mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
E : ι → Type u_4
inst✝⁹ : Fintype ι
inst✝⁸ : RCLike 𝕜
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → InnerProductSpace 𝕜 (E i)
𝕝 : Type u_5
inst✝⁵ : CommSemiring 𝕝
inst✝⁴ : StarRing 𝕝
inst✝³ : Algebra 𝕝 𝕜
inst✝² : StarModule 𝕝 𝕜
inst✝¹ : (i : ι) → Module 𝕝 (E i)
inst✝ : ∀ (i : ι), IsScalarTower 𝕝 𝕜 (E i)
c : 𝕝
w : ι → ℝ
f g : (i : ι) → E i
⊢ ⟪f, c • g⟫_[𝕜, w] = c • ⟪f, g⟫_[𝕜, w]

Tactic:
[

Goal state after:
ι : Type u_1
𝕜 : Type u_3
E : ι → Type u_4
inst✝⁹ : Fintype ι
inst✝⁸ : RCLike 𝕜
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → InnerProductSpace 𝕜 (E i)
𝕝 : Type u_5
inst✝⁵ : CommSemiring 𝕝
inst✝⁴ : StarRing 𝕝
inst✝³ : Algebra 𝕝 𝕜
inst✝² : StarModule 𝕝 𝕜
inst✝¹ : (i : ι) → Module 𝕝 (E i)
inst✝ : ∀ (i : ι), IsScalarTower 𝕝 𝕜 (E i)
c : 𝕝
w : ι → ℝ
f g : (i : ι) → E i
⊢ ⟪f, c • g⟫_[𝕜, w] = c • ⟪f, g⟫_[𝕜, w]
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: ι : Type u_1
κ : Type u_2
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
f : κ → ι → 𝕜
hf : ∀ (k : κ), f k ≠ 0
hinner : Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫_[𝕜] = 0
⊢ LinearIndependent 𝕜 f

Tactic:
[

Goal state after:
ι : Type u_1
κ : Type u_2
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
f : κ → ι → 𝕜
hf : ∀ (k : κ), f k ≠ 0
hinner : Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫_[𝕜] = 0
⊢ LinearIndependent 𝕜 f
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
h : ∫⁻ (a : E), ENNReal.ofReal ‖(1 + ‖a‖ ^ 2) ^ (-s)‖ < ∞
⊢ ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s / 2)‖ₑ ^ 2 < ∞

Tactic:
[

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
h : ∫⁻ (a : E), ENNReal.ofReal ‖(1 + ‖a‖ ^ 2) ^ (-s)‖ < ∞
⊢ ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s / 2)‖ₑ ^ 2 < ∞
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
h : ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s)‖ₑ < ∞
⊢ ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s / 2)‖ₑ ^ 2 < ∞

Tactic:
[

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
h : ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s)‖ₑ < ∞
⊢ ∫⁻ (a : E), ‖(1 + ‖a‖ ^ 2) ^ (-s / 2)‖ₑ ^ 2 < ∞
Module: Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality

Goal state before:
pp: G : Type u_1
R : Type u_3
inst✝² : AddGroup G
inst✝¹ : RCLike R
inst✝ : Fintype G
ψ : AddChar G R
⊢ ⟪⇑ψ, ⇑ψ⟫ₙ_[R] = 1

Tactic:
simp [wInner_cWeight_eq_expect, ψ.norm_apply]

Goal state after:
Module: Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality

Goal state before:
pp: case neg
G : Type u_1
R : Type u_3
inst✝² : AddCommGroup G
inst✝¹ : RCLike R
inst✝ : Fintype G
ψ₁ ψ₂ : AddChar G R
h : ¬ψ₁ = ψ₂
this : ψ₂ * ψ₁⁻¹ ≠ 1
⊢ ⟪⇑ψ₁, ⇑ψ₂⟫ₙ_[R] = 0

Tactic:
[

Goal state after:
case neg
G : Type u_1
R : Type u_3
inst✝² : AddCommGroup G
inst✝¹ : RCLike R
inst✝ : Fintype G
ψ₁ ψ₂ : AddChar G R
h : ¬ψ₁ = ψ₂
this : ψ₂ * ψ₁⁻¹ ≠ 1
⊢ ⟪⇑ψ₁, ⇑ψ₂⟫ₙ_[R] = 0
Module: Mathlib.GroupTheory.FiniteAbelian.Basic

Goal state before:
pp: case neg
ι : Type
inst✝ : DecidableEq ι
p n : ι → ℕ
i : ι
x : ZMod (p i ^ n i)
h : ¬n i = 0
⊢ (directSumNeZeroMulHom p n) ((DirectSum.of (fun j ↦ ZMod (p ↑j ^ n ↑j)) ⟨i, h⟩) x) =
    (DirectSum.of (fun i ↦ ZMod (p i ^ n i)) i) x

Tactic:
[

Goal state after:
case neg
ι : Type
inst✝ : DecidableEq ι
p n : ι → ℕ
i : ι
x : ZMod (p i ^ n i)
h : ¬n i = 0
⊢ (directSumNeZeroMulHom p n) ((DirectSum.of (fun j ↦ ZMod (p ↑j ^ n ↑j)) ⟨i, h⟩) x) =
    (DirectSum.of (fun i ↦ ZMod (p i ^ n i)) i) x
Module: Mathlib.GroupTheory.FiniteAbelian.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CommGroup A
B : Subgroup A
hB : B.FG
n : ℕ
hn : n ≠ 0
this : (map (powMonoidHom n) B).subgroupOf B = (powMonoidHom n).range
⊢ ((map (powMonoidHom n) B).subgroupOf B).FiniteIndex

Tactic:
this

Goal state after:
A : Type u_1
inst✝ : CommGroup A
B : Subgroup A
hB : B.FG
n : ℕ
hn : n ≠ 0
this : (map (powMonoidHom n) B).subgroupOf B = (powMonoidHom n).range
⊢ (powMonoidHom n).range.FiniteIndex
Module: Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ Injective (zmod n)

Tactic:
[

Goal state after:
n : ℕ
inst✝ : NeZero n
⊢ Injective (zmod n)
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
this : ∀ (x : ℝ), ‖e x‖ = 1
K : Compacts ℝ
g : C(ℝ, ℂ)
⊢ ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖

Tactic:
[

Goal state after:
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
this : ∀ (x : ℝ), ‖e x‖ = 1
K : Compacts ℝ
g : C(ℝ, ℂ)
⊢ ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ fourierCoeff ⋯.lift m = ∫ (x : ℝ) in 0..1, e x * (∑' (n : ℤ), f.comp (ContinuousMap.addRight ↑n)) x

Tactic:
[

Goal state after:
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ fourierCoeff ⋯.lift m = ∫ (x : ℝ) in 0..1, e x * (∑' (n : ℤ), f.comp (ContinuousMap.addRight ↑n)) x
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ ∫ (x : ℝ) in 0..1, e x * (∑' (n : ℤ), f.comp (ContinuousMap.addRight ↑n)) x =
    ∫ (x : ℝ) in 0..1, ∑' (n : ℤ), (e * f.comp (ContinuousMap.addRight ↑n)) x

Tactic:
[

Goal state after:
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ ∫ (x : ℝ) in 0..1, e x * (∑' (n : ℤ), f.comp (ContinuousMap.addRight ↑n)) x =
    ∫ (x : ℝ) in 0..1, ∑' (n : ℤ), (e * f.comp (ContinuousMap.addRight ↑n)) x
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ ∑' (n : ℤ), ∫ (x : ℝ) in 0..1, (e * f.comp (ContinuousMap.addRight ↑n)) x =
    ∑' (n : ℤ), ∫ (x : ℝ) in 0..1, (e.comp (ContinuousMap.addRight ↑n) * f.comp (ContinuousMap.addRight ↑n)) x

Tactic:
[

Goal state after:
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ ∑' (n : ℤ), ∫ (x : ℝ) in 0..1, (e * f.comp (ContinuousMap.addRight ↑n)) x =
    ∑' (n : ℤ), ∫ (x : ℝ) in 0..1, (e.comp (ContinuousMap.addRight ↑n) * f.comp (ContinuousMap.addRight ↑n)) x
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ (fun n ↦ ‖ContinuousMap.restrict (Icc 0 1) ((e * f).comp (ContinuousMap.addRight ↑n))‖) = fun n ↦
    ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := ⋯ }) (f.comp (ContinuousMap.addRight ↑n))‖

Tactic:
[

Goal state after:
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ (fun n ↦ ‖ContinuousMap.restrict (Icc 0 1) ((e * f).comp (ContinuousMap.addRight ↑n))‖) = fun n ↦
    ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := ⋯ }) (f.comp (ContinuousMap.addRight ↑n))‖
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ (fun n ↦
      ‖ContinuousMap.restrict (Icc 0 1) (e.comp (ContinuousMap.addRight ↑n) * f.comp (ContinuousMap.addRight ↑n))‖) =
    fun n ↦ ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := ⋯ }) (f.comp (ContinuousMap.addRight ↑n))‖

Tactic:
[

Goal state after:
f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ (fun n ↦
      ‖ContinuousMap.restrict (Icc 0 1) (e.comp (ContinuousMap.addRight ↑n) * f.comp (ContinuousMap.addRight ↑n))‖) =
    fun n ↦ ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := ⋯ }) (f.comp (ContinuousMap.addRight ↑n))‖
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hiw : ⟪i w, w⟫ = 1 / 2
v : V
⊢ 𝐞 (-⟪v, w⟫) • f (v + (1 / (2 * ‖w‖ ^ 2)) • w) =
    -(𝐞 (-⟪v + (1 / (2 * ‖w‖ ^ 2)) • w, w⟫) • f (v + (1 / (2 * ‖w‖ ^ 2)) • w))

Tactic:
[

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hiw : ⟪i w, w⟫ = 1 / 2
v : V
⊢ 𝐞 (-⟪v, w⟫) • f (v + (1 / (2 * ‖w‖ ^ 2)) • w) =
    -(𝐞 (-⟪v + (1 / (2 * ‖w‖ ^ 2)) • w, w⟫) • f (v + (1 / (2 * ‖w‖ ^ 2)) • w))
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: case e'_3
f : C(ℝ, ℂ)
h_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
h_sum : Summable fun n ↦ 𝓕 ⇑f ↑n
x : ℝ
F : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }
this : Summable (fourierCoeff ⇑F)
⊢ ∑' (n : ℤ), 𝓕 ⇑f ↑n * (fourier n) ↑x = ∑' (b : ℤ), fourierCoeff (⇑F) b • (fourier b) ↑x

Tactic:
[

Goal state after:
case e'_3
f : C(ℝ, ℂ)
h_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
h_sum : Summable fun n ↦ 𝓕 ⇑f ↑n
x : ℝ
F : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }
this : Summable (fourierCoeff ⇑F)
⊢ ∑' (n : ℤ), 𝓕 ⇑f ↑n * (fourier n) ↑x = ∑' (b : ℤ), fourierCoeff (⇑F) b • (fourier b) ↑x
Module: Mathlib.Data.PNat.Prime

Goal state before:
pp: m n : ℕ+
⊢ m.gcd n = m ↔ m ∣ n

Tactic:
rw [dvd_iff, ← Nat.gcd_eq_left_iff_dvd, ← coe_inj]
  simp

Goal state after:
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ ∫ (v : V), 𝐞 (-⟪v, w⟫) • f v = (1 / 2) • ∫ (v : V), 𝐞 (-⟪v, w⟫) • (f v - f (v + (1 / (2 * ‖w‖ ^ 2)) • w))

Tactic:
[

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ ∫ (v : V), 𝐞 (-⟪v, w⟫) • f v = (1 / 2) • ∫ (v : V), 𝐞 (-⟪v, w⟫) • (f v - f (v + (1 / (2 * ‖w‖ ^ 2)) • w))
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ ∫ (v : V), 𝐞 (-⟪v, w⟫) • f v =
    (1 / 2) • ((∫ (a : V), 𝐞 (-⟪a, w⟫) • f a) - ∫ (a : V), 𝐞 (-⟪a, w⟫) • f (a + (1 / (2 * ‖w‖ ^ 2)) • w))

pp: case hf
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ Integrable (fun v ↦ 𝐞 (-⟪v, w⟫) • f v) volume

pp: case hg
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ Integrable (fun v ↦ 𝐞 (-⟪v, w⟫) • f (v + (1 / (2 * ‖w‖ ^ 2)) • w)) volume

Tactic:
fourierIntegral_half_period_translate hw,

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ ∫ (v : V), 𝐞 (-⟪v, w⟫) • f v = (1 / 2) • ((∫ (a : V), 𝐞 (-⟪a, w⟫) • f a) - -∫ (v : V), 𝐞 (-⟪v, w⟫) • f v)

case hf
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ Integrable (fun v ↦ 𝐞 (-⟪v, w⟫) • f v) volume

case hg
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hf : Integrable f volume
⊢ Integrable (fun v ↦ 𝐞 (-⟪v, w⟫) • f (v + (1 / (2 * ‖w‖ ^ 2)) • w)) volume
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
this : ∃ T, ∀ (w : V), T ≤ ‖w‖ → ‖∫ (v : V), 𝐞 (-⟪v, w⟫) • f v‖ < ε
⊢ ∀ᶠ (x : V) in cocompact V, ‖∫ (v : V), 𝐞 (-⟪v, x⟫) • f v‖ < ε

Tactic:
[

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
this : ∃ T, ∀ (w : V), T ≤ ‖w‖ → ‖∫ (v : V), 𝐞 (-⟪v, w⟫) • f v‖ < ε
⊢ ∀ᶠ (x : V) in cocompact V, ‖∫ (v : V), 𝐞 (-⟪v, x⟫) • f v‖ < ε
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
f : ArithmeticFunction R
h_mult : f.IsMultiplicative
n : ℕ
hn : Squarefree n
⊢ ∏ p ∈ n.primeFactors, (1 + f p) = (ArithmeticFunction.prodPrimeFactors fun p ↦ (↑ζ + f) p) n

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
f : ArithmeticFunction R
h_mult : f.IsMultiplicative
n : ℕ
hn : Squarefree n
⊢ ∏ p ∈ n.primeFactors, (1 + f p) = (ArithmeticFunction.prodPrimeFactors fun p ↦ (↑ζ + f) p) n
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : ArithmeticFunction R
hf : f.IsMultiplicative
n : ℕ
hn : Squarefree n
⊢ ∑ d ∈ n.divisors, ((↑μ).pmul f) d = ∑ d ∈ n.divisors, ↑(μ d) * f d

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : ArithmeticFunction R
hf : f.IsMultiplicative
n : ℕ
hn : Squarefree n
⊢ ∑ d ∈ n.divisors, ((↑μ).pmul f) d = ∑ d ∈ n.divisors, ↑(μ d) * f d
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
this : A = Metric.closedBall 0 (R + 1)
⊢ MeasurableSet A

Tactic:
this

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
this : A = Metric.closedBall 0 (R + 1)
⊢ MeasurableSet (Metric.closedBall 0 (R + 1))
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
⊢ A = Metric.closedBall 0 (R + 1)

Tactic:
[

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
⊢ A = Metric.closedBall 0 (R + 1)
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x ↦ |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a ↦ -a, continuous_toFun := ⋯ }) =O[atTop] fun x ↦ |x| ^ (-b)
h2 :
  ((fun x ↦ ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a ↦ -a, continuous_toFun := ⋯ })‖) ∘
      Neg.neg) =O[atBot]
    ((fun x ↦ |x| ^ (-b)) ∘ Neg.neg)
this : (fun x ↦ |x| ^ (-b)) ∘ Neg.neg = fun x ↦ |x| ^ (-b)
⊢ (fun x ↦ ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x ↦ |x| ^ (-b)

Tactic:
this

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x ↦ |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a ↦ -a, continuous_toFun := ⋯ }) =O[atTop] fun x ↦ |x| ^ (-b)
h2 :
  ((fun x ↦ ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a ↦ -a, continuous_toFun := ⋯ })‖) ∘
      Neg.neg) =O[atBot]
    fun x ↦ |x| ^ (-b)
this : (fun x ↦ |x| ^ (-b)) ∘ Neg.neg = fun x ↦ |x| ^ (-b)
⊢ (fun x ↦ ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x ↦ |x| ^ (-b)