⊢ (σ A) ((σ B) (f (A • B • τ))) * ↑|↑(Matrix.GeneralLinearGroup.det A * Matrix.GeneralLinearGroup.det B)| ^ (k - 1) *
      ((σ B) (denom A ↑(B • τ)) ^ (-k) * denom B ↑τ ^ (-k)) =
    (σ A) ((σ B) (f (A • B • τ))) *
        (↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1) * ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1)) *
      ((σ B) (denom A ↑(B • τ) ^ (-k)) * denom B ↑τ ^ (-k))
Module: Mathlib.NumberTheory.ModularForms.SlashActions

Goal state before:
pp: k : ℤ
A B : GL (Fin 2) ℝ
f : ℍ → ℂ
τ : ℍ
⊢ (σ B) ((σ A) (f (A • B • τ))) *
        (↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1) * ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1)) *
      ((σ B) (denom A ↑(B • τ) ^ (-k)) * denom B ↑τ ^ (-k)) =
    (σ B) ((σ A) (f (A • B • τ)) * ↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1) * denom A ↑(B • τ) ^ (-k)) *
        ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1) *
      denom B ↑τ ^ (-k)

Tactic:
map_mul,

Goal state after:
k : ℤ
A B : GL (Fin 2) ℝ
f : ℍ → ℂ
τ : ℍ
⊢ (σ B) ((σ A) (f (A • B • τ))) *
        (↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1) * ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1)) *
      ((σ B) (denom A ↑(B • τ) ^ (-k)) * denom B ↑τ ^ (-k)) =
    (σ B) ((σ A) (f (A • B • τ)) * ↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1)) * (σ B) (denom A ↑(B • τ) ^ (-k)) *
        ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1) *
      denom B ↑τ ^ (-k)
Module: Mathlib.NumberTheory.ModularForms.SlashActions

Goal state before:
pp: k : ℤ
A B : GL (Fin 2) ℝ
f : ℍ → ℂ
τ : ℍ
⊢ (σ B) ((σ A) (f (A • B • τ))) *
        (↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1) * ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1)) *
      ((σ B) (denom A ↑(B • τ)) ^ (-k) * denom B ↑τ ^ (-k)) =
    (σ B) ((σ A) (f (A • B • τ)) * ↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1)) * (σ B) (denom A ↑(B • τ)) ^ (-k) *
        ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1) *
      denom B ↑τ ^ (-k)

Tactic:
map_mul,

Goal state after:
k : ℤ
A B : GL (Fin 2) ℝ
f : ℍ → ℂ
τ : ℍ
⊢ (σ B) ((σ A) (f (A • B • τ))) *
        (↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1) * ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1)) *
      ((σ B) (denom A ↑(B • τ)) ^ (-k) * denom B ↑τ ^ (-k)) =
    (σ B) ((σ A) (f (A • B • τ))) * (σ B) (↑|↑(Matrix.GeneralLinearGroup.det A)| ^ (k - 1)) *
          (σ B) (denom A ↑(B • τ)) ^ (-k) *
        ↑|↑(Matrix.GeneralLinearGroup.det B)| ^ (k - 1) *
      denom B ↑τ ^ (-k)
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)

Tactic:
rintro ⟨g, ⟨-, hg, -, -⟩, hx⟩
    rw [show x = g 0 1 by simpa using congr_arg (· 0 1) hx.symm]
    apply AddSubgroup.mem_map_of_mem
    rwa [Int.mem_zmultiples_iff, ← ZMod.intCast_zmod_eq_zero_iff_dvd]

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

Goal state before:
pp: f : ℍ → ℂ
hf : MDiff f
k : ℤ
⊢ MDiff (f ∣[k] J)

Tactic:
by
  simp only [mdifferentiable_iff, slash_J, Function.comp_def] at hf ⊢
  have : {z | 0 < z.im}.EqOn (fun x ↦ conj (f <| ofComplex <| -conj ↑(ofComplex x)))
      (fun x ↦ conj (f <| ofComplex <| -conj x)) := fun z h ↦ by simp [ofComplex_apply_of_im_pos h]
  refine .congr (fun z hz ↦ DifferentiableAt.differentiableWithinAt ?_) this
  have : 0 < (-conj z).im := by simpa using! hz
  have := hf.differentiableAt (isOpen_upperHalfPlaneSet.mem_nhds this)
  simpa using! (this.comp _ differentiable_neg.differentiableAt).star_star.neg

Goal state after:
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 * (↑z * ((starRingEnd ℂ) (↑(c 0) * ↑z) + (starRingEnd ℂ) ↑(c 1))).im = c 1

Tactic:
map_mul,

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 * (↑z * ((starRingEnd ℂ) ↑(c 0) * (starRingEnd ℂ) ↑z + (starRingEnd ℂ) ↑(c 1))).im = c 1
Module: Mathlib.NumberTheory.ModularForms.QExpansion

Goal state before:
pp: F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
hh : 0 < h
hfper : Periodic (⇑f ∘ ↑ofComplex) ↑h
hfhol : MDiff ⇑f
hfbdd : IsBoundedAtImInfty ⇑f
r : NNReal
hr : ↑r < 1
⊢ Summable fun n ↦ ‖(PowerSeries.coeff n) (qExpansion h ⇑f)‖ * ‖↑r‖ ^ n

Tactic:
← Complex.norm_real,

Goal state after:
F : Type u_1
inst✝ : FunLike F ℍ ℂ
h : ℝ
f : F
hh : 0 < h
hfper : Periodic (⇑f ∘ ↑ofComplex) ↑h
hfhol : MDiff ⇑f
hfbdd : IsBoundedAtImInfty ⇑f
r : NNReal
hr : ↑r < 1
⊢ Summable fun n ↦ ‖(PowerSeries.coeff n) (qExpansion h ⇑f)‖ * ‖↑↑r‖ ^ n
Module: Mathlib.NumberTheory.ModularForms.Petersson

Goal state before:
pp: case right
F : Type u_1
F' : Type u_2
inst✝⁶ : FunLike F ℍ ℂ
inst✝⁵ : FunLike F' ℍ ℂ
k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝⁴ : Fact (IsCusp OnePoint.infty Γ)
inst✝³ : Γ.HasDetPlusMinusOne
inst✝² : DiscreteTopology ↥Γ
inst✝¹ : ModularFormClass F Γ k
inst✝ : ModularFormClass F' Γ k
f : F
h_bd : IsZeroAtImInfty ⇑f
f' : F'
b : ℝ
hb : b > 0
hbf : ⇑f =O[atImInfty] fun τ ↦ Real.exp (-b * τ.im)
a : ℝ
ha : 0 < a
ha' : a < b
⊢ (fun x ↦ ‖(starRingEnd ℂ) (f x)‖ * ‖f' x‖ * ‖↑x.im ^ k‖) =O[atImInfty] fun τ ↦ Real.exp (-a * τ.im)

Tactic:
Complex.norm_conj,

Goal state after:
case right
F : Type u_1
F' : Type u_2
inst✝⁶ : FunLike F ℍ ℂ
inst✝⁵ : FunLike F' ℍ ℂ
k : ℤ
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝⁴ : Fact (IsCusp OnePoint.infty Γ)
inst✝³ : Γ.HasDetPlusMinusOne
inst✝² : DiscreteTopology ↥Γ
inst✝¹ : ModularFormClass F Γ k
inst✝ : ModularFormClass F' Γ k
f : F
h_bd : IsZeroAtImInfty ⇑f
f' : F'
b : ℝ
hb : b > 0
hbf : ⇑f =O[atImInfty] fun τ ↦ Real.exp (-b * τ.im)
a : ℝ
ha : 0 < a
ha' : a < b
⊢ (fun x ↦ ‖f x‖ * ‖f' x‖ * ‖↑x.im ^ k‖) =O[atImInfty] fun τ ↦ Real.exp (-a * τ.im)
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
f : F
c : ℂ
hf : ⇑f = Function.const ℍ c
this✝ : SlashInvariantFormClass F (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) Γ(1)) k
hI : c = ↑UpperHalfPlane.I ^ k * c
h2I2 : c = ↑(⟨2, ⋯⟩ • UpperHalfPlane.I) ^ k * c
this : 2 ^ k = 1 ↔ k = 0
⊢ k = 0 ∨ c = 0

Tactic:
by
    simpa [mul_zpow, zpow_ne_zero, this] using h2I2.symm.trans hI

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«0».«0»
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
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«0».«1»
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
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«1».«0»
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
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«1».«1»
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
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(T ^ ↑g 0 1) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«0».«0»
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
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«0».«1»
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
⊢ ↑g ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«1».«0»
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
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

Goal state after:
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: case h.«1».«1»
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
⊢ ↑g ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) = ↑(-T ^ (-↑g 0 1)) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
simp [ha, hc, hd, coe_T_zpow, show (1 : Fin (0 + 2)) = (1 : Fin 2) from rfl]

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 : SL(2, ℤ)
hj : j ∈ Γ
hj' : h = g * j
⊢ f (g⁻¹ • τ) = f (h⁻¹ • τ)

Tactic:
simp [-sl_moeb, hj', mul_smul, hf_inv j⁻¹ (inv_mem hj)]

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
  -- marshall the info we have in terms of a function on the quotient
  let f' τ : SL(2, ℤ) ⧸ Γ → E := Quotient.lift (fun g ↦ f (g⁻¹ • τ)) fun g h hgh ↦ by
    obtain ⟨j, hj, hj'⟩ : ∃ j ∈ Γ, h = g * j := by
      rw [← Quotient.eq_iff_equiv, Quotient.eq, QuotientGroup.leftRel_apply] at hgh
      exact ⟨g⁻¹ * h, hgh, (mul_inv_cancel_left g h).symm⟩
    simp [-sl_moeb, hj', mul_smul, hf_inv j⁻¹ (inv_mem hj)]
  have hf'_cont γ : Continuous (f' · γ) := QuotientGroup.induction_on γ fun g ↦ by
    simp only [sl_moeb, f']
    fun_prop
  have hf'_inv τ (g : SL(2, ℤ)) γ : f' (g • τ) (g • γ) = f' τ γ := by
    induction γ using QuotientGroup.induction_on
    simp [-sl_moeb, f', mul_smul]
  have hf'_infty γ : (f' · γ) =O[_] _ := γ.induction_on fun h ↦ hf_infinity h⁻¹
  -- now take the sum over the quotient
  have : Fintype (SL(2, ℤ) ⧸ Γ) := Subgroup.fintypeQuotientOfFiniteIndex
  -- Now the conclusion is very simple.
  obtain ⟨C, hC⟩ := exists_bound_of_invariant_of_isBigO (by fun_prop) ht
    (.fun_sum fun i _ ↦ (hf'_infty i).norm_left)
    (fun g τ ↦ (Fintype.sum_equiv (MulAction.toPerm g) _ _ (by simp [-sl_moeb, hf'_inv])).symm)
  refine ⟨C, fun τ ↦ le_trans ?_ (hC τ)⟩
  simpa [Real.norm_of_nonneg <| show 0 ≤ ∑ γ, ‖f' τ γ‖ by positivity, -sl_moeb, f'] using
    Finset.univ.single_le_sum (fun γ _ ↦ norm_nonneg (f' τ γ)) (Finset.mem_univ ⟦1⟧)

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

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
a : Fin 2 → ZMod N
k : ℤ
hk : 3 ≤ k
A : SL(2, ℤ)
z : ℍ
hz : 2 ≤ z.im
n : ℤ
hn : ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im
⊢ ‖(⇑{ toFun := eisensteinSeries a k, slash_action_eq' := ⋯ } ∣[k] A) z‖ ≤
    ∑' (x : Fin 2 → ℤ), r { coe := { re := ↑N, im := 2 }, coe_im_pos := ⋯ } ^ (-k) * ‖x‖ ^ (-k)

Tactic:
rw [SlashInvariantForm.coe_mk, eisensteinSeries_slash_apply, ← eisensteinSeriesSIF_apply,
    ← T_zpow_width_invariant N k n (eisensteinSeriesSIF (a ᵥ* A) k) z]

Goal state after:
N : ℕ
inst✝ : NeZero N
a : Fin 2 → ZMod N
k : ℤ
hk : 3 ≤ k
A : SL(2, ℤ)
z : ℍ
hz : 2 ≤ z.im
n : ℤ
hn : ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im
⊢ ‖(eisensteinSeriesSIF (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) A)) k)
        (ModularGroup.T ^ (↑N * n) • z)‖ ≤
    ∑' (x : Fin 2 → ℤ), r { coe := { re := ↑N, im := 2 }, coe_im_pos := ⋯ } ^ (-k) * ‖x‖ ^ (-k)
Module: Mathlib.NumberTheory.ModularForms.Bounds

Goal state before:
pp: Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
f : F
C : ℝ
τ : ℍ
k : ℕ
inst✝ : ModularFormClass F Γ ↑k
hC : ‖(starRingEnd ℂ) (f τ)‖ * ‖f τ‖ * ‖↑τ.im ^ ↑k‖ ≤ C * max τ.im (1 / τ.im) ^ ↑k
hC' : 0 ≤ C
h : 0 < ‖↑τ.im ^ ↑k‖
⊢ ‖f τ‖ ≤ √C * max 1 (1 / τ.im ^ ↑k)

Tactic:
Complex.norm_conj,

Goal state after:
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
F : Type u_2
inst✝¹ : FunLike F ℍ ℂ
f : F
C : ℝ
τ : ℍ
k : ℕ
inst✝ : ModularFormClass F Γ ↑k
hC : ‖f τ‖ * ‖f τ‖ * ‖↑τ.im ^ ↑k‖ ≤ C * max τ.im (1 / τ.im) ^ ↑k
hC' : 0 ≤ C
h : 0 < ‖↑τ.im ^ ↑k‖
⊢ ‖f τ‖ ≤ √C * max 1 (1 / τ.im ^ ↑k)
Module: Mathlib.NumberTheory.Modular

Goal state before:
pp: g✝ : SL(2, ℤ)
z✝ : ℍ
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟
hg : g • z ∈ 𝒟
hc : ↑g 1 0 = 0
hd' : ↑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 ^ b
⊢ (g • z).re = ↑b + z.re

Tactic:
hgz,

Goal state after:
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 ^ b
⊢ (T ^ b • z).re = ↑b + z.re