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

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
e : ℝ
f : ℝ → F
⊢ UniqueDiffOn ℝ Set.univ

Tactic:
by simp

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

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
e : ℝ
f : ℝ → F
⊢ e ∈ Set.univ

Tactic:
by simp

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

Goal state before:
pp: E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
⊢ 2 ≠ ↑⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
this : (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z
⊢ (fun w ↦ (∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) - (w - z).im • f z) = fun w ↦
    (∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) - ∫ (x : ℝ) in z.im..w.im, f z

Tactic:
by simp

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

Goal state before:
pp: E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
h₁ : ContDiffAt ℝ 2 f₁ x
h₂ : ContDiffAt ℝ 2 f₂ x
⊢ 2 ≠ ↑⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
⊢ (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z

Tactic:
have : (fun w ↦ f w - f z) =o[𝓝 z] fun _ ↦ (1 : ℝ) := by
    rw [Asymptotics.isLittleO_one_iff, tendsto_sub_nhds_zero_iff]
    exact f_cont.continuousAt <| _root_.mem_nhds_iff.mpr ⟨ball c r, le_refl _, isOpen_ball, hz⟩

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
this : (fun w ↦ f w - f z) =o[𝓝 z] fun x ↦ 1
⊢ (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
ε : ℝ
ε_pos : 0 < ε
this : ∀ᶠ (w : ℂ) in 𝓝 z, ∀ y ∈ Ι z.im w.im, ‖f (↑w.re + ↑y * I) - f z‖ ≤ ε
w : ℂ
hw : ∀ y ∈ Ι z.im w.im, ‖f (↑w.re + ↑y * I) - f z‖ ≤ ε
⊢ ε * ‖w.im - z.im‖ = ε * ‖(w - z).im‖

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : IsScalarTower ℝ 𝕜 F
f : E → F
x : E
s : Set E
v : 𝕜
hf : ContDiffWithinAt ℝ 2 f s x
hs : UniqueDiffOn ℝ s
⊢ 2 ≠ ↑⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
h : IsConservativeOn f (ball c r)
⊢ ∀ (x : ℂ),
    c.wedgeIntegral x f - c.wedgeIntegral z f = z.wedgeIntegral x f →
      (fun x' ↦ c.wedgeIntegral x' f - c.wedgeIntegral z f - (x' - z) • f z) x =
        (fun w ↦ z.wedgeIntegral w f - (w - z) • f z) x

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : IsScalarTower ℝ 𝕜 F
f : E → F
x : E
s : Set E
v : 𝕜
hf : ContDiffWithinAt ℝ 2 f s x
hs : UniqueDiffOn ℝ s
⊢ Δ[s] v • f =ᶠ[𝓝[s] x] v • Δ[s] f

Tactic:
by
  filter_upwards [(hf.eventually (