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

Goal state before:
pp: z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
⊢ (2 * ↑n + 1) / (2 * ↑n + 2) * (1 / 2) * ∫ (x : ℝ) in 0..π, sin x ^ (2 * n) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * (1 / 2) * ∫ (x : ℝ) in 0..π, sin x ^ (2 * n)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case inr.succ
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
⊢ ↑π * z * A * B / ↑C =
    (↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * (n + 1))) /
      ↑((2 * ↑n + 1) / (2 * ↑n + 2) * C)

Tactic:
]

Goal state after:
case inr.succ
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
⊢ ↑π * z * A * B / ↑C =
    (↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * (n + 1))) /
      ↑((2 * ↑n + 1) / (2 * ↑n + 2) * C)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_1
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
⊢ Tendsto (fun n ↦ f x - f 1 - x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f x - f 1))

Tactic:
have : f x - f 1 = f x - f 1 - x * 0 := by ring

Goal state after:
case refine_1
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
this : f x - f 1 = f x - f 1 - x * 0
⊢ Tendsto (fun n ↦ f x - f 1 - x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f x - f 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_1
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
this : f x - f 1 = f x - f 1 - x * 0
⊢ Tendsto (fun n ↦ f x - f 1 - x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f x - f 1 - x * 0))

Tactic:
]

Goal state after:
case refine_1
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
this : f x - f 1 = f x - f 1 - x * 0
⊢ Tendsto (fun n ↦ f x - f 1 - x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f x - f 1 - x * 0))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_2
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
n : ℕ
⊢ f x ≤ f 1 + (x * (log (↑n + 1) - log ↑n) + logGammaSeq x n)

Tactic:
]

Goal state after:
case refine_2
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
n : ℕ
⊢ f x ≤ f 1 + (x * (log (↑n + 1) - log ↑n) + logGammaSeq x n)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case refine_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
⊢ 0 < x

Tactic:
]

Goal state after:
case refine_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
⊢ 0 < x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case inr.succ
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this✝ : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
⊢ ↑π * z * A * B / ↑C =
    ↑π * z * A *
        ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ)) /
      ↑((2 * ↑n + 1) / (2 * ↑n + 2) * C)

Tactic:
]

Goal state after:
case inr.succ
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this✝ : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
⊢ ↑π * z * A * B / ↑C =
    ↑π * z * A *
        ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ)) /
      ↑((2 * ↑n + 1) / (2 * ↑n + 2) * C)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_1.inl
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
this : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
h✝ : x < 1
⊢ 0 < x

Tactic:
]

Goal state after:
case refine_1.inl
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
this : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
h✝ : x < 1
⊢ 0 < x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this✝¹ : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this✝ :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
this :
  (1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * B
⊢ ↑π * z * A * B / ↑C = ↑π * z * A * ((2 * ↑n + 1) / (2 * ↑n + 2) * B) / (↑(2 * ↑n + 1) / ↑(2 * ↑n + 2) * ↑C)

Tactic:
]

Goal state after:
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this✝¹ : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this✝ :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
this :
  (1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * B
⊢ ↑π * z * A * B / ↑C = ↑π * z * A * ((2 * ↑n + 1) / (2 * ↑n + 2) * B) / (↑(2 * ↑n + 1) / ↑(2 * ↑n + 2) * ↑C)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_2
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
this : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
⊢ x - 1 ≤ ↑⌈x - 1⌉₊

Tactic:
]

Goal state after:
case refine_2
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
this : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
⊢ x - 1 ≤ ↑⌈x - 1⌉₊
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case refine_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
U : HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x
⊢ x < 1

Tactic:
]

Goal state after:
case refine_1
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
x : ℝ
hx : x ∈ Ioo 0 1
U : HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x
⊢ x < 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case zero
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
x : ℝ
hx : 0 < x
⊢ 0 < x → x ≤ 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
]

Goal state after:
case zero
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
x : ℝ
hx : 0 < x
⊢ 0 < x → x ≤ 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
hm : ∀ {x : ℝ}, 0 < x → ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
x : ℝ
hx : 0 < x
hy : ↑(m + 1) < x
hy' : x - 1 ≤ ↑m + 1
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
]

Goal state after:
case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
hm : ∀ {x : ℝ}, 0 < x → ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
x : ℝ
hx : 0 < x
hy : ↑(m + 1) < x
hy' : x - 1 ≤ ↑m + 1
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
hm : ∀ {x : ℝ}, 0 < x → ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
]

Goal state after:
case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
hm : ∀ {x : ℝ}, 0 < x → ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
n : ℕ
hn : 1 ≤ n
this : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1))
⊢ logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)

Tactic:
]

Goal state after:
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
n : ℕ
hn : 1 ≤ n
this : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1))
⊢ logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
n : ℕ
hn : 1 ≤ n
this : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1))
⊢ logGammaSeq (x - 1) n =
    logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1)) +
        x * (log (↑(n - 1) + 1) - log ↑(n - 1)) -
      log (x - 1)

Tactic:
]

Goal state after:
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
n : ℕ
hn : 1 ≤ n
this : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1))
⊢ logGammaSeq (x - 1) n =
    logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1)) +
        x * (log (↑(n - 1) + 1) - log ↑(n - 1)) -
      log (x - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v

Tactic:
]

Goal state after:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_5
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this✝ : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
⊢ ↑n + 1 = ↑n + 1

Tactic:
]

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

Goal state before:
pp: u v : ℂ
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = F 1 - F 0
hu : u = 0
⊢ 0 ≤ 0

Tactic:
]

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

Goal state before:
pp: f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm :
  Tendsto
    ((fun x_1 ↦ logGammaSeq x (x_1 - 1) + x * (log (↑(x_1 - 1) + 1) - log ↑(x_1 - 1)) - log (x - 1) + log (x - 1)) ∘
      fun a ↦ a + 1)
    atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
n : ℕ
⊢ logGammaSeq x n + x * (log (↑n + 1) - log ↑n) = logGammaSeq x n + x * (log (↑n + 1) - log ↑n)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
h4 : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), ∀ z ∈ Metric.ball s v, ‖F' z t‖ ≤ bound t
h5 : IntegrableOn bound (Ioi 0) volume
t : ℝ
ht : t ∈ Ioi 0
y : ℂ
x✝ : y ∈ Metric.ball s v
ht' : ↑t ≠ 0
⊢ ↑t ^ (y - 1) * Complex.log ↑t = ↑t ^ (y - 1) * Complex.log ↑t * 1

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
h4 : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), ∀ z ∈ Metric.ball s v, ‖F' z t‖ ≤ bound t
h5 : IntegrableOn bound (Ioi 0) volume
t : ℝ
ht : t ∈ Ioi 0
y : ℂ
x✝ : y ∈ Metric.ball s v
ht' : ↑t ≠ 0
⊢ ↑t ^ (y - 1) * Complex.log ↑t = ↑t ^ (y - 1) * Complex.log ↑t * 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
]

Goal state after:
case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = F 1 - F 0
hF0 : F 0 = 0
hv : v = 0
⊢ 0 ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
⊢ 0 < 1

Tactic:
]

Goal state after:
f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
⊢ 0 < 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
⊢ 0 ≤ π / 2

Tactic:
]

Goal state after:
f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
⊢ 0 ≤ π / 2
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: s : ℂ
aux1 : -1 < (s - 1).re
hs : s = 0
⊢ 0 ≤ 0

Tactic:
]

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

Goal state before:
pp: case e'_5
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝¹ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this✝ :
  ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
this : f x = f (x - 1) + log (x - 1)
⊢ f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0

Tactic:
]

Goal state after:
case e'_5
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝¹ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this✝ :
  ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
this : f x = f (x - 1) + log (x - 1)
⊢ f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
A :
  Tendsto
    (fun n ↦
      ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
          ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
        ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)))
    atTop (𝓝 (Complex.sin (↑π * z)))
⊢ 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z))

Tactic:
]

Goal state after: