mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: case e'_9
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
hf' : Integrable (fun x ↦ x • f x) volume
w : ℝ
hf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume
L : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := (ContinuousLinearMap.mul ℝ ℝ).flip
h_int : Integrable (fun v ↦ fourierSMulRight L f v) volume
⊢ ∫ (v : ℝ), 𝐞 (-(v * w)) • (-2 * ↑π * I * ↑v) • f v =
    ∫ (v : ℝ), 𝐞 (-((ContinuousLinearMap.toLinearMap₁₂ L) v) w) • (fourierSMulRight L f v) 1

Tactic:
]

Goal state after:
case e'_9
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
hf' : Integrable (fun x ↦ x • f x) volume
w : ℝ
hf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume
L : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := (ContinuousLinearMap.mul ℝ ℝ).flip
h_int : Integrable (fun v ↦ fourierSMulRight L f v) volume
⊢ ∫ (v : ℝ), 𝐞 (-(v * w)) • (-2 * ↑π * I * ↑v) • f v =
    ∫ (v : ℝ), 𝐞 (-((ContinuousLinearMap.toLinearMap₁₂ L) v) w) • (fourierSMulRight L f v) 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
x : ℝ
hx : x ∈ Ioi 0
⊢ ↑(rexp (-x)) * ↑x ^ (↑s - 1) = ↑(rexp (-x)) * ↑x ^ ↑(s - 1)

Tactic:
]

Goal state after:
s : ℝ
this : ∀ (r : ℝ), ↑r = ↑r
x : ℝ
hx : x ∈ Ioi 0
⊢ ↑(rexp (-x)) * ↑x ^ (↑s - 1) = ↑(rexp (-x)) * ↑x ^ ↑(s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
X : ℝ
hs : 0 < s.re
hX : 0 ≤ X
⊢ IntegrableOn (fun x ↦ ↑(rexp (-x)) * ↑x ^ (s - 1)) (Ioc 0 X) volume

Tactic:
]

Goal state after:
s : ℂ
X : ℝ
hs : 0 < s.re
hX : 0 ≤ X
⊢ IntegrableOn (fun x ↦ ↑(rexp (-x)) * ↑x ^ (s - 1)) (Ioc 0 X) volume
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
x : ℝ
I : Integrable (fun x ↦ fderiv ℝ f x) volume
this : 𝓕 (deriv f) x = (𝓕 (fderiv ℝ f) x) 1
⊢ (fourierSMulRight (-innerSL ℝ) (𝓕 f) x) 1 = (2 * ↑π * Complex.I * ↑x) • 𝓕 f x

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
x : ℝ
I : Integrable (fun x ↦ fderiv ℝ f x) volume
this : 𝓕 (deriv f) x = (𝓕 (fderiv ℝ f) x) 1
⊢ (fourierSMulRight (-innerSL ℝ) (𝓕 f) x) 1 = (2 * ↑π * Complex.I * ↑x) • 𝓕 f x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ IntegrableOn (fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))) (Ioc 0 Y) volume

Tactic:
]

Goal state after:
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ IntegrableOn (fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))) (Ioc 0 Y) volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p s : ℝ
hs : -1 < s
hp : 0 < p
ht : 0 < (s + 1) / p
x : ℝ
hx : x ∈ Ioi 0
⊢ x ^ (p - 1 + p * ((s + 1) / p - 1)) * rexp (-x ^ p) = x ^ s * rexp (-x ^ p)

Tactic:
]

Goal state after:
p s : ℝ
hs : -1 < s
hp : 0 < p
ht : 0 < (s + 1) / p
x : ℝ
hx : x ∈ Ioi 0
⊢ x ^ (p - 1 + p * ((s + 1) / p - 1)) * rexp (-x ^ p) = x ^ s * rexp (-x ^ p)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: x y : ℝ
a✝ : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
⊢ √(x ^ 2 + y ^ 2) = √(x ^ 2 + y ^ 2)

Tactic:
]

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

Goal state before:
pp: p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
⊢ IntegrableOn (fun x ↦ (b ^ (-p⁻¹) * x) ^ s * rexp (-b * (b ^ (-p⁻¹) * x) ^ p)) (Ioi 0) volume

Tactic:
]

Goal state after:
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
⊢ IntegrableOn (fun x ↦ (b ^ (-p⁻¹) * x) ^ s * rexp (-b * (b ^ (-p⁻¹) * x) ^ p)) (Ioi 0) volume
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ HasFiniteIntegral (fun a ↦ ‖s * (↑(rexp (-a)) * ↑a ^ (s - 1))‖) (volume.restrict (Ioc 0 Y))

Tactic:
]

