mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
⊢ ∀ x ∈ Ioo 0 X,
    HasDerivAt (fun x ↦ ↑(rexp (-x)) * ↑x ^ s) (-(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) x

Tactic:
by

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

Goal state before:
pp: p : ℝ × ℝ
⊢ HasFDerivAt (↑polarCoord.symm) (fderivPolarCoordSymm p) p

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: b : ℝ
hb : 0 < b
⊢ -1 < 0

Tactic:
by

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

Goal state before:
pp: b : ℝ
hb : 0 < b
⊢ Integrable (fun x ↦ rexp (-b * x ^ 2)) volume

Tactic:
by

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

Goal state before:
pp: b : ℝ
h : IntegrableOn (fun x ↦ rexp (-b * x ^ 2)) (Ioi 0) volume
hb : b ≤ 0
⊢ ∫⁻ (x : ℝ) in Ioi 0, 1 ≤ ∫⁻ (x : ℝ) in Ioi 0, ↑‖rexp (-b * x ^ 2)‖₊

Tactic:
by

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

Goal state before:
pp: b : ℝ
⊢ IntegrableOn (fun x ↦ rexp (-b * x ^ 2)) (Ioi 0) volume ↔ 0 < b

Tactic:
by

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

Goal state before:
pp: p : ℝ × ℝ
⊢ (fderivPolarCoordSymm p).det = p.1

Tactic:
by

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

Goal state before:
pp: b : ℝ
hb : 0 < b
⊢ -1 < 1

Tactic:
by

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

Goal state before:
pp: b : ℝ
hb : 0 < b
⊢ Integrable (fun x ↦ x * rexp (-b * x ^ 2)) volume

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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
⊢ (LinearMap.snd ℝ ℝ ℝ) (0, 1) = 0 (0, 1)

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: ⊢ polarCoord.source =ᵐ[volume] univ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : NormedCommRing 𝕜
E : ι → Type u_3
f : (i : ι) → E i → 𝕜
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hf : ∀ (i : ι), Integrable (f i) (μ i)
⊢ Integrable (fun x ↦ ∏ i, f i (x i)) (Measure.pi μ)

Tactic:
by

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

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

Tactic:
by

Goal state after:
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
⊢ ∫ (p : ℝ × ℝ), f p = ∫ (p : ℝ × ℝ) in polarCoord.source, f p

Tactic:
by

Goal state after:
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
⊢ ∫ (p : ℝ × ℝ) in polarCoord.source, f p = ∫ (p : ℝ × ℝ) in polarCoord.target, |p.1| • f (↑polarCoord.symm p)

Tactic:
by

Goal state after:
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
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, |p.1| • f (↑polarCoord.symm p) =
    ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p)

Tactic:
by

Goal state after:
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
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p) = ∫ (p : ℝ × ℝ), f p

Tactic:
by

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 : (i : Fin (n + 1)) → E i), ∏ i, f i (x i) ∂Measure.pi μ =
    ∫ (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:
by

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

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
⊢ ∫⁻ (p : ℝ × ℝ), f p = ∫⁻ (p : ℝ × ℝ) in ↑polarCoord.symm '' polarCoord.target, f p

Tactic:
by

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) =
    (∫ (x : E 0), f 0 x ∂μ 0) * ∏ i, ∫ (x : E i.succ), f i.succ x ∂μ i.succ

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ b ≠ 0

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
hb' : b ≠ 0
⊢ ∀ (x : ℂ), HasDerivAt (fun x ↦ -(2 * b)⁻¹ * cexp (-b * x ^ 2)) (x * cexp (-b * x ^ 2)) x

Tactic:
by

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), f 0 x ∂μ 0) * ∏ i, ∫ (x : E i.succ), f i.succ x ∂μ i.succ = ∏ i, ∫ (x : E i), f i x ∂μ i

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
hb' : b ≠ 0
A : ∀ (x : ℂ), HasDerivAt (fun x ↦ -(2 * b)⁻¹ * cexp (-b * x ^ 2)) (x * cexp (-b * x ^ 2)) x
⊢ Tendsto (fun y ↦ -(2 * b)⁻¹ * cexp (-b * ↑y ^ 2)) atTop (𝓝 (-(2 * b)⁻¹ * 0))

Tactic:
by

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

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
⊢ ∫⁻ (p : ℝ × ℝ) in ↑polarCoord.symm '' polarCoord.target, f p =
    ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal |p.1| • f (↑polarCoord.symm p)

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ ∫ (r : ℝ) in Ioi 0, ↑r * cexp (-b * ↑r ^ 2) = (2 * b)⁻¹

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : ℕ
E : Fin n → Type u_3
mE : (i : Fin n) → MeasurableSpace (E i)
μ : (i : Fin n) → Measure (E i)
inst✝ : ∀ (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

Tactic:
by

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

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
⊢ ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal |p.1| • f (↑polarCoord.symm p) =
    ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑polarCoord.symm p)

Tactic:
by

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

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
⊢ ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑polarCoord.symm p) = ∫⁻ (p : ℝ × ℝ), f p

