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

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

pp: case hf
f : ℝ × ℝ → ℝ≥0∞
⊢ InjOn (↑polarCoord.symm) polarCoord.target

pp: f : ℝ × ℝ → ℝ≥0∞
⊢ MeasurableSet polarCoord.target

Tactic:
]

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

case hf
f : ℝ × ℝ → ℝ≥0∞
⊢ InjOn (↑polarCoord.symm) polarCoord.target

f : ℝ × ℝ → ℝ≥0∞
⊢ MeasurableSet polarCoord.target
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 =
    (∫ (x : E 0), f 0 x ∂μ 0) * ∏ i, ∫ (x : E i.succ), f i.succ x ∂μ i.succ

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: f : ℝ × ℝ → ℝ≥0∞
x : ℝ × ℝ
hx : x ∈ polarCoord.target
⊢ ENNReal.ofReal x.1 • f (↑polarCoord.symm x) = ENNReal.ofReal x.1 • f (↑polarCoord.symm x)

Tactic:
]

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

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

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
⊢ ∫ (z : ℝ × ℝ), cexp (-b * ↑z.1 ^ 2) * cexp (-b * ↑z.2 ^ 2) ∂volume.prod volume =
    ∫ (p : ℝ × ℝ), cexp (-b * ↑p.1 ^ 2) * cexp (-b * ↑p.2 ^ 2)
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)
e : Fin (card ι) ≃ ι := (equivFin ι).symm
⊢ (∫ (x : (b : Fin (card ι)) → E (e b)),
      ∏ i, f i ((MeasurableEquiv.piCongrLeft E e) x i) ∂Measure.pi fun i' ↦ μ (e i')) =
    ∏ i, ∫ (x : E i), f i x ∂μ i

Tactic:
]

Goal state after:
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)
e : Fin (card ι) ≃ ι := (equivFin ι).symm
⊢ (∫ (x : (b : Fin (card ι)) → E (e b)),
      ∏ i, f i ((MeasurableEquiv.piCongrLeft E e) x i) ∂Measure.pi fun i' ↦ μ (e i')) =
    ∏ i, ∫ (x : E i), f i x ∂μ i
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)) * (s * ↑x ^ (s - 1))) + ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) 0 -
      ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) X =
    (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)) * (s * ↑x ^ (s - 1))) + ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) 0 -
      ((ofReal ∘ fun y ↦ rexp (-y)) * fun x ↦ ↑x ^ s) X =
    (s * ∫ (x : ℝ) in 0..X, ↑(rexp (-x)) * ↑x ^ (s - 1)) - ↑(rexp (-X)) * ↑X ^ s
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 : ℝ × ℝ) in polarCoord.target, p.1 • f (measurableEquivRealProd.symm (↑polarCoord.symm p))

Tactic:
]

Goal state after:
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 : ℝ × ℝ) in polarCoord.target, p.1 • f (measurableEquivRealProd.symm (↑polarCoord.symm p))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

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

Tactic:
]

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 : E), f x ∂μ) ^ Finset.univ.card = (∫ (x : E), f x ∂μ) ^ Finset.univ.card

Tactic:
]

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 f ((∏ j ∈ Finset.univ.erase i, (μ j) Set.univ) • μ i)

Tactic:
]

Goal state after:
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 f ((∏ j ∈ Finset.univ.erase i, (μ j) Set.univ) • μ i)
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 : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (measurableEquivRealProd.symm (↑polarCoord.symm p))

Tactic:
]

Goal state after:
f : ℂ → ℝ≥0∞
⊢ ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑Complex.polarCoord.symm p) =
    ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (measurableEquivRealProd.symm (↑polarCoord.symm p))
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 * (↑(↑polarCoord.symm p).1 ^ 2 + ↑(↑polarCoord.symm 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:
]

Goal state after:
b : ℂ
hb : 0 < b.re
⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • cexp (-b * (↑(↑polarCoord.symm p).1 ^ 2 + ↑(↑polarCoord.symm 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))
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
⊢ 0 ≤ 0

Tactic:
]

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 : ι) → X x), f (Function.eval i x) ∂Measure.pi μ

pp: case hφ
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)
⊢ AEMeasurable (Function.eval i) (Measure.pi μ)

pp: case hfm
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)
⊢ AEStronglyMeasurable f (map (Function.eval i) (Measure.pi μ))

Tactic:
]

Goal state after:
case hφ
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)
⊢ AEMeasurable (Function.eval i) (Measure.pi μ)

case hfm
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)
⊢ AEStronglyMeasurable f (map (Function.eval i) (Measure.pi μ))
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
p : ι → ℝ × ℝ
this : Fintype ι
⊢ ∀ (i : ι), HasFDerivAt (fun x ↦ ↑polarCoord.symm (x i)) (fderivPolarCoordSymm (p i) ∘SL ContinuousLinearMap.proj i) p

Tactic:
]

Goal state after:
ι : Type u_1
inst✝ : Finite ι
p : ι → ℝ × ℝ
this : Fintype ι
⊢ ∀ (i : ι), HasFDerivAt (fun x ↦ ↑polarCoord.symm (x i)) (fderivPolarCoordSymm (p i) ∘SL ContinuousLinearMap.proj i) p
Module: Mathlib.MeasureTheory.Integral.Pi

Goal state before:
pp: case hfm
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)
⊢ AEStronglyMeasurable f (μ i)

Tactic:
]

Goal state after:
case hfm
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)
⊢ AEStronglyMeasurable f (μ i)
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
⊢ (Set.univ.pi fun i ↦ polarCoord.source) =ᵐ[Measure.pi fun x ↦ volume] Set.pi ?m.45 fun i ↦ Set.univ

pp: ι : Type u_1
inst✝ : Fintype ι
⊢ Set ι

Tactic:
]

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
⊢ (Set.univ.pi fun i ↦ polarCoord.source) =ᵐ[Measure.pi fun x ↦ volume] Set.pi ?m.45 fun i ↦ Set.univ

ι : Type u_1
inst✝ : Fintype ι
⊢ Set ι
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)) =
    ∫ (z : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, ↑z.1 * cexp (-b * ↑z.1 ^ 2) * 1 ∂volume.prod volume

Tactic:
]

Goal state after:
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)) =
    ∫ (z : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, ↑z.1 * cexp (-b * ↑z.1 ^ 2) * 1 ∂volume.prod volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

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

Tactic:
]

Goal state after:
case e_f
b : ℂ
hb : 0 < b.re
p : ℝ × ℝ
⊢ p.1 • cexp (-b * ((↑p.1 * Complex.cos ↑p.2) ^ 2 + (↑p.1 * Complex.sin ↑p.2) ^ 2)) = ↑p.1 * cexp (-b * ↑p.1 ^ 2)
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)) =
    ∫ (x : ι → ℝ × ℝ) in (Pi.map fun x ↦ ↑polarCoord.symm) '' Set.univ.pi fun x ↦ polarCoord.target, f x

Tactic:
]

Goal state after:
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)) =
    ∫ (x : ι → ℝ × ℝ) in (Pi.map fun x ↦ ↑polarCoord.symm) '' Set.univ.pi fun x ↦ polarCoord.target, f x
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))
⊢ 0 < s.re + 1

Tactic:
]

Goal state after:
s : ℂ
hs : 0 < s.re
this : Tendsto (s + 1).partialGamma atTop (𝓝 (s * s.GammaIntegral))
⊢ 0 < s.re + 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
hs : 0 < s.re
X : ℝ
hX : X ∈ Ici 0
⊢ (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) X = s * s.partialGamma X - ↑(rexp (-X)) * ↑X ^ s

Tactic:
]

Goal state after:
s : ℂ
hs : 0 < s.re
X : ℝ
hX : X ∈ Ici 0
⊢ (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) X = s * s.partialGamma X - ↑(rexp (-X)) * ↑X ^ s
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

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

Tactic:
]

Goal state after:
b : ℂ
c T : ℝ
⊢ rexp (-((↑b.re + ↑b.im * I) * (↑T + ↑c * I) ^ 2).re) =
    rexp (-((↑b.re + ↑b.im * I).re * T ^ 2 - 2 * (↑b.re + ↑b.im * I).im * c * T - (↑b.re + ↑b.im * I).re * c ^ 2))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
⊢ (2 * b)⁻¹ * ↑(π - -π) = ↑π / b

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
⊢ (2 * b)⁻¹ * ↑(π - -π) = ↑π / b
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:

Tactic:
(∫ x : ℝ, cexp (-b * (x : ℂ) ^ 2)) ^ 2 =
        ∫ p : ℝ × ℝ, cexp (-b * (p.1 : ℂ) ^ 2) * cexp (-b * (p.2 : ℂ) ^ 2) := by
      rw [pow_two, ← integral_prod_mul]; rfl
    _ = ∫ p : ℝ × ℝ, cexp (-b * ((p.1 : ℂ) ^ 2 + (p.2 : ℂ) ^ 2)) := by
      congr
      ext1 p
      rw [← Complex.exp_add, mul_add]
    _ = ∫ p in polarCoord.target, p.1 •
        cexp (-b * ((p.1 * Complex.cos p.2) ^ 2 + (p.1 * Complex.sin p.2) ^ 2)) := by
      rw [← integral_comp_polarCoord_symm]
      simp only [polarCoord_symm_apply, ofReal_mul, ofReal_cos, ofReal_sin]
    _ = (∫ r in Ioi (0 : ℝ), r * cexp (-b * (r : ℂ) ^ 2)) * ∫ θ in Ioo (-π) π, 1 := by
      rw [← setIntegral_prod_mul]
      congr with p : 1
      rw [mul_one]
      congr
      conv_rhs => rw [← one_mul ((p.1 : ℂ) ^ 2), ← sin_sq_add_cos_sq (p.2 : ℂ)]
      ring
    _ = ↑π / b := by
      simp only [integral_const, MeasurableSet.univ, measureReal_restrict_apply,
        univ_inter, real_smul, mul_one, integral_mul_cexp_neg_mul_sq hb]
      rw [volume_real_Ioo_of_le (by linarith [pi_nonneg])]
      simp
      ring

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
⊢ Tendsto (fun x ↦ ‖-↑x ^ s * ↑(rexp (-x))‖) atTop (𝓝 0)

Tactic:
]

Goal state after:
s : ℂ
hs : 0 < s.re
this : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma
⊢ Tendsto (fun x ↦ ‖-↑x ^ s * ↑(rexp (-x))‖) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: case inl
b : ℝ
hb : b ≤ 0
⊢ 0 = 0

pp: case inl
b : ℝ
hb : b ≤ 0
⊢ π / b ≤ 0

pp: case inl
b : ℝ
hb : b ≤ 0
⊢ ¬Integrable (fun x ↦ rexp (-b * x ^ 2)) volume

Tactic:
]

Goal state after:
case inl
b : ℝ
hb : b ≤ 0
⊢ π / b ≤ 0

case inl
b : ℝ
hb : b ≤ 0
⊢ ¬Integrable (fun x ↦ rexp (-b * x ^ 2)) volume
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
e : (ι → ℝ × ℝ) ≃ᵐ (ι → ℂ) := MeasurableEquiv.piCongrRight fun x ↦ measurableEquivRealProd.symm
this : MeasurePreserving (fun a i ↦ measurableEquivRealProd.symm (a i)) volume volume
⊢ (∫ (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)) =
    ∫ (x : ι → ℝ × ℝ), f fun i ↦ measurableEquivRealProd.symm (x i) ∂volume

Tactic:
]

Goal state after:
ι : Type u_1
inst✝² : Fintype ι
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : (ι → ℂ) → E
e : (ι → ℝ × ℝ) ≃ᵐ (ι → ℂ) := MeasurableEquiv.piCongrRight fun x ↦ measurableEquivRealProd.symm
this : MeasurePreserving (fun a i ↦ measurableEquivRealProd.symm (a i)) volume volume
⊢ (∫ (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)) =
    ∫ (x : ι → ℝ × ℝ), f fun i ↦ measurableEquivRealProd.symm (x i) ∂volume
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

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

Tactic:
]

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)) =
    ∫⁻ (x : ι → ℝ × ℝ) in (Pi.map fun x ↦ ↑polarCoord.symm) '' Set.univ.pi fun x ↦ polarCoord.target, f x

Tactic:
]

Goal state after:
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)) =
    ∫⁻ (x : ι → ℝ × ℝ) in (Pi.map fun x ↦ ↑polarCoord.symm) '' Set.univ.pi fun x ↦ polarCoord.target, f x
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
x : ℝ
hx : x ∈ Ioi 0
⊢ x ^ s.re * rexp (-x) = x ^ s.re * rexp (-x)

Tactic:
]

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

Goal state before:
pp: case inr
b : ℝ
hb : 0 < b
⊢ (∫ (x : ℝ), ↑(rexp (-b * x ^ 2))) ^ 2 = ↑π / ↑b

Tactic:
]

Goal state after:
case inr
b : ℝ
hb : 0 < b
⊢ (∫ (x : ℝ), ↑(rexp (-b * x ^ 2))) ^ 2 = ↑π / ↑b
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero
s : ℂ
h1 : 0 < s.re
⊢ s.GammaIntegral = s * s.GammaIntegral / s

Tactic:
]

Goal state after:
case zero
s : ℂ
h1 : 0 < s.re
⊢ s.GammaIntegral = s * s.GammaIntegral / s
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

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

Tactic:
]

Goal state after:
b : ℝ
hb : 0 < b
⊢ 0 < b
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero
s : ℂ
h1 : 0 < s.re
⊢ s.GammaIntegral = s.GammaIntegral

pp: case zero.hb
s : ℂ
h1 : 0 < s.re
⊢ s ≠ 0

Tactic:
]

Goal state after:
case zero.hb
s : ℂ
h1 : 0 < s.re
⊢ s ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero.hb
s : ℂ
h1 : s = 0
⊢ re 0 ≤ 0

Tactic:
]

Goal state after:
case zero.hb
s : ℂ
h1 : s = 0
⊢ re 0 ≤ 0
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:
]

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

Goal state before:
pp: case e'_7
b : ℝ
hb : 0 < b
x : ℝ
⊢ cexp (-↑b * ↑x ^ 2) = cexp (-↑b * ↑x ^ 2)

Tactic:
]

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.re + 1) < ↑n

Tactic:
]

Goal state after:
n : ℕ
hn : ∀ (s : ℂ), -s.re < ↑n → GammaAux n s = GammaAux n (s + 1) / s
s : ℂ
h1 : -s.re < ↑n + 1
⊢ -(s.re + 1) < ↑n
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
n : ℕ
hn : ∀ (s : ℂ), -s.re < ↑n → GammaAux n s = GammaAux n (s + 1) / s
s : ℂ
h1 : -s.re < ↑(n + 1)
hh1 : -(s + 1).re < ↑n
⊢ GammaAux n (s + 1) / s = GammaAux n (s + 1) / s

Tactic:
]

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
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ rexp (-(b.re * T ^ 2 - 2 * b.im * y * T - b.re * y ^ 2)) ≤
    rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ rexp (-(b.re * T ^ 2 - 2 * b.im * y * T - b.re * y ^ 2)) ≤
    rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero
s : ℂ
h1 : 0 < s.re
⊢ s.GammaIntegral = s.GammaIntegral

pp: case zero.ha
s : ℂ
h1 : 0 < s.re
⊢ s ≠ 0

Tactic:
]

Goal state after:
case zero.ha
s : ℂ
h1 : 0 < s.re
⊢ s ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.PolarCoord

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℂ) → ℝ≥0∞
e : (ι → ℝ × ℝ) ≃ᵐ (ι → ℂ) := MeasurableEquiv.piCongrRight fun x ↦ measurableEquivRealProd.symm
this : MeasurePreserving (fun a i ↦ measurableEquivRealProd.symm (a i)) volume volume
⊢ (∫⁻ (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)) =
    ∫⁻ (a : ι → ℝ × ℝ), f fun i ↦ measurableEquivRealProd.symm (a i) ∂volume

Tactic:
]

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℂ) → ℝ≥0∞
e : (ι → ℝ × ℝ) ≃ᵐ (ι → ℂ) := MeasurableEquiv.piCongrRight fun x ↦ measurableEquivRealProd.symm
this : MeasurePreserving (fun a i ↦ measurableEquivRealProd.symm (a i)) volume volume
⊢ (∫⁻ (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)) =
    ∫⁻ (a : ι → ℝ × ℝ), f fun i ↦ measurableEquivRealProd.symm (a i) ∂volume