mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) =
    ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
suffices (C : ℝ) * γ = eLpNormLESNormFDerivOfEqInnerConst μ p by
        rw [eLpNorm_nnreal_eq_lintegral h0p]
        congr
        norm_cast at this ⊢

Goal state after:
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ ↑C * ↑γ = ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p)
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ mapL f g ∘SL ↑↑(commIsometry 𝕜 G E) = ↑↑(commIsometry 𝕜 H F) ∘SL mapL g f

Tactic:
ext; simp [map_comm]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup U
inst✝³ : InnerProductSpace 𝕜 U
inst✝² : FiniteDimensional 𝕜 U
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace 𝕜 V
f : U →ₗ[𝕜] V
h : f.ker ≠ ⊥
this : ¬Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range)
⊢ f.normDet = 0

Tactic:
simp [normDet, this]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : 0 < r
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
⊢ (fun r ↦ (mkMetric'.pre m r) (s ∪ t)) =ᶠ[𝓝[>] 0] fun x ↦ (mkMetric'.pre m x) s + (mkMetric'.pre m x) t

Tactic:
filter_upwards [Ioo_mem_nhdsGT r0]

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : 0 < r
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
⊢ ∀ a ∈ Ioo 0 r, (mkMetric'.pre m a) (s ∪ t) = (mkMetric'.pre m a) s + (mkMetric'.pre m a) t
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
⊢ ⊤ ≤ ⨆ r, ⨆ (_ : r > 0), ⊤

Tactic:
rw [le_iSup_iff]

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
⊢ ∀ (b : OuterMeasure X), (∀ (i : ℝ≥0∞), ⨆ (_ : i > 0), ⊤ ≤ b) → ⊤ ≤ b
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
⊢ inner_ 𝕜 (I • x) y = (starRingEnd 𝕜) I * inner_ 𝕜 x y

Tactic:
rw [conj_I, inner_, inner_, mul_left_comm, smul_smul, hI', neg_one_smul]

Goal state after:
case neg
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : I * I = -1
⊢ 4⁻¹ *
      (𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ + I * 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ -
        I * 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖) =
    4⁻¹ *
      (-I *
        (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
          I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖))
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
x y : E
⊢ inner_ 𝕜 (↑(re r) • x) y + inner_ 𝕜 ((↑(im r) * I) • x) y = (starRingEnd 𝕜) (↑(re r) + ↑(im r) * I) * inner_ 𝕜 x y

Tactic:
real_prop _ x,

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
x y : E
⊢ (starRingEnd 𝕜) ↑(re r) * inner_ 𝕜 x y + inner_ 𝕜 ((↑(im r) * I) • x) y =
    (starRingEnd 𝕜) (↑(re r) + ↑(im r) * I) * inner_ 𝕜 x y
Module: Mathlib.Analysis.LocallyConvex.WeakSpace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : Convex ℝ s
x : E
hx : (toWeakSpace 𝕜 E).toEquiv x ∈ (⇑(toWeakSpace 𝕜 E) '' closure s)ᶜ
⊢ (toWeakSpace 𝕜 E).toEquiv x ∈ (closure (⇑(toWeakSpace 𝕜 E) '' s))ᶜ

Tactic:
have : ContinuousSMul ℝ E := IsScalarTower.continuousSMul 𝕜

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : Convex ℝ s
x : E
hx : (toWeakSpace 𝕜 E).toEquiv x ∈ (⇑(toWeakSpace 𝕜 E) '' closure s)ᶜ
this : ContinuousSMul ℝ E
⊢ (toWeakSpace 𝕜 E).toEquiv x ∈ (closure (⇑(toWeakSpace 𝕜 E) '' s))ᶜ
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ ↑⟪x, y⟫ + (o.areaForm x) y • Complex.I = (starRingEnd ℂ) (↑⟪y, x⟫ + (o.areaForm y) x • Complex.I)

Tactic:
rw [real_inner_comm, areaForm_swap]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ ↑⟪y, x⟫ + -(o.areaForm y) x • Complex.I = (starRingEnd ℂ) (↑⟪y, x⟫ + (o.areaForm y) x • Complex.I)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this : EqOn (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) (Ioi 0)
⊢ ↑a ^ (1 - s) • a⁻¹ • ∫ (x : ℝ) in Ioi (a * 0), ↑x ^ (s - 1) • f x = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t

Tactic:
mul_zero,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this : EqOn (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) (Ioi 0)
⊢ ↑a ^ (1 - s) • a⁻¹ • ∫ (x : ℝ) in Ioi 0, ↑x ^ (s - 1) • f x = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
⊢ HasDerivAt (fun x ↦ x ^ (n + 1) / (↑n + 1)) (z ^ n) z

Tactic:
simp_rw [div_eq_mul_inv]

Goal state after:
z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
⊢ HasDerivAt (fun x ↦ x ^ (n + 1) * (↑n + 1)⁻¹) (z ^ n) z
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) -
        ↑(cos 0) ^ n * (Complex.sin (2 * z * 0) / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))

Tactic:
mul_zero,

Goal state after:
case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) - ↑(cos 0) ^ n * (Complex.sin 0 / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) - ↑(cos 0) ^ n * 0 -
      ∫ (x : ℝ) in 0..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))

Tactic:
mul_zero,

Goal state after:
case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) - 0 -
      ∫ (x : ℝ) in 0..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
⊢ (fun z ↦ log (1 + z) - logTaylor (n + 1) z) =O[𝓝 0] fun z ↦ z ^ (n + 1)

Tactic:
rw [Asymptotics.isBigO_iff]

Goal state after:
n : ℕ
⊢ ∃ c, ∀ᶠ (x : ℂ) in 𝓝 0, ‖log (1 + x) - logTaylor (n + 1) x‖ ≤ c * ‖x ^ (n + 1)‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2) =
    ↑(sin (π / 2)) * 0 * (-Complex.cos (2 * z * ↑(π / 2)) / (2 * z)) - 0 * (-Complex.cos (2 * z * 0) / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2,
        (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z))

pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0

Tactic:
mul_zero,

Goal state after:
case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2) =
    0 * (-Complex.cos (2 * z * ↑(π / 2)) / (2 * z)) - 0 * (-Complex.cos (2 * z * 0) / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2,
        (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z))

case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case e'_4
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))
⊢ (∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b =
    (∫ (a_1 : ℝ) in Ioi 0, f a s a_1 ^ (1 / a)) ^ (1 / (1 / a)) *
      (∫ (a : ℝ) in Ioi 0, f b t a ^ (1 / b)) ^ (1 / (1 / b))

Tactic:
· rw [one_div_one_div, one_div_one_div]
    congr 2 <;> exact setIntegral_congr_fun measurableSet_Ioi fun x hx => fpow (by assumption) _ hx

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

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
nne : ↑n ≠ 0
this :
  ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n =
    ↑n / (2 * z) *
      (-((↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)) +
        ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n)
⊢ (1 - 4 * z ^ 2 / ↑n ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n =
    (↑n - 1) / ↑n * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)

Tactic:
mul_add,

Goal state after:
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
nne : ↑n ≠ 0
this :
  ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n =
    ↑n / (2 * z) * -((↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)) +
      ↑n / (2 * z) * (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n)
⊢ (1 - 4 * z ^ 2 / ↑n ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n =
    (↑n - 1) / ↑n * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case succ
f : ℝ → ℝ
x : ℝ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
n : ℕ
hn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)
this : x + ↑n.succ = x + ↑n + 1
⊢ f x + ∑ m ∈ Finset.range n, log (x + ↑m) + log (x + ↑n) = f x + ∑ m ∈ Finset.range (n + 1), log (x + ↑m)

Tactic:
rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
      abel

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

Goal state before:
pp: case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) =
    ((↑π * z * ∏ j ∈ Finset.range 0, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * 0)) /
      ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * 0))

Tactic:
mul_zero,

Goal state after:
case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) =
    ((↑π * z * ∏ j ∈ Finset.range 0, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ 0) /
      ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ 0)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) =
    ↑π * z * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z) - Complex.sin (2 * z * 0) / (2 * z)) / ↑(π / 2)

Tactic:
mul_zero,

Goal state after:
case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) = ↑π * z * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z) - Complex.sin 0 / (2 * z)) / ↑(π / 2)
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
⊢ 2 * (n + 1) = 2 * n + 2

Tactic:
mul_add,

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
⊢ 2 * n + 2 * 1 = 2 * n + 2
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 ≤ ↑(m + 1) + 1
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
Nat.cast_succ,

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 ≤ ↑m + 1 + 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:
Nat.cast_succ,

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.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 < cos 0

Tactic:
rw [cos_zero]

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.Meromorphic.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f g : 𝕜 → 𝕜'
U : Set 𝕜
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
h'g : ∀ x ∈ U, meromorphicOrderAt g x ≠ ⊤
y : 𝕜
h₁y : AnalyticAt 𝕜 f y
h₂y : AnalyticAt 𝕜 g y
h₃y : f y ≠ 0
h₄y : g y ≠ 0
⊢ logDeriv (f * g) y = (logDeriv f + logDeriv g) y

Tactic:
rw [Pi.add_apply, Pi.mul_def]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f g : 𝕜 → 𝕜'
U : Set 𝕜
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
h'g : ∀ x ∈ U, meromorphicOrderAt g x ≠ ⊤
y : 𝕜
h₁y : AnalyticAt 𝕜 f y
h₂y : AnalyticAt 𝕜 g y
h₃y : f y ≠ 0
h₄y : g y ≠ 0
⊢ logDeriv (fun i ↦ f i * g i) y = logDeriv f y + logDeriv g y
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_5
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
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) * 1))
this : ContinuousOn (fun x ↦ Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
⊢ 𝓝 1 = 𝓝 (Complex.cos (2 * z * 0))

Tactic:
mul_zero,

Goal state after:
case e'_5
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
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) * 1))
this : ContinuousOn (fun x ↦ Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
⊢ 𝓝 1 = 𝓝 (Complex.cos 0)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ ↑n.succ * (u + 1).betaIntegral ↑n.succ / u = ↑(n + 1)! / ((∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) * (u + ↑0))

Tactic:
Nat.cast_succ,

Goal state after:
case succ.refine_2
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ (↑n + 1) * (u + 1).betaIntegral (↑n + 1) / u = ↑(n + 1)! / ((∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) * (u + ↑0))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s : ℝ
hs : 0 < s
⊢ Γ s * Γ (s + 1 / 2) = Γ (2 * s) * 2 ^ (1 - 2 * s) * √π

Tactic:
← doublingGamma_eq_Gamma (mul_pos two_pos hs),

Goal state after:
s : ℝ
hs : 0 < s
⊢ Γ s * Γ (s + 1 / 2) = (2 * s).doublingGamma * 2 ^ (1 - 2 * s) * √π
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
s : Set P
⊢ asymptoticCone k (closure[inst✝¹] s) = asymptoticCone k s

Tactic:
by
  ext
  simp_rw [mem_asymptoticCone_iff, mem_closure_iff_frequently, ← frequently_bind,
    asymptoticNhds_bind_nhds]

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

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
⊢ ↑n ^ z * ↑n ^ (1 - z) = ↑n

Tactic:
rw [← cpow_add _ _ (Nat.cast_ne_zero.mpr hn), add_sub_cancel, cpow_one]

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

Goal state before:
pp: case pos
z : ℂ
pi_ne : ↑π ≠ 0
hs : sin (-(↑π * z)) = 0
⊢ Gamma z * Gamma (1 - z) = 0

Tactic:
← mul_neg,

Goal state after:
case pos
z : ℂ
pi_ne : ↑π ≠ 0
hs : sin (↑π * -z) = 0
⊢ Gamma z * Gamma (1 - z) = 0