Tactic:
by

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

Goal state before:
pp: a : ℂ
⊢ ↑Complex.polarCoord a = (‖a‖, a.arg)

Tactic:
by

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

Goal state before:
pp: p : ℝ × ℝ
⊢ ↑Complex.polarCoord.symm p = ↑p.1 * (↑(Real.cos p.2) + ↑(Real.sin p.2) * I)

Tactic:
by

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

Goal state before:
pp: p : ℝ × ℝ
⊢ ‖↑Complex.polarCoord.symm p‖ = |p.1|

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ (∫ (x : ℝ), cexp (-b * ↑x ^ 2)) ^ 2 = ∫ (p : ℝ × ℝ), cexp (-b * ↑p.1 ^ 2) * cexp (-b * ↑p.2 ^ 2)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : RCLike 𝕜
E : ι → Type u_3
f : (i : ι) → E i → 𝕜
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
⊢ ∫ (x : (i : ι) → E i), ∏ i, f i (x i) ∂Measure.pi μ = ∏ i, ∫ (x : E i), f i x ∂μ i

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : RCLike 𝕜
E : Type u_3
f : E → 𝕜
mE : MeasurableSpace E
μ : Measure E
inst✝ : SigmaFinite μ
⊢ (∫ (x : ι → E), ∏ i, f (x i) ∂Measure.pi fun x ↦ μ) = (∫ (x : E), f x ∂μ) ^ card ι

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : ∀ (i : ι), IsFiniteMeasure (μ i)
i : ι
f : X i → E
hf : Integrable f (μ i)
⊢ Measurable fun x ↦ x i

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ ∫ (p : ℝ × ℝ), cexp (-b * ↑p.1 ^ 2) * cexp (-b * ↑p.2 ^ 2) = ∫ (p : ℝ × ℝ), cexp (-b * (↑p.1 ^ 2 + ↑p.2 ^ 2))

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : ∀ (i : ι), IsFiniteMeasure (μ i)
i : ι
f : X i → E
hf : Integrable f (μ i)
⊢ ∀ a ∈ Finset.univ.erase i, (μ a) Set.univ ≠ ⊤

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : ∀ (i : ι), IsFiniteMeasure (μ i)
i : ι
f : X i → E
hf : Integrable f (μ i)
⊢ Integrable (fun x ↦ f (x i)) (Measure.pi μ)

Tactic:
by

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

Goal state before:
pp: f : ℂ → ℝ≥0∞
⊢ ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑Complex.polarCoord.symm p) = ∫⁻ (p : ℂ), f p

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
p : ι → ℝ × ℝ
⊢ HasFDerivAt (fun x i ↦ ↑polarCoord.symm (x i)) (fderivPiPolarCoordSymm p) p

Tactic:
by

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)
⊢ s ≠ 0

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝³ : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
f : X i → E
hf : AEStronglyMeasurable f (μ i)
⊢ Measurable (Function.eval i)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
p : ι → ℝ × ℝ
⊢ (fderivPiPolarCoordSymm p).det = ∏ i, (p i).1

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝³ : Fintype ι
X : ι → Type u_3
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)
i : ι
f : X i → E
hf : AEStronglyMeasurable f (μ i)
⊢ ∫ (x : (i : ι) → X i), f (x i) ∂Measure.pi μ = ∫ (x : X i), f x ∂μ i

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
⊢ ((Pi.map fun x ↦ ↑polarCoord.symm) '' Set.univ.pi fun x ↦ polarCoord.target) =ᵐ[volume] Set.univ

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ ∫ (p : ℝ × ℝ), cexp (-b * (↑p.1 ^ 2 + ↑p.2 ^ 2)) =
    ∫ (p : ℝ × ℝ) in polarCoord.target,
      p.1 • cexp (-b * ((↑p.1 * Complex.cos ↑p.2) ^ 2 + (↑p.1 * Complex.sin ↑p.2) ^ 2))

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
this : Tendsto (s + 1).partialGamma atTop (𝓝 (s * s.GammaIntegral))
⊢ (s + 1).GammaIntegral = s * s.GammaIntegral

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target,
      p.1 • cexp (-b * ((↑p.1 * Complex.cos ↑p.2) ^ 2 + (↑p.1 * Complex.sin ↑p.2) ^ 2)) =
    (∫ (r : ℝ) in Ioi 0, ↑r * cexp (-b * ↑r ^ 2)) * ∫ (θ : ℝ) in Ioo (-π) π, 1

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ -π ≤ π

Tactic:
by

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
⊢ (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : (ι → ℝ × ℝ) → E
⊢ (∫ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ polarCoord.target, (∏ i, (p i).1) • f fun i ↦ ↑polarCoord.symm (p i)) =
    ∫ (p : ι → ℝ × ℝ), f p

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ (∫ (r : ℝ) in Ioi 0, ↑r * cexp (-b * ↑r ^ 2)) * ∫ (θ : ℝ) in Ioo (-π) π, 1 = ↑π / b

Tactic:
by

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
this✝ : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
this : Tendsto (fun X ↦ -↑X ^ s * ↑(rexp (-X))) atTop (𝓝 0)
⊢ Tendsto (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) atTop (𝓝 (s * s.GammaIntegral))

Tactic:
by

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

Goal state before:
pp: b : ℂ
c T : ℝ
⊢ ‖cexp (-b * (↑T + ↑c * I) ^ 2)‖ = rexp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2))

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ (∫ (x : ℝ), cexp (-b * ↑x ^ 2)) ^ 2 = ↑π / b

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : (ι → ℂ) → E
⊢ (∫ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ Complex.polarCoord.target,
      (∏ i, (p i).1) • f fun i ↦ ↑Complex.polarCoord.symm (p i)) =
    ∫ (p : ι → ℂ), f p

Tactic:
by

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

Goal state before:
pp: b : ℝ
hb : 0 < b
⊢ 0 ≤ ∫ (x : ℝ), rexp (-b * x ^ 2)

Tactic:
by

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

Goal state before:
pp: b : ℝ
hb : 0 < b
⊢ 0 ≤ √(π / b)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
this : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
⊢ (fun e ↦ ‖-↑e ^ s * ↑(rexp (-e))‖) =ᶠ[atTop] fun e ↦ e ^ s.re * rexp (-1 * e)

Tactic:
by

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

Goal state before:
pp: s : ℂ
hs : 0 < s.re
⊢ (s + 1).GammaIntegral = s * s.GammaIntegral

Tactic:
by

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

Goal state before:
pp: b : ℝ
hb : 0 < b
⊢ 0 < (↑b).re

Tactic:
by

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

Goal state before:
pp: b : ℝ
⊢ ∫ (x : ℝ), rexp (-b * x ^ 2) = √(π / b)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℝ × ℝ) → ℝ≥0∞
⊢ (∫⁻ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ polarCoord.target,
      (∏ i, ENNReal.ofReal (p i).1) * f fun i ↦ ↑polarCoord.symm (p i)) =
    ∫⁻ (p : ι → ℝ × ℝ), f p

Tactic:
by

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

Goal state before:
pp: n : ℕ
hn : ∀ (s : ℂ), -s.re < ↑n → GammaAux n s = GammaAux n (s + 1) / s
s : ℂ
h1 : -s.re < ↑(n + 1)
⊢ -(s + 1).re < ↑n

Tactic:
by

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

Goal state before:
pp: s : ℂ
n : ℕ
h1 : -s.re < ↑n
⊢ GammaAux n s = GammaAux n (s + 1) / s

Tactic:
by

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

Goal state before:
pp: s : ℂ
n : ℕ
h1 : -s.re < ↑(n + 1)
⊢ -(s + 1).re < ↑n

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℂ) → ℝ≥0∞
⊢ (∫⁻ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ Complex.polarCoord.target,
      (∏ i, ENNReal.ofReal (p i).1) * f fun i ↦ ↑Complex.polarCoord.symm (p i)) =
    ∫⁻ (p : ι → ℂ), f p

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x ↦ cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
⊢ ∀ᶠ (c : ℂ) in 𝓝 b, ∀ᵐ (x : ℝ), ‖f c x‖ ≤ rexp (-d * x ^ 2)

Tactic:
by

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

Goal state before:
pp: s : ℂ
n : ℕ
h1 : -s.re < ↑(n + 1)
⊢ GammaAux n (s + 1 + 1) / (s + 1) = GammaAux n (s + 1)

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x ↦ cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_le_bd : ∀ᶠ (c : ℂ) in 𝓝 b, ∀ᵐ (x : ℝ), ‖f c x‖ ≤ rexp (-d * x ^ 2)
⊢ ∀ (x : ℂ), AEStronglyMeasurable (fun x_1 ↦ cexp (-x * ↑x_1 ^ 2)) volume

Tactic:
by

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

Goal state before:
pp: s : ℂ
n : ℕ
h1 : -s.re < ↑n
⊢ GammaAux n s = GammaAux (n + 1) s

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
f : ℂ → ℝ → ℂ := fun c x ↦ cexp (-c * ↑x ^ 2)
d : ℝ
hd : 0 < d
hd' : d < b.re
f_le_bd : ∀ᶠ (c : ℂ) in 𝓝 b, ∀ᵐ (x : ℝ), ‖f c x‖ ≤ rexp (-d * x ^ 2)
⊢ ∀ (a : ℝ), ContinuousAt (fun x ↦ cexp (-x * ↑a ^ 2)) b

Tactic:
by

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

Goal state before:
pp: V : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : V → E
hf : Integrable f volume
c : ℝ
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun a ↦ cexp (-↑c⁻¹ * ↑‖a‖ ^ 2)

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ ContinuousAt (fun c ↦ ∫ (x : ℝ), cexp (-c * ↑x ^ 2)) b

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c T : ℝ
hT : 0 ≤ T
⊢ ∀ {T : ℝ},
    0 ≤ T →
      ∀ {c y : ℝ},
        |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))

Tactic:
by

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ ∀ {b : ℂ}, 0 < b.re → b ≠ 0

Tactic:
by

Goal state after: