mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
B

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν = ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: 𝕜 : Type u_2
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
inst✝⁶ : NormedAddCommGroup D
inst✝⁵ : NormedSpace ℝ D
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
hf : HasTemperateGrowth f
hg : HasTemperateGrowth g
⊢ HasTemperateGrowth fun x ↦ (B (f x)) (g x)

Tactic:
rw [Function.hasTemperateGrowth_iff_isBigO]
  constructor
  · apply (B.bilinearRestrictScalars ℝ).isBoundedBilinearMap.contDiff.comp (hf.1.prodMk hg.1)
  intro n
  rcases hf.isBigO_uniform n with ⟨k1, h1⟩
  rcases hg.isBigO_uniform n with ⟨k2, h2⟩
  use k1 + k2
  have estimate (x : D) : ‖iteratedFDeriv ℝ n (fun x ↦ B (f x) (g x)) x‖ ≤
      ‖B‖ * ∑ i ∈ Finset.range (n + 1), (n.choose i) *
        ‖iteratedFDeriv ℝ i f x‖ * ‖iteratedFDeriv ℝ (n - i) g x‖ :=
    (B.bilinearRestrictScalars ℝ).norm_iteratedFDeriv_le_of_bilinear hf.1 hg.1 x (mod_cast le_top)
  refine (IsBigO.of_norm_le estimate).trans (.const_mul_left (.fun_sum fun i hi ↦ ?_) _)
  simp_rw [mul_assoc, pow_add]
  refine .const_mul_left (.mul (h1 i ?_).norm_left (h2 (n - i) ?_).norm_left) _ <;>
  grind

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
x : E
⊢ |re ⟪T x, x⟫_𝕜 / ‖x‖ ^ 2| ≤ ‖T‖

Tactic:
abs_div,

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
x : E
⊢ |re ⟪T x, x⟫_𝕜| / |‖x‖ ^ 2| ≤ ‖T‖
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : (↑T).IsSymmetric
M : ℝ := ⨆ x, |T.rayleighQuotient x|
nonneg : 0 ≤ M
x : E
hM : |T.reApplyInnerSelf x / ‖x‖ ^ 2| ≤ M
hx : 0 < ‖x‖ ^ 2
⊢ |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ^ 2

Tactic:
abs_div,

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : (↑T).IsSymmetric
M : ℝ := ⨆ x, |T.rayleighQuotient x|
nonneg : 0 ≤ M
x : E
hM : |T.reApplyInnerSelf x| / |‖x‖ ^ 2| ≤ M
hx : 0 < ‖x‖ ^ 2
⊢ |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ^ 2
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • x₀ + b • T x₀ = 0
hx₀ : ¬x₀ = 0
hb : ¬b = 0
⊢ 1 - b⁻¹ * (b * 1) = 0

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • x₀ + b • T x₀ = 0
hx₀ : ¬x₀ = 0
hb : ¬b = 0
⊢ -(-b⁻¹ * a * 1 + b⁻¹ * (a * 1)) = 0

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
j : n
⊢ star ↑hA.eigenvectorUnitary *ᵥ (hA.eigenvectorBasis j).ofLp = Pi.single j 1

Tactic:
rw [← eigenvectorUnitary_mulVec, mulVec_mulVec, Unitary.coe_star_mul_self, one_mulVec]

Goal state after:
Module: Mathlib.Analysis.Matrix.PosDef

Goal state before:
pp: m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike 𝕜
A M : Matrix n n 𝕜
hM : M.PosSemidef
x y : n → 𝕜
⊢ (starRingEnd 𝕜) (M *ᵥ x ⬝ᵥ star y) = M *ᵥ y ⬝ᵥ star x

Tactic:
dotProduct_comm,

Goal state after:
m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : RCLike 𝕜
A M : Matrix n n 𝕜
hM : M.PosSemidef
x y : n → 𝕜
⊢ (starRingEnd 𝕜) (star y ⬝ᵥ M *ᵥ x) = M *ᵥ y ⬝ᵥ star x
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
x : Matrix n n 𝕜
hx : x.PosSemidef
h : IsUnit x
v : n → 𝕜
hv : v ≠ 0
⊢ 0 < star v ⬝ᵥ x *ᵥ v

Tactic:
obtain ⟨y, rfl⟩ := CStarAlgebra.nonneg_iff_eq_star_mul_self.mp hx.nonneg

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
hv : v ≠ 0
y : Matrix n n 𝕜
hx : (star y * y).PosSemidef
h : IsUnit (star y * y)
⊢ 0 < star v ⬝ᵥ (star y * y) *ᵥ v
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x y : Matrix n n 𝕜
hy : IsUnit y
hM : (star y * y).PosDef
__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace
hx : (xᴴᴴ * yᴴ * y * xᴴᴴᴴ).trace = 0
⊢ x = 0

Tactic:
← conjTranspose_mul,

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x y : Matrix n n 𝕜
hy : IsUnit y
hM : (star y * y).PosDef
__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace
hx : ((y * xᴴ)ᴴ * y * xᴴᴴᴴ).trace = 0
⊢ x = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b : A
h₁ : Commute a b
h₂ : Commute a (star b)
hab : Commute (abs a) (abs b)
⊢ sqrt (star (a * b) * (a * b)) = abs a * abs b

Tactic:
CFC.sqrt_eq_iff _ _ (star_mul_self_nonneg _)
    (hab.mul_nonneg (abs_nonneg a) (abs_nonneg b)),

Goal state after:
A : Type u_2
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b : A
h₁ : Commute a b
h₂ : Commute a (star b)
hab : Commute (abs a) (abs b)
⊢ abs a * abs b * (abs a * abs b) = star (a * b) * (a * b)
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: case neg.inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
P : ℕ → X →L[𝕜] X
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
i j : ℕ
x : X
h : ¬i = j
h' : i > j
⊢ (P (min i j + 1)) x - (P (min i (j + 1))) x - ((P (min (i + 1) j)) x - (P (min i j)) x) = 0

Tactic:
· rw [min_eq_right_of_lt h', min_eq_right (Nat.succ_le_of_lt h'),
        min_eq_right_of_lt (Nat.lt_succ_of_lt h')]
      abel

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
f : ↥(lp (fun x ↦ 𝕜) 2)
i : ι
this✝ : NormedSpace 𝕜 ↥(lp (fun _i ↦ 𝕜) 2) := inferInstance
this : lp.single 2 i (↑f i) = ↑f i • lp.single 2 i 1
⊢ b.repr (↑f i • b i) = b.repr (↑b.repr.symm ((fun i ↦ lp.single 2 i (↑f i)) i))

Tactic:
rw [map_smul, b.repr_self, ← this, LinearIsometryEquiv.coe_toContinuousLinearEquiv]

Goal state after:
ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
f : ↥(lp (fun x ↦ 𝕜) 2)
i : ι
this✝ : NormedSpace 𝕜 ↥(lp (fun _i ↦ 𝕜) 2) := inferInstance
this : lp.single 2 i (↑f i) = ↑f i • lp.single 2 i 1
⊢ lp.single 2 i (↑f i) = b.repr (b.repr.symm ((fun i ↦ lp.single 2 i (↑f i)) i))
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
⊢ HasSum (fun i ↦ ↑(fourierBasis.repr f) i • fourierLp 2 i) f

Tactic:
rw [← coe_fourierBasis]

Goal state after:
T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
⊢ HasSum (fun i ↦ ↑(fourierBasis.repr f) i • fourierBasis i) f
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
h : Summable (fourierCoeff ⇑f)
⊢ HasSum (fun i ↦ fourierCoeff (⇑f) i • fourier i) f

Tactic:
have sum_L2 := hasSum_fourier_series_L2 (toLp (E := ℂ) 2 haarAddCircle ℂ f)

Goal state after:
T : ℝ
hT : Fact (0 < T)
f : C(AddCircle T, ℂ)
h : Summable (fourierCoeff ⇑f)
sum_L2 : HasSum (fun i ↦ fourierCoeff (↑↑((toLp 2 haarAddCircle ℂ) f)) i • fourierLp 2 i) ((toLp 2 haarAddCircle ℂ) f)
⊢ HasSum (fun i ↦ fourierCoeff (⇑f) i • fourier i) f
Module: Mathlib.MeasureTheory.Integral.PeakFunction

Goal state before:
pp: α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : BorelSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst✝³] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε
δpos : 0 < δ
δone : δ < 1
u : Set α
u_open : IsOpen[inst✝³] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball 0 δ
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < δ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < δ
hφpos : ∀ x ∈ s, 0 ≤ φ i x
h''i : IntegrableOn (fun x ↦ φ i x • g x) s μ
I : IntegrableOn (φ i) t μ
⊢ (∫ (x : α) in t, φ i x ∂μ) * δ ≤ 2 * δ

Tactic:
gcongr; linarith [(le_abs_self _).trans h'i.le]

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
x : E
⊢ ((smulLeftCLM F fun x ↦ ↑(g x)) f) x = ((smulLeftCLM F g) f) x

Tactic:
smulLeftCLM_apply_apply (by fun_prop),

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
x : E
⊢ ↑(g x) • f x = ((smulLeftCLM F g) f) x
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
x : E
⊢ ↑(g x) • f x = ((smulLeftCLM F g) f) x

Tactic:
smulLeftCLM_apply_apply (by fun_prop),

Goal state after:
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
𝕜' : Type u_10
inst✝¹ : RCLike 𝕜'
inst✝ : NormedSpace 𝕜' F
g : E → ℝ
hg : Function.HasTemperateGrowth g
f : 𝓢(E, F)
x : E
⊢ ↑(g x) • f x = g x • f x
Module: Mathlib.Analysis.SpecialFunctions.ImproperIntegrals

Goal state before:
pp: case e'_9
c : ℝ
hc : 1 < c
t : ℝ
a✝ : t ∈ Ici c
⊢ t⁻¹ / log t ^ 2 = -(-t⁻¹ / log t ^ 2)

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case left
s : ℝ
h : 0 < s
⊢ IntegrableOn (fun x ↦ rexp (-x) * x ^ (s - 1)) (Ioc 0 1) volume

Tactic:
rw [← integrableOn_Icc_iff_integrableOn_Ioc]

Goal state after:
case left
s : ℝ
h : 0 < s
⊢ IntegrableOn (fun x ↦ rexp (-x) * x ^ (s - 1)) (Icc 0 1) volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p b : ℝ
hb : 0 < b
hp : 1 < p
this : Tendsto (fun x ↦ x * (b * x ^ (p - 1) + -1)) atTop atTop
x : ℝ
hx : 0 < x
⊢ x * (b * (x ^ p / x) + -1) = -x - -b * x ^ p

Tactic:
field

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝¹ : NormedCommRing 𝕜
E : Fin 0 → Type u_3
mE : (i : Fin 0) → MeasurableSpace (E i)
μ : (i : Fin 0) → Measure (E i)
inst✝ : ∀ (i : Fin 0), SigmaFinite (μ i)
f : (i : Fin 0) → E i → 𝕜
hf : ∀ (i : Fin 0), Integrable (f i) (μ i)
⊢ Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)

Tactic:
simp only [Finset.univ_eq_empty, Finset.prod_empty, isFiniteMeasure_iff,
      integrable_const_iff, pi_empty_univ, ENNReal.one_lt_top, or_true]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : b = 0
⊢ re 0 ≤ 0

Tactic:
zero_re

Goal state after:
b : ℂ
hb : b = 0
⊢ 0 ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: case e'_9
b : ℂ
hb : 0 < b.re
hb' : b ≠ 0
x : ℂ
⊢ x * cexp (-b * x ^ 2) = -(2 * b)⁻¹ * (cexp (-b * x ^ 2) * (-b * (↑2 * x ^ (2 - 1))))

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
X : ℝ
hX : 0 ≤ X
F_der_I :
  ∀ x ∈ Ioo 0 X,
    HasDerivAt (fun x ↦ ↑(rexp (-x)) * ↑x ^ s) (-(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) x
cont : Continuous ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s)
der_ible : IntervalIntegrable (fun x ↦ -(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) volume 0 X
int_eval :
  (∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ s) + -∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * (s * ↑x ^ (s - 1)) =
    -(((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) X - ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) 0)
hs : s = 0
⊢ re 0 ≤ 0

Tactic:
zero_re

Goal state after:
s : ℂ
X : ℝ
hX : 0 ≤ X
F_der_I :
  ∀ x ∈ Ioo 0 X,
    HasDerivAt (fun x ↦ ↑(rexp (-x)) * ↑x ^ s) (-(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) x
cont : Continuous ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s)
der_ible : IntervalIntegrable (fun x ↦ -(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) volume 0 X
int_eval :
  (∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ s) + -∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * (s * ↑x ^ (s - 1)) =
    -(((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) X - ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) 0)
hs : s = 0
⊢ 0 ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : b.re ≠ 0
c T : ℝ
⊢ b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2 = b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re)

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero.ha
h1 : 0 < re 0
⊢ False

Tactic:
zero_re

Goal state after:
case zero.ha
h1 : 0 < 0
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
h2 : s ≠ 0
n : ℕ := ⌊1 - s.re⌋₊
t1 : -s.re < ↑n
t2 : -(s + 1).re < ↑n
⊢ GammaAux n (s + 1) = s * (GammaAux n (s + 1) / s)

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero
⊢ Gamma (-↑0) = 0

Tactic:
rw [Nat.cast_zero, neg_zero, Gamma_zero]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
aux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
⊢ ∫ (t : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * (↑r * ↑t) ^ (a - 1) * cexp (-(↑r * ↑t)) =
    1 / ↑r * ∫ (t : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑t ^ (a - 1) * cexp (-↑t)

Tactic:
simp_rw [← ofReal_mul]

Goal state after:
a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
aux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
⊢ ∫ (t : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑(r * t) ^ (a - 1) * cexp (-↑(r * t)) =
    1 / ↑r * ∫ (t : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑t ^ (a - 1) * cexp (-↑t)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
c : ℂ
hc : c ∈ {c | 0 < c.re}
⊢ ¬↑π / c = 0 ∨ ¬1 / 2 ≠ 0

Tactic:
exact Or.inl (div_ne_zero (ofReal_ne_zero.mpr pi_ne_zero) (nv hc))

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b c d : ℂ
hb : b = 0
⊢ 0 ≤ re 0

Tactic:
zero_re

Goal state after:
b c d : ℂ
hb : b = 0
⊢ 0 ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : b.re < 0
c d : ℂ
hb' : b ≠ 0
x : ℝ
⊢ b * ↑x ^ 2 + c * ↑x + d = - -b * (↑x + c / (2 * b)) ^ 2 + (d - c ^ 2 / (4 * b))

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : b.re < 0
c d : ℂ
hb' : b ≠ 0
h : ∀ (x : ℝ), cexp (b * ↑x ^ 2 + c * ↑x + d) = cexp (- -b * (↑x + c / (2 * b)) ^ 2) * cexp (d - c ^ 2 / (4 * b))
⊢ (∫ (a : ℝ), cexp (- -b * (↑a + c / (2 * b)) ^ 2)) * cexp (d - c ^ 2 / (4 * b)) =
    (↑π / -b) ^ (1 / 2) * cexp (d - c ^ 2 / (4 * b))

Tactic:
← re_add_im (c / (2 * b))

Goal state after:
b : ℂ
hb : b.re < 0
c d : ℂ
hb' : b ≠ 0
h : ∀ (x : ℝ), cexp (b * ↑x ^ 2 + c * ↑x + d) = cexp (- -b * (↑x + c / (2 * b)) ^ 2) * cexp (d - c ^ 2 / (4 * b))
⊢ (∫ (a : ℝ), cexp (- -b * (↑a + (↑(c / (2 * b)).re + ↑(c / (2 * b)).im * I)) ^ 2)) * cexp (d - c ^ 2 / (4 * b)) =
    (↑π / -b) ^ (1 / 2) * cexp (d - c ^ 2 / (4 * b))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b c : ℂ
hb : b = 0
⊢ re 0 ≤ 0

Tactic:
zero_re

Goal state after:
b c : ℂ
hb : b = 0
⊢ 0 ≤ 0
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
h :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
⊢ 𝓕 (𝓕⁻ f) = f

