mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
⊢ T.graph.adjoint.toLinearPMap.graph = T†.graph

Tactic:
adjoint_graph_eq_graph_adjoint hT

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
⊢ T.graph.adjoint.toLinearPMap.graph = T.graph.adjoint
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x x' : E
y y' : F
⊢ dist (x ⊗ₜ[𝕜] y) (x' ⊗ₜ[𝕜] y') ≤ ‖x‖ * ‖y‖ + ‖x'‖ * ‖y'‖

Tactic:
grw [dist_eq_norm, norm_sub_le]; simp

Goal state after:
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 =
    ↑(eLpNormLESNormFDerivOneConst μ ↑(↑(finrank ℝ E)).conjExponent *
        (↑p * (↑(finrank ℝ E) - 1) / (↑(finrank ℝ E) - ↑p)).toNNReal)

Tactic:
γ

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 * ↑(NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯) =
    ↑(eLpNormLESNormFDerivOneConst μ ↑(↑(finrank ℝ E)).conjExponent *
        (↑p * (↑(finrank ℝ E) - 1) / (↑(finrank ℝ E) - ↑p)).toNNReal)
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: case inr
X : Type u_1
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℕ
h✝ : d₁ < d₂
s : Set X
h : μH[↑d₁] s = ⊤
⊢ (if volume.addHaarScalarFactor μH[↑d₁] = 0 then 0 else ⊤) = ⊤

Tactic:
simp [addHaarScalarFactor_volume_hausdorffMeasure_ne_zero]

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
s : Set X
h : s.Subsingleton
⊢ dimH s = 0

Tactic:
borelize X
  rw [← nonpos_iff_eq_zero]
  apply dimH_le_of_hausdorffMeasure_ne_top
  exact ((hausdorffMeasure_le_one_of_subsingleton h le_rfl).trans_lt ENNReal.one_lt_top).ne

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
⊢ [f.normDet ≠ 0, f.ker = ⊥, finrank 𝕜 ↥f.range = finrank 𝕜 U,
      Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range), Function.Injective ⇑f].TFAE

Tactic:
tfae_have 1 ↔ 2 := f.normDet_eq_zero_iff_ker_ne_bot.not_left
  tfae_have 1 ↔ 3 := f.normDet_eq_zero_iff_rank_range_ne.not_left
  tfae_have 3 → 4 := by
    intro h
    rw [← h]
    exact ⟨stdOrthonormalBasis 𝕜 f.range⟩
  tfae_have 4 → 3 := by
    rintro ⟨b⟩
    simpa using Module.finrank_eq_card_basis b.toBasis
  tfae_have 2 ↔ 5 := ker_eq_bot
  tfae_finish

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

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
⊢ Monotone m → Monotone fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s

Tactic:
intro h_mono s t hst
    simp only [extend, le_iInf_iff]
    intro ht
    apply le_trans _ (h_mono (ediam_mono hst))
    simp only [(ediam_mono hst).trans ht, le_refl, ciInf_pos]

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

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
⊢ (mkMetric fun x ↦ ∞) = ⊤

Tactic:
by
  apply toOuterMeasure_injective
  rw [mkMetric_toOuterMeasure, OuterMeasure.mkMetric_top, toOuterMeasure_top]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : ℝ → E
hf : Continuous f
hg : Continuous g
⊢ Continuous fun x ↦ _root_.inner_ 𝕜 (f x) (g x)

Tactic:
unfold _root_.inner_
  fun_prop

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

Goal state before:
pp: case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : f.ker = ⊥
bv : OrthonormalBasis ι 𝕜 ↥f.range := orthonormalBasis_range h b
⊢ ↑(f.normDet ^ 2) =
    (starRingEnd 𝕜) (Matrix.of fun i j ↦ (bv.repr (f.rangeRestrict (b j))).ofLp i).det *
      (Matrix.of fun i j ↦ (bv.repr (f.rangeRestrict (b j))).ofLp i).det

Tactic:
RCLike.conj_mul,

Goal state after:
case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : f.ker = ⊥
bv : OrthonormalBasis ι 𝕜 ↥f.range := orthonormalBasis_range h b
⊢ ↑(f.normDet ^ 2) = ↑‖(Matrix.of fun i j ↦ (bv.repr (f.rangeRestrict (b j))).ofLp i).det‖ ^ 2
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
⊢ μH[d] (f '' s) ≤ ↑C ^ d * μH[↑r * d] s

Tactic:
have hCd : (C : ℝ≥0∞) ^ d ≠ ∞ := by simp [hd]

Goal state after:
case inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
hCd : ↑C ^ d ≠ ∞
⊢ μH[d] (f '' s) ≤ ↑C ^ d * μH[↑r * d] s
Module: Mathlib.Analysis.InnerProductSpace.StarOrder

Goal state before:
pp: 𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f g : H →L[𝕜] H
h : (g - f).IsPositive
p : H →L[𝕜] H
hp₁ : IsSelfAdjoint p
hp₃ : p * p = g - f
⊢ f + p ^ 2 = g

Tactic:
add_comm,

Goal state after:
𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f g : H →L[𝕜] H
h : (g - f).IsPositive
p : H →L[𝕜] H
hp₁ : IsSelfAdjoint p
hp₃ : p * p = g - f
⊢ p ^ 2 + f = g
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
x y : E
⊢ μH[1] (segment ℝ x y) = edist x y

Tactic:
by
  rw [← affineSegment_eq_segment, hausdorffMeasure_affineSegment]

Goal state after:
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 :
  IntegrableOn (fun x ↦ ↑(a * x) ^ (s - 1) • f (a * x)) (Ioi 0) volume ↔
    IntegrableOn (fun t ↦ ↑t ^ (s - 1) • f t) (Ioi 0) volume
h1 : EqOn (fun t ↦ ↑(a * t) ^ (s - 1) • f (a * t)) (↑a ^ (s - 1) • fun t ↦ ↑t ^ (s - 1) • f (a * t)) (Ioi 0)
h2 : ↑a ^ (s - 1) ≠ 0
⊢ MellinConvergent (fun t ↦ f (a * t)) s ↔ MellinConvergent f s

Tactic:
rw [MellinConvergent, MellinConvergent, ← this, integrableOn_congr_fun h1 measurableSet_Ioi,
    IntegrableOn, IntegrableOn, integrable_smul_iff h2]

Goal state after:
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 : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (↑a - 1 + (s - ↑a)) • f (t ^ a)

Tactic:
add_comm,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (s - ↑a + (↑a - 1)) • f (t ^ a)
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 : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (s - ↑a + ↑a - 1) • f (t ^ a)

Tactic:
sub_add_cancel

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (s - 1) • f (t ^ a)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (↑a - 1 + (s - ↑a * 1)) • f (t ^ a)

Tactic:
add_comm,

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (s - ↑a * 1 + (↑a - 1)) • f (t ^ a)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (s - ↑a + ↑a - 1) • f (t ^ a)

Tactic:
sub_add_cancel

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (s - 1) • f (t ^ a)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : 1 + z ∈ slitPlane
⊢ -z ≠ 1

Tactic:
intro H

Goal state after:
n : ℕ
z : ℂ
hz : 1 + z ∈ slitPlane
H : -z = 1
⊢ False
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
hd' : ∃ ε > 0, ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c) volume

Tactic:
gt_iff_lt

Goal state after:
b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
hd' : ∃ ε, 0 < ε ∧ ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c) volume
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

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

Tactic:
intro x _

Goal state after:
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : x ∈ uIcc 0 (π / 2)
⊢ HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
⊢ ∀ 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

Tactic:
intro x _

Goal state after:
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : 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
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
⊢ (cexp ∘ fun x ↦ ↑x * log (1 + g x)) =ᶠ[atTop] fun x ↦ (1 + g x) ^ ↑x

Tactic:
have hg0 := tendsto_zero_of_isBoundedUnder_smul_of_tendsto_cobounded
    hg.norm.isBoundedUnder_le (RCLike.tendsto_ofReal_atTop_cobounded ℂ)

Goal state after:
g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
hg0 : Tendsto g atTop (𝓝 0)
⊢ (cexp ∘ fun x ↦ ↑x * log (1 + g x)) =ᶠ[atTop] fun x ↦ (1 + g x) ^ ↑x
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))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ ‖F' z t‖ ≤ bound t

Tactic:
simp_rw [F', bound, norm_smul, norm_mul, norm_real, mul_assoc, norm_eq_abs]

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))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ ‖↑t ^ (z - 1)‖ * (|log t| * ‖f t‖) ≤ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * (|log t| * ‖f t‖)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
n : ℕ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
⊢ f ↑n = f 1 + log ↑(n - 1)!

Tactic:
refine Nat.le_induction (by simp) (fun m hm IH => ?_) n (Nat.one_le_iff_ne_zero.2 hn)

Goal state after:
f : ℝ → ℝ
n : ℕ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
m : ℕ
hm : 1 ≤ m
IH : f ↑m = f 1 + log ↑(m - 1)!
⊢ f ↑(m + 1) = f 1 + log ↑(m + 1 - 1)!
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case hbc.inr
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))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
h : t < 1
⊢ s.re - v ≤ z.re - 1 + 1

Tactic:
sub_add_cancel

Goal state after:
case hbc.inr
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))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
h : t < 1
⊢ s.re - v ≤ z.re
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u : ℂ
hu : 0 < u.re
⊢ 1 / (u - 1 + 1) = 1 / u

pp: u : ℂ
hu : 0 < u.re
⊢ u - 1 + 1 ≠ 0

Tactic:
sub_add_cancel

Goal state after:
u : ℂ
hu : 0 < u.re
⊢ 1 / u = 1 / u

u : ℂ
hu : 0 < u.re
⊢ u - 1 + 1 ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u : ℂ
hu : 0 < u.re
⊢ u - 1 + 1 ≠ 0

Tactic:
sub_add_cancel

Goal state after:
u : ℂ
hu : 0 < u.re
⊢ u ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u : ℂ
hu : 0 < u.re
⊢ 0 < u.re - 1 + 1

Tactic:
sub_add_cancel

Goal state after:
u : ℂ
hu : 0 < u.re
⊢ 0 < u.re
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))

Tactic:
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
      mul_assoc]

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

Goal state before:
pp: s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ s.GammaIntegral * t.GammaIntegral = Gamma (s + t) * s.betaIntegral t

Tactic:
Gamma_eq_integral hst,

Goal state after:
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ s.GammaIntegral * t.GammaIntegral = (s + t).GammaIntegral * s.betaIntegral t
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : f ↑n - f (↑n - 1) ≤ (f (↑n + x) - f ↑n) / x
⊢ f (↑n - 1 + 1) = f ↑n

Tactic:
sub_add_cancel

Goal state after:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : 2 ≤ n
hx : 0 < x
npos : 0 < ↑n - 1
c : f ↑n - f (↑n - 1) ≤ (f (↑n + x) - f ↑n) / x
⊢ f ↑n = f ↑n
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 - 1 + 1) = f (x - 1) + log (x - 1)
⊢ f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0

Tactic:
sub_add_cancel

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.MellinTransform

Goal state before:
pp: s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ HasMellin ((Ioc 0 1).indicator fun x ↦ 1) s (1 / s)

Tactic:
simp_rw [HasMellin, mellin, MellinConvergent, ← indicator_smul, IntegrableOn,
    integrable_indicator_iff aux3, smul_eq_mul, integral_indicator aux3, mul_one, IntegrableOn,
    Measure.restrict_restrict_of_subset Ioc_subset_Ioi_self]

Goal state after:
s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ Integrable (fun a ↦ ↑a ^ (s - 1)) (volume.restrict (Ioc 0 1)) ∧ ∫ (a : ℝ) in Ioc 0 1, ↑a ^ (s - 1) ∂volume = 1 / s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: x : ℝ
hx : 0 < x
this : log (Γ x) = (log ∘ Γ) x - (log ∘ Γ) 1
y : ℝ
hy : 0 < y
⊢ log y + log (Γ y) = (log ∘ Γ) y + log y

Tactic:
add_comm,

Goal state after:
x : ℝ
hx : 0 < x
this : log (Γ x) = (log ∘ Γ) x - (log ∘ Γ) 1
y : ℝ
hy : 0 < y
⊢ log (Γ y) + log y = (log ∘ Γ) y + log y