Goal state after:
case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ HasFiniteIntegral (fun a ↦ ‖s * (↑(rexp (-a)) * ↑a ^ (s - 1))‖) (volume.restrict (Ioc 0 Y))
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
N : ℕ∞
n : ℕ
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun x ↦ x ^ n • f x) volume
hn : ↑n ≤ N
x : ℝ
A : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) volume
B : AEStronglyMeasurable f volume
⊢ (∫ (v : ℝ), 𝐞 (-inner ℝ v x) • (fourierPowSMulRight (innerSL ℝ) f v n) fun x ↦ 1) =
    ∫ (v : ℝ), 𝐞 (-inner ℝ v x) • (-2 * ↑π * I * ↑v) ^ n • f v

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
N : ℕ∞
n : ℕ
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun x ↦ x ^ n • f x) volume
hn : ↑n ≤ N
x : ℝ
A : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) volume
B : AEStronglyMeasurable f volume
⊢ (∫ (v : ℝ), 𝐞 (-inner ℝ v x) • (fourierPowSMulRight (innerSL ℝ) f v n) fun x ↦ 1) =
    ∫ (v : ℝ), 𝐞 (-inner ℝ v x) • (-2 * ↑π * I * ↑v) ^ n • f v
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ ∀ᵐ (x : ℝ), x ∈ Ioc 0 Y → rexp (-x) * x ^ (s.re - 1) = ‖↑(rexp (-x))‖ * ‖↑x ^ (s - 1)‖

pp: case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ MeasurableSet (Ioc 0 Y)

Tactic:
]

Goal state after:
case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ ∀ᵐ (x : ℝ), x ∈ Ioc 0 Y → rexp (-x) * x ^ (s.re - 1) = ‖↑(rexp (-x))‖ * ‖↑x ^ (s - 1)‖

case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
⊢ MeasurableSet (Ioc 0 Y)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
x : ℝ
hx : x ∈ Ioc 0 Y
⊢ rexp (-x) * x ^ (s.re - 1) = ‖↑(rexp (-x))‖ * ‖↑x ^ (s - 1)‖

Tactic:
Complex.norm_of_nonneg (exp_pos _).le,

Goal state after:
case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
x : ℝ
hx : x ∈ Ioc 0 Y
⊢ rexp (-x) * x ^ (s.re - 1) = rexp (-x) * ‖↑x ^ (s - 1)‖
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ (b ^ (-p⁻¹)) ^ s * x✝ ^ s * rexp (-x✝ ^ p) = (b ^ (-p⁻¹)) ^ s * x✝ ^ s * rexp (-x✝ ^ p)

pp: case h
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ b ≠ 0

pp: p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ p ≠ 0

pp: case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b

pp: case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b ^ (-p⁻¹)

pp: case hy
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ x✝

pp: case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b ^ (-p⁻¹)

pp: case hy
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ x✝

Tactic:
]

Goal state after:
case h
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ b ≠ 0

p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ p ≠ 0

case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b

case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b ^ (-p⁻¹)

case hy
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ x✝

case hx
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ b ^ (-p⁻¹)

case hy
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
this : IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * rexp (-x ^ p))) (Ioi 0) volume
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ 0 ≤ x✝
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
x : ℝ
hx : x ∈ Ioc 0 Y
⊢ rexp (-x) * x ^ (s.re - 1) = rexp (-x) * x ^ (s - 1).re

Tactic:
]

Goal state after:
case right
s : ℂ
hs : 0 < s.re
Y : ℝ
hY : 0 ≤ Y
this : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))
x : ℝ
hx : x ∈ Ioc 0 Y
⊢ rexp (-x) * x ^ (s.re - 1) = rexp (-x) * x ^ (s - 1).re
Module: Mathlib.Analysis.Fourier.FourierTransformDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
N : ℕ∞
n✝ : ℕ
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedDeriv n f) volume
hn✝ : ↑n✝ ≤ N
x : ℝ
n : ℕ
hn : ↑n ≤ N
⊢ Integrable (⇑(ContinuousMultilinearMap.piFieldEquiv ℝ (Fin n) E) ∘ iteratedDeriv n f) volume

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
N : ℕ∞
n✝ : ℕ
hf : ContDiff ℝ (↑N) f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedDeriv n f) volume
hn✝ : ↑n✝ ≤ N
x : ℝ
n : ℕ
hn : ↑n ≤ N
⊢ Integrable (⇑(ContinuousMultilinearMap.piFieldEquiv ℝ (Fin n) E) ∘ iteratedDeriv n f) volume
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
⊢ ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ s =
    (s * ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ (s - 1)) - ↑(rexp (-X)) * ↑X ^ s

Tactic:
]

Goal state after:
s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
⊢ ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ s =
    (s * ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ (s - 1)) - ↑(rexp (-X)) * ↑X ^ s
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-x ^ p)) (Ioi 0) volume

Tactic:
]

Goal state after:
p s b : ℝ
hs : -1 < s
hp : 0 < p
hb : 0 < b
hib : 0 < b ^ (-p⁻¹)
⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-x ^ p)) (Ioi 0) volume
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: p : ℝ × ℝ
⊢ HasFDerivAt (↑polarCoord.symm)
    ((cos p.2 • ContinuousLinearMap.fst ℝ ℝ ℝ + (-p.1 * sin p.2) • ContinuousLinearMap.snd ℝ ℝ ℝ).prod
      (sin p.2 • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 * cos p.2) • ContinuousLinearMap.snd ℝ ℝ ℝ))
    p

Tactic:
]

Goal state after:
p : ℝ × ℝ
⊢ HasFDerivAt (↑polarCoord.symm)
    ((cos p.2 • ContinuousLinearMap.fst ℝ ℝ ℝ + (-p.1 * sin p.2) • ContinuousLinearMap.snd ℝ ℝ ℝ).prod
      (sin p.2 • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 * cos p.2) • ContinuousLinearMap.snd ℝ ℝ ℝ))
    p
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-b * x ^ 2)) (Iio 0) volume ∧
    IntegrableOn (fun x ↦ x ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume

Tactic:
]

Goal state after:
b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-b * x ^ 2)) (Iio 0) volume ∧
    IntegrableOn (fun x ↦ x ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ IntegrableOn ((fun x ↦ x ^ s * rexp (-b * x ^ 2)) ∘ Neg.neg) (Neg.neg ⁻¹' Iio 0) volume

Tactic:
]

Goal state after:
b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
⊢ IntegrableOn ((fun x ↦ x ^ s * rexp (-b * x ^ 2)) ∘ Neg.neg) (Neg.neg ⁻¹' Iio 0) volume
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝¹ : NormedCommRing 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] {f : (i : Fin n) → E i → 𝕜},
    (∀ (i : Fin n), Integrable (f i) (μ i)) → Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
hf : ∀ (i : Fin (n + 1)), Integrable (f i) (μ i)
this :
  MeasurePreserving (⇑(MeasurableEquiv.piFinSuccAbove E 0).symm) ((μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)))
    (Measure.pi μ)
⊢ Integrable ((fun x ↦ ∏ i, f i (x i)) ∘ ⇑(MeasurableEquiv.piFinSuccAbove E 0).symm)
    ((μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)))

Tactic:
]

Goal state after:
case succ
𝕜 : Type u_1
inst✝¹ : NormedCommRing 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] {f : (i : Fin n) → E i → 𝕜},
    (∀ (i : Fin n), Integrable (f i) (μ i)) → Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
hf : ∀ (i : Fin (n + 1)), Integrable (f i) (μ i)
this :
  MeasurePreserving (⇑(MeasurableEquiv.piFinSuccAbove E 0).symm) ((μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)))
    (Measure.pi μ)
⊢ Integrable ((fun x ↦ ∏ i, f i (x i)) ∘ ⇑(MeasurableEquiv.piFinSuccAbove E 0).symm)
    ((μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
this : MeasurableSet (Ioi 0)
x : ℝ
hx : x ∈ Ioi 0
h'x : 0 ≤ x
⊢ |(-x) ^ s| * rexp (-b * x ^ 2) ≤ x ^ s * rexp (-b * x ^ 2)

Tactic:
]

Goal state after:
b : ℝ
hb : 0 < b
s : ℝ
hs : -1 < s
this : MeasurableSet (Ioi 0)
x : ℝ
hx : x ∈ Ioi 0
h'x : 0 ≤ x
⊢ |(-x) ^ s| * rexp (-b * x ^ 2) ≤ x ^ s * rexp (-b * x ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.snd ℝ ℝ ℝ).ker
⊢ ¬LinearMap.snd ℝ ℝ ℝ = 0

Tactic:
]

Goal state after:
A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.snd ℝ ℝ ℝ).ker
⊢ ¬LinearMap.snd ℝ ℝ ℝ = 0
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
x : ℝ
⊢ rexp (-b.re * x ^ 2) = rexp (-b.re * x ^ 2)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: A : polarCoord.sourceᶜ ⊆ ↑(LinearMap.snd ℝ ℝ ℝ).ker
h : LinearMap.snd ℝ ℝ ℝ = 0
⊢ 0 (0, 1) = 0 (0, 1)

Tactic:
]

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ HasFiniteIntegral (fun a ↦ ‖cexp (-b * ↑a ^ 2)‖) volume

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
⊢ HasFiniteIntegral (fun a ↦ ‖cexp (-b * ↑a ^ 2)‖) volume
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
⊢ ∫ (x : ℝ × ℝ) in univ, f x = ∫ (p : ℝ × ℝ) in polarCoord.source, f p

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
⊢ ∫ (x : ℝ × ℝ) in univ, f x = ∫ (p : ℝ × ℝ) in polarCoord.source, f p
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] (f : (i : Fin n) → E i → 𝕜),
    ∫ (x : (i : Fin n) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
⊢ ∫ (x : E 0 × ((j : Fin n) → E (Fin.succAbove 0 j))),
      ∏ i, f i ((MeasurableEquiv.piFinSuccAbove E 0).symm x i) ∂(μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)) =
    ∫ (x : E 0 × ((i : Fin n) → E i.succ)), f 0 x.1 * ∏ i, f i.succ (x.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ)

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] (f : (i : Fin n) → E i → 𝕜),
    ∫ (x : (i : Fin n) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
⊢ ∫ (x : E 0 × ((j : Fin n) → E (Fin.succAbove 0 j))),
      ∏ i, f i ((MeasurableEquiv.piFinSuccAbove E 0).symm x i) ∂(μ 0).prod (Measure.pi fun j ↦ μ (Fin.succAbove 0 j)) =
    ∫ (x : E 0 × ((i : Fin n) → E i.succ)), f 0 x.1 * ∏ i, f i.succ (x.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ)
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
⊢ ∫ (x : ℝ × ℝ) in polarCoord.target, |(fderivPolarCoordSymm x).det| • f (↑polarCoord.symm x) =
    ∫ (p : ℝ × ℝ) in polarCoord.target, |p.1| • f (↑polarCoord.symm p)

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
⊢ ∫ (x : ℝ × ℝ) in polarCoord.target, |(fderivPolarCoordSymm x).det| • f (↑polarCoord.symm x) =
    ∫ (p : ℝ × ℝ) in polarCoord.target, |p.1| • f (↑polarCoord.symm p)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: case refine_2
b : ℂ
hb : 0 < b.re
this : HasFiniteIntegral (fun a ↦ ‖a * rexp (-b.re * a ^ 2)‖) volume
⊢ HasFiniteIntegral (fun a ↦ ‖↑a * cexp (-b * ↑a ^ 2)‖) volume

Tactic:
]

Goal state after:
case refine_2
b : ℂ
hb : 0 < b.re
this : HasFiniteIntegral (fun a ↦ ‖a * rexp (-b.re * a ^ 2)‖) volume
⊢ HasFiniteIntegral (fun a ↦ ‖↑a * cexp (-b * ↑a ^ 2)‖) volume
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ × ℝ → E
x : ℝ × ℝ
hx : x ∈ polarCoord.target
⊢ x.1 • f (↑polarCoord.symm x) = x.1 • f (↑polarCoord.symm x)

Tactic:
]

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
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)
⊢ ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ s =
    (s * ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ (s - 1)) - ↑(rexp (-X)) * ↑X ^ s

Tactic:
]

Goal state after:
s : ℂ
hs : 0 < s.re
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)
⊢ ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ s =
    (s * ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ (s - 1)) - ↑(rexp (-X)) * ↑X ^ s
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
⊢ ∫⁻ (x : ℝ × ℝ) in polarCoord.source, f x = ∫⁻ (p : ℝ × ℝ) in polarCoord.source, f p

Tactic:
]

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
this : HasFiniteIntegral (fun a ↦ ‖a * rexp (-b.re * a ^ 2)‖) volume
x✝ : ℝ
⊢ ‖x✝‖ * rexp (-b.re * x✝ ^ 2) = ‖x✝‖ * rexp (-b.re * x✝ ^ 2)

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : ℕ
n_ih :
  ∀ {E : Fin n → Type u_3} {mE : (i : Fin n) → MeasurableSpace (E i)} {μ : (i : Fin n) → Measure (E i)}
    [∀ (i : Fin n), SigmaFinite (μ i)] (f : (i : Fin n) → E i → 𝕜),
    ∫ (x : (i : Fin n) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i
E : Fin (n + 1) → Type u_3
mE : (i : Fin (n + 1)) → MeasurableSpace (E i)
μ : (i : Fin (n + 1)) → Measure (E i)
inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)
f : (i : Fin (n + 1)) → E i → 𝕜
⊢ ∫ (x : E 0 × ((i : Fin n) → E i.succ)), f 0 x.1 * ∏ i, f i.succ (x.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ) =
    ∫ (z : E 0 × ((i : Fin n) → E i.succ)), f 0 z.1 * ∏ i, f i.succ (z.2 i) ∂(μ 0).prod (Measure.pi fun i ↦ μ i.succ)

Tactic:
]

Goal state after: