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

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

Tactic:
]

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

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

Tactic:
]

Goal state after:
s : ℂ
n : ℕ
h1 : -s.re < ↑n + 1
⊢ -(s + 1).re < ↑n
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
c : ℂ
hc : c ∈ re ⁻¹' Ioi d
x : ℝ
⊢ rexp (-c.re * x ^ 2) ≤ rexp (-d * x ^ 2)

Tactic:
]

Goal state after:
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
c : ℂ
hc : c ∈ re ⁻¹' Ioi d
x : ℝ
⊢ rexp (-c.re * x ^ 2) ≤ rexp (-d * x ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

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

Tactic:
]

Goal state after:
s : ℂ
n : ℕ
h1 : -s.re < ↑n + 1
⊢ -(s.re + 1) < ↑n
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

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

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ GammaAux (⌊1 - s.re⌋₊ + k + 1) s = GammaAux (⌊1 - s.re⌋₊ + k) s

Tactic:
]

Goal state after:
case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ GammaAux (⌊1 - s.re⌋₊ + k + 1) s = GammaAux (⌊1 - s.re⌋₊ + k) s
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ |b.im| * |y| ≤ |b.im| * |c|

Tactic:
]

Goal state after:
case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ |b.im| * |y| ≤ |b.im| * |c|
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ -s.re < ↑⌊1 - s.re⌋₊ + ↑k

Tactic:
]

Goal state after:
case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ -s.re < ↑⌊1 - s.re⌋₊ + ↑k
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ |y| ≤ |c|

Tactic:
]

Goal state after:
case hbc
b : ℂ
hb : 0 < b.re
c✝ T✝ : ℝ
hT✝ : 0 ≤ T✝
T : ℝ
hT : 0 ≤ T
c y : ℝ
hy : |y| ≤ |c|
⊢ |y| ≤ |c|
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

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

Tactic:
]

Goal state after:
b✝ : ℂ
hb✝ : 0 < b✝.re
b : ℂ
hb : b = 0
⊢ re 0 ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
i0 : -s.re < ↑⌊1 - s.re⌋₊
⊢ 0 ≤ k

Tactic:
]

Goal state after:
case succ
s : ℂ
n : ℕ
h1 : -s.re < ↑n
k : ℕ
hk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
i0 : -s.re < ↑⌊1 - s.re⌋₊
⊢ 0 ≤ k
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ n = n

pp: s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ ⌊1 - s.re⌋₊ ≤ n

Tactic:
]

Goal state after:
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
⊢ ⌊1 - s.re⌋₊ ≤ n
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c T : ℝ
hT : 0 ≤ T
vert_norm_bound :
  ∀ {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))
⊢ ?m.420 * |c| ≤ 2 * |c| * 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
vert_norm_bound :
  ∀ {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))
⊢ ?m.420 * |c| ≤ 2 * |c| * 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 neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 0 ≤ n

pp: case neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 1 - s.re ≤ 0

Tactic:
]

Goal state after:
case neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 0 ≤ n

case neg
s : ℂ
n : ℕ
h1 : -s.re < ↑n
u : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s
h : ¬0 ≤ 1 - s.re
⊢ 1 - s.re ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b✝ : ℂ
hb✝ : 0 < b✝.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
b : ℂ
hb : b ∈ {c | 0 < c.re}
⊢ 0 < π * b.re / normSq b

Tactic:
]

Goal state after:
b✝ : ℂ
hb✝ : 0 < b✝.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
b : ℂ
hb : b ∈ {c | 0 < c.re}
⊢ 0 < π * b.re / normSq b
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

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

Tactic:
]

Goal state after:
s : ℂ
h2 : s ≠ 0
n : ℕ := ⌊1 - s.re⌋₊
t1 : -s.re < ↑n
⊢ -(s.re + 1) < ↑n
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:
]

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

Goal state before:
pp: ⊢ GammaIntegral 1 = 1

pp: ⊢ 0 < re 1

Tactic:
]

Goal state after:
⊢ GammaIntegral 1 = 1

⊢ 0 < re 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
n : ℕ
hn : Gamma (↑n + 1) = ↑n !
⊢ ↑n.succ * Gamma ↑n.succ = ↑(n + 1)!

Tactic:
]

Goal state after:
case succ
n : ℕ
hn : Gamma (↑n + 1) = ↑n !
⊢ ↑n.succ * Gamma ↑n.succ = ↑(n + 1)!
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: ⊢ GammaAux 1 0 = 0

Tactic:
GammaAux,

Goal state after:
⊢ (0 + 1).GammaIntegral / 0 = 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: n : ℕ
IH : Gamma (-↑n) = 0
⊢ n.succ ≠ 0

Tactic:
]

Goal state after:
n : ℕ
IH : Gamma (-↑n) = 0
⊢ n.succ ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
n : ℕ
IH : -↑n.succ * Gamma (-↑n.succ) = 0
A : -↑n.succ ≠ 0
this : -↑n = -↑n.succ + 1
⊢ Gamma (-↑(n + 1)) = 0

Tactic:
]

Goal state after:
case succ
n : ℕ
IH : -↑n.succ * Gamma (-↑n.succ) = 0
A : -↑n.succ ≠ 0
this : -↑n = -↑n.succ + 1
⊢ Gamma (-↑(n + 1)) = 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case zero
s : ℂ
⊢ ∀ (s : ℂ), ((starRingEnd ℂ) s).GammaIntegral = (starRingEnd ℂ) s.GammaIntegral

Tactic:
]

Goal state after:
case zero
s : ℂ
⊢ ∀ (s : ℂ), ((starRingEnd ℂ) s).GammaIntegral = (starRingEnd ℂ) s.GammaIntegral
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (fun s ↦ GammaAux n (s + 1) / s) ((starRingEnd ℂ) s) = (starRingEnd ℂ) ((fun s ↦ GammaAux n (s + 1) / s) s)

Tactic:
]

Goal state after:
case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (fun s ↦ GammaAux n (s + 1) / s) ((starRingEnd ℂ) s) = (starRingEnd ℂ) ((fun s ↦ GammaAux n (s + 1) / s) s)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux n ((starRingEnd ℂ) s + 1) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s

Tactic:
]

Goal state after:
case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ GammaAux n ((starRingEnd ℂ) s + 1) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
this : (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) (s + 1)
⊢ (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s = (starRingEnd ℂ) (GammaAux n (s + 1)) / (starRingEnd ℂ) s

Tactic:
]

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

Goal state before:
pp: case succ
s✝ : ℂ
n : ℕ
IH : ∀ (s : ℂ), GammaAux n ((starRingEnd ℂ) s) = (starRingEnd ℂ) (GammaAux n s)
s : ℂ
⊢ (starRingEnd ℂ) s + 1 = (starRingEnd ℂ) s + 1

Tactic:
]

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

Goal state before:
pp: a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
⊢ (1 / ↑r) ^ a = (1 / ↑r) ^ a

Tactic:
]

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

Goal state before:
pp: case e'_3
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
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
c : ℝ
w : V
⊢ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪w, v⟫) • 𝓕 f w = cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + ↑(2 * π * ⟪w, v⟫) * I) • 𝓕 f w

Tactic:
]

Goal state after:
case e'_3
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
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
this : Integrable (fun w ↦ 𝐞 ⟪w, v⟫ • 𝓕 f w) volume
c : ℝ
w : V
⊢ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪w, v⟫) • 𝓕 f w = cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + ↑(2 * π * ⟪w, v⟫) * I) • 𝓕 f w
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)
x : ℝ
hx : 0 < x
⊢ ↑x ^ (a - 1) * cexp (-(↑r * ↑x)) = (1 / ↑r) ^ (a - 1) * (↑r * ↑x) ^ (a - 1) * cexp (-(↑r * ↑x))

Tactic:
]

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ ∫ (x : ℝ), rexp (-1 * x ^ 2) = (π / 1) ^ (1 / 2)

pp: case hx
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ 0 ≤ π / 1

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ ∫ (x : ℝ), rexp (-1 * x ^ 2) = (π / 1) ^ (1 / 2)

case hx
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ 0 ≤ π / 1
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)
x : ℝ
hx : 0 < x
⊢ ↑x ^ (a - 1) * cexp (-(↑r * ↑x)) = ↑x ^ (a - 1) * cexp (-(↑r * ↑x))

Tactic:
]

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ ↑(∫ (x : ℝ), rexp (-1 * x ^ 2)) = ↑(π / 1) ^ ↑(1 / 2)

Tactic:
rw [← ofReal_cpow, ofReal_inj]

Goal state after:
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ ∫ (x : ℝ), rexp (-1 * x ^ 2) = (π / 1) ^ (1 / 2)

case hx
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ 0 ≤ π / 1
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c T : ℝ
hT : 0 ≤ T
vert_norm_bound :
  ∀ {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))
y : ℝ
hy : y ∈ uIoc 0 c
absy : |y| ≤ |c|
⊢ ‖cexp (-b * (↑T + ↑y * I) ^ 2) - cexp (-b * (↑T - ↑y * I) ^ 2)‖ ≤
    rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 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
vert_norm_bound :
  ∀ {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))
y : ℝ
hy : y ∈ uIoc 0 c
absy : |y| ≤ |c|
⊢ ‖cexp (-b * (↑T + ↑y * I) ^ 2) - cexp (-b * (↑T - ↑y * I) ^ 2)‖ ≤
    rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2)) +
      rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral

Goal state before:
pp: b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ (π / 1) ^ (1 / 2) = (π / 1) ^ (1 / 2)

Tactic:
]

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

Goal state before:
pp: case hx
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ 0 ≤ π

Tactic:
]

Goal state after:
case hx
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
this : ∀ (x : ℝ), cexp (-1 * ↑x ^ 2) = ↑(rexp (-1 * x ^ 2))
⊢ 0 ≤ π
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c T : ℝ
hT : 0 ≤ T
vert_norm_bound :
  ∀ {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))
y : ℝ
hy : y ∈ uIoc 0 c
absy : |(-y)| ≤ |c|
⊢ ‖cexp (-b * (↑T - ↑y * I) ^ 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
vert_norm_bound :
  ∀ {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))
y : ℝ
hy : y ∈ uIoc 0 c
absy : |(-y)| ≤ |c|
⊢ ‖cexp (-b * (↑T - ↑y * I) ^ 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: a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
aux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
⊢ 1 / ↑r * ∫ (x : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑x ^ (a - 1) * cexp (-↑x) =
    1 / ↑r * ∫ (t : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑t ^ (a - 1) * cexp (-↑t)

Tactic:
]

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun x ↦ ‖verticalIntegral b c x‖) atTop (𝓝 0)

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun x ↦ ‖verticalIntegral b c x‖) atTop (𝓝 0)
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 = (↑π / c) ^ (1 / 2) * (↑π / c) ^ (1 / 2)

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
c : ℂ
hc : c ∈ {c | 0 < c.re}
⊢ ↑π / c = (↑π / c) ^ (1 / 2) * (↑π / c) ^ (1 / 2)
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ 0 = 2 * |c| * 0

Tactic:
by ring

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

Goal state before:
pp: b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun T ↦ 2 * |c| * rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))) atTop (𝓝 (2 * |c| * 0))

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
c : ℝ
⊢ Tendsto (fun T ↦ 2 * |c| * rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))) atTop (𝓝 (2 * |c| * 0))
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) ^ 1 = (↑π / c) ^ (1 / 2 + 1 / 2)

Tactic:
]

Goal state after:
b : ℂ
hb : 0 < b.re
nv : ∀ {b : ℂ}, 0 < b.re → b ≠ 0
c : ℂ
hc : c ∈ {c | 0 < c.re}
⊢ (↑π / c) ^ 1 = (↑π / c) ^ (1 / 2 + 1 / 2)
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)
⊢ 1 / ↑r * (1 / ↑r) ^ (a - 1) * ∫ (t : ℝ) in Ioi 0, ↑t ^ (a - 1) * cexp (-↑t) =
    1 / ↑r * (1 / ↑r) ^ (a - 1) * a.GammaIntegral

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: case e_a.e_f
a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
aux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
x : ℝ
⊢ cexp (-↑x) * ↑x ^ (a - 1) = cexp (-↑x) * ↑x ^ (a - 1)

Tactic:
]

Goal state after:
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:
]

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

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1) = ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)

Tactic:
]

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

Goal state before:
pp: s : ℝ
hs : s ≠ 0
⊢ s * (Complex.Gamma ↑s).re = s * (Complex.Gamma ↑s).re

pp: case h2
s : ℝ
hs : s ≠ 0
⊢ ↑s ≠ 0

Tactic:
]

Goal state after:
case h2
s : ℝ
hs : s ≠ 0
⊢ ↑s ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: case h2
s : ℝ
hs : s ≠ 0
⊢ s ≠ 0

Tactic:
]

Goal state after:
case h2
s : ℝ
hs : s ≠ 0
⊢ s ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: ⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: s : ℝ
⊢ Complex.Gamma ↑s = Complex.Gamma ↑s

Tactic:
]

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

Goal state before:
pp: n : ℕ
⊢ ↑n ! = ↑n !

Tactic:
]

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

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ 0 < ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)

Tactic:
]

Goal state after:
s : ℝ
hs : 0 < s
⊢ 0 < ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ Ioi 0 ⊆ Function.support fun x ↦ rexp (-x) * x ^ (s - 1)

Tactic:
]

Goal state after:
s : ℝ
hs : 0 < s
⊢ Ioi 0 ⊆ Function.support fun x ↦ rexp (-x) * x ^ (s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
hs : 0 < s
x : ℝ
hx : x ∈ Ioi 0
⊢ rexp (-x) * x ^ (s - 1) ≠ 0

Tactic:
]

Goal state after:
s : ℝ
hs : 0 < s
x : ℝ
hx : x ∈ Ioi 0
⊢ rexp (-x) * x ^ (s - 1) ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ 0 < volume ((Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0)

pp: case hf
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ 0 ≤ᵐ[volume.restrict (Ioi 0)] fun x ↦ rexp (-x) * x ^ (s - 1)

pp: case hfi
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ IntegrableOn (fun x ↦ rexp (-x) * x ^ (s - 1)) (Ioi 0) volume

Tactic:
]

Goal state after:
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ 0 < volume ((Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0)

case hf
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ 0 ≤ᵐ[volume.restrict (Ioi 0)] fun x ↦ rexp (-x) * x ^ (s - 1)

case hfi
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ IntegrableOn (fun x ↦ rexp (-x) * x ^ (s - 1)) (Ioi 0) volume
Module: Mathlib.Analysis.Fourier.Inversion

Goal state before:
pp: case e_f
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
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
A : Tendsto (fun c ↦ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
B :
  Tendsto (fun c ↦ ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
c : ℝ
hc : 0 < c
w : V
⊢ ((↑π / ↑c⁻¹) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖v - w‖ ^ 2 / ↑c⁻¹)) • f w =
    ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w

Tactic:
]

Goal state after:
case e_f
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
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
A : Tendsto (fun c ↦ ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
B :
  Tendsto (fun c ↦ ∫ (w : V), 𝓕 (fun w ↦ cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))
c : ℝ
hc : 0 < c
w : V
⊢ ((↑π / ↑c⁻¹) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖v - w‖ ^ 2 / ↑c⁻¹)) • f w =
    ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Basic

Goal state before:
pp: s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ ENNReal.ofReal 0 < ⊤

Tactic:
]

Goal state after:
s : ℝ
hs : 0 < s
this : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0
⊢ ENNReal.ofReal 0 < ⊤