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

Goal state before:
pp: case ha
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
⊢ 0 ≤ ‖(o.kahler x) y‖

pp: case hb
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
⊢ 0 ≤ ‖x‖ * ‖y‖

Tactic:
·

Goal state after:
case ha
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
⊢ 0 ≤ ‖(o.kahler x) y‖
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case hb
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
⊢ 0 ≤ ‖x‖ * ‖y‖

Tactic:
·

Goal state after:
case hb
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
⊢ 0 ≤ ‖x‖ * ‖y‖
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case inl
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
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖x‖ = 0
⊢ x = 0 ∨ y = 0

pp: case inr
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
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖y‖ = 0
⊢ x = 0 ∨ y = 0

Tactic:
·

Goal state after:
case inl
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
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖x‖ = 0
⊢ x = 0 ∨ y = 0
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case inr
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
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖y‖ = 0
⊢ x = 0 ∨ y = 0

Tactic:
·

Goal state after:
case inr
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
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖y‖ = 0
⊢ x = 0 ∨ y = 0
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
⊢ (o.kahler x) y = 0 ↔ x = 0 ∨ y = 0

Tactic:
by
  refine ⟨o.eq_zero_or_eq_zero_of_kahler_eq_zero, ?_⟩
  rintro (rfl | rfl) <;> simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakSpace

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ↑(e_dual f) = e.dualMap ↑f
he' : ∀ (f : StrongDual 𝕜 E), ↑(e_dual.symm f) = e.symm.dualMap ↑f
⊢ ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)

pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ↑(e_dual f) = e.dualMap ↑f
he' : ∀ (f : StrongDual 𝕜 E), ↑(e_dual.symm f) = e.symm.dualMap ↑f
⊢ ∀ (f : StrongDual 𝕜 E), Continuous ⇑(e.symm.dualMap ↑f)

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ↑(e_dual f) = e.dualMap ↑f
he' : ∀ (f : StrongDual 𝕜 E), ↑(e_dual.symm f) = e.symm.dualMap ↑f
⊢ ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
Module: Mathlib.Analysis.LocallyConvex.WeakSpace

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ↑(e_dual f) = e.dualMap ↑f
he' : ∀ (f : StrongDual 𝕜 E), ↑(e_dual.symm f) = e.symm.dualMap ↑f
⊢ ∀ (f : StrongDual 𝕜 E), Continuous ⇑(e.symm.dualMap ↑f)

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ↑(e_dual f) = e.dualMap ↑f
he' : ∀ (f : StrongDual 𝕜 E), ↑(e_dual.symm f) = e.symm.dualMap ↑f
⊢ ∀ (f : StrongDual 𝕜 E), Continuous ⇑(e.symm.dualMap ↑f)
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
⊢ cexp (-s * ↑u) • f (rexp (-u)) = cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))

pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
⊢ cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u)) =
    cexp (↑(-2 * π * (u * (s.im / (2 * π)))) * I) • rexp (-s.re * u) • f (rexp (-u))

Tactic:
·

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
⊢ cexp (-s * ↑u) • f (rexp (-u)) = cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)

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
⊢ mellin (fun t ↦ f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)

Tactic:
·

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hE : CompleteSpace E
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)

pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hE : ¬CompleteSpace E
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)

Tactic:
·

Goal state after:
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hE : CompleteSpace E
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hE : ¬CompleteSpace E
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)

Tactic:
·

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hE : ¬CompleteSpace E
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hf : MellinConvergent f s
R : Type u_2
inst✝³ : NormedRing R
inst✝² : Module R E
inst✝¹ : IsBoundedSMul R E
inst✝ : SMulCommClass ℂ R E
c : R
⊢ mellin (fun t ↦ c • f t) s = c • mellin f s

Tactic:
by simp [mellin, smul_comm, hf.integral_smul]

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

Goal state before:
pp: case convert_2
s : ℂ
hs : 0 < s.re
⊢ MeasureTheory.LocallyIntegrableOn (fun x ↦ ↑(rexp (-x))) (Ioi 0) MeasureTheory.volume

pp: case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ↑(rexp (-x))) =O[atTop] fun x ↦ x ^ (-(s.re + 1))

pp: case convert_4
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ↑(rexp (-x))) =O[𝓝[>] 0] fun x ↦ x ^ (-0)

Tactic:
·

Goal state after:
case convert_2
s : ℂ
hs : 0 < s.re
⊢ MeasureTheory.LocallyIntegrableOn (fun x ↦ ↑(rexp (-x))) (Ioi 0) MeasureTheory.volume
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ↑(rexp (-x))) =O[atTop] fun x ↦ x ^ (-(s.re + 1))

pp: case convert_4
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ↑(rexp (-x))) =O[𝓝[>] 0] fun x ↦ x ^ (-0)

Tactic:
·

Goal state after:
case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ↑(rexp (-x))) =O[atTop] fun x ↦ x ^ (-(s.re + 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ‖rexp (-x)‖) =O[atTop] fun x ↦ x ^ (-(s.re + 1))

Tactic:
isBigO_norm_left

Goal state after:
case convert_3
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ rexp (-x)) =O[atTop] fun x ↦ x ^ (-(s.re + 1))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case convert_4
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ↑(rexp (-x))) =O[𝓝[>] 0] fun x ↦ x ^ (-0)

Tactic:
·

Goal state after:
case convert_4
s : ℂ
hs : 0 < s.re
⊢ (fun x ↦ ↑(rexp (-x))) =O[𝓝[>] 0] fun x ↦ x ^ (-0)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_1
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
⊢ AEStronglyMeasurable (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioi (max e 1)))

pp: case refine_2
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioi (max e 1)))

Tactic:
·

Goal state after:
case refine_1
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
⊢ AEStronglyMeasurable (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioi (max e 1)))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case zero
s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
hsre : 0 < s.re
this : IsOpen {s | 0 < s.re}
⊢ DifferentiableAt ℂ Gamma s

Tactic:
apply (hasDerivAt_GammaIntegral (by simpa using hsre)).differentiableAt.congr_of_eventuallyEq

Goal state after:
case zero
s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
hsre : 0 < s.re
this : IsOpen {s | 0 < s.re}
⊢ Gamma =ᶠ[𝓝 s] GammaIntegral
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioi (max e 1)))

Tactic:
·

Goal state after:
case refine_2
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioi (max e 1)))
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case e'_2
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
⊢ f z = f (0 + z) - f 0

pp: case e'_3
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
⊢ z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹ = z • ∫ (t : ℝ) in 0..1, f' (0 + t • z)

Tactic:
·

Goal state after:
case e'_2
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
⊢ f z = f (0 + z) - f 0
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case e'_3
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
⊢ z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹ = z • ∫ (t : ℝ) in 0..1, f' (0 + t • z)

Tactic:
·

Goal state after:
case e'_3
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
⊢ z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹ = z • ∫ (t : ℝ) in 0..1, f' (0 + t • z)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case pos
s : ℂ
hs : s ≠ 0
h : ∃ m, s = -↑m
⊢ deriv Gamma (s + 1) = Gamma s + s * deriv Gamma s

pp: case neg
s : ℂ
hs : s ≠ 0
h : ∀ (m : ℕ), s ≠ -↑m
⊢ deriv Gamma (s + 1) = Gamma s + s * deriv Gamma s

Tactic:
·

Goal state after:
case pos
s : ℂ
hs : s ≠ 0
h : ∃ m, s = -↑m
⊢ deriv Gamma (s + 1) = Gamma s + s * deriv Gamma s
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_1
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ AEStronglyMeasurable (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioo 0 ε))

pp: case refine_2
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioo 0 ε))

Tactic:
·

Goal state after:
case refine_1
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ AEStronglyMeasurable (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioo 0 ε))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case neg
s : ℂ
hs : s ≠ 0
h : ∀ (m : ℕ), s ≠ -↑m
⊢ deriv Gamma (s + 1) = Gamma s + s * deriv Gamma s

Tactic:
·

Goal state after:
case neg
s : ℂ
hs : s ≠ 0
h : ∀ (m : ℕ), s ≠ -↑m
⊢ deriv Gamma (s + 1) = Gamma s + s * deriv Gamma s
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioo 0 ε))

Tactic:
·

Goal state after:
case refine_2
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioo 0 ε))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2.hg
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ HasFiniteIntegral (fun t ↦ d * t ^ (s - b - 1)) (volume.restrict (Ioo 0 ε))

pp: case refine_2.h
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioo 0 ε), ‖a ^ (s - 1) * f a‖ ≤ d * a ^ (s - b - 1)

Tactic:
·

Goal state after:
case refine_2.hg
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ HasFiniteIntegral (fun t ↦ d * t ^ (s - b - 1)) (volume.restrict (Ioo 0 ε))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2.h
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioo 0 ε), ‖a ^ (s - 1) * f a‖ ≤ d * a ^ (s - b - 1)

Tactic:
·

Goal state after:
case refine_2.h
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioo 0 ε), ‖a ^ (s - 1) * f a‖ ≤ d * a ^ (s - b - 1)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_2.e'_4
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
⊢ (fun x ↦ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) = fun x ↦ ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)

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))

pp: case convert_1
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
⊢ IntervalIntegrable (fun x ↦ -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) volume 0 (π / 2)

pp: case convert_2
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
⊢ IntervalIntegrable (fun x ↦ Complex.cos (2 * z * ↑x)) volume 0 (π / 2)

Tactic:
·

Goal state after:
case e'_2.e'_4
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
⊢ (fun x ↦ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) = fun x ↦ ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2.h
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
t : ℝ
ht : t ∈ Ioo 0 ε
⊢ dist t 0 < ε

pp: case refine_2.h
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
ε : ℝ
hε : 0 < ε
t : ℝ
hε' : ‖f t‖ ≤ d * ‖t ^ (-b)‖
ht : t ∈ Ioo 0 ε
⊢ ‖f t‖ * ‖t ^ (s - 1)‖ ≤ d * t ^ (s - b - 1)

Tactic:
·

Goal state after:
case refine_2.h
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
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
t : ℝ
ht : t ∈ Ioo 0 ε
⊢ dist t 0 < ε
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))

pp: case convert_1
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
⊢ IntervalIntegrable (fun x ↦ -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) volume 0 (π / 2)

pp: case convert_2
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
⊢ IntervalIntegrable (fun x ↦ Complex.cos (2 * z * ↑x)) volume 0 (π / 2)

Tactic:
·

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 (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))