Tactic:
ext v
  exact hf.fourier_fourierInv_eq h'f h.continuousAt

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
x w : V
hb : b = 0
⊢ re 0 ≤ 0

Tactic:
zero_re

Goal state after:
b : ℂ
V : Type u_1
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
x w : V
hb : b = 0
⊢ 0 ≤ 0
Module: Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module ℝ V
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup W
inst✝¹ : Module ℝ W
inst✝ : TopologicalSpace W
e : AddChar ℝ Circle
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL : Continuous fun p ↦ (L p.1) p.2
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
⊢ (StarSubalgebra.map (toContinuousMapStarₐ ℂ) (charPoly he hL)).SeparatesPoints

Tactic:
intro v v' hvv'
  obtain ⟨w, hw⟩ : ∃ w, char he hL w v ≠ char he hL w v' := by
    contrapose! hvv'
    exact ext_of_char_eq he he' hL hL' hvv'
  use char he hL w
  simp only [StarSubalgebra.coe_toSubalgebra, StarSubalgebra.coe_map, Set.mem_image,
    SetLike.mem_coe, exists_exists_and_eq_and, ne_eq]
  exact ⟨⟨char he hL w, char_mem_charPoly w, rfl⟩, hw⟩

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

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
a : d → ℝ
this :
  Measure.map (fun x i ↦ ↑(x i)) (Measure.map Subtype.val (Measure.comap Subtype.val volume)) =
    Measure.map (fun x i ↦ ↑(↑x i)) (Measure.comap Subtype.val volume)
⊢ volume = Measure.map (⇑(measurableEquivPiIoc a).symm) (Measure.comap Subtype.val volume)

Tactic:
simp_rw [coe_symm_measurableEquivPiIoc, ← this]

Goal state after:
d : Type u_1
inst✝ : Fintype d
a : d → ℝ
this :
  Measure.map (fun x i ↦ ↑(x i)) (Measure.map Subtype.val (Measure.comap Subtype.val volume)) =
    Measure.map (fun x i ↦ ↑(↑x i)) (Measure.comap Subtype.val volume)
⊢ volume = Measure.map (fun x i ↦ ↑(x i)) (Measure.map Subtype.val (Measure.comap Subtype.val volume))
Module: Mathlib.Analysis.Fourier.AddCircleMulti

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
f : ↥(Lp ℂ 2 volume)
i : d → ℤ
⊢ ↑(mFourierBasis.repr f) i = ∫ (t : UnitAddTorus d), (starRingEnd ℂ) (↑↑(mFourierLp 2 i) t) * ↑↑f t

Tactic:
rw [mFourierBasis.repr_apply_apply f i, MeasureTheory.L2.inner_def, coe_mFourierBasis]

Goal state after:
d : Type u_1
inst✝ : Fintype d
f : ↥(Lp ℂ 2 volume)
i : d → ℤ
⊢ ∫ (a : UnitAddTorus d), inner ℂ (↑↑(mFourierLp 2 i) a) (↑↑f a) =
    ∫ (t : UnitAddTorus d), (starRingEnd ℂ) (↑↑(mFourierLp 2 i) t) * ↑↑f t
Module: Mathlib.Analysis.Fourier.AddCircleMulti

Goal state before:
pp: d : Type u_1
inst✝ : Fintype d
f g : ↥(Lp ℂ 2 volume)
⊢ HasSum (fun i ↦ mFourierCoeff (↑↑g) i * (starRingEnd ℂ) (mFourierCoeff (↑↑f) i))
    (∫ (t : UnitAddTorus d), ↑↑g t * (starRingEnd ℂ) (↑↑f t))

Tactic:
refine HasSum.congr_fun (mFourierBasis.hasSum_inner_mul_inner f g) (fun n ↦ ?_)

Goal state after:
d : Type u_1
inst✝ : Fintype d
f g : ↥(Lp ℂ 2 volume)
n : d → ℤ
⊢ mFourierCoeff (↑↑g) n * (starRingEnd ℂ) (mFourierCoeff (↑↑f) n) =
    inner ℂ f (mFourierBasis n) * inner ℂ (mFourierBasis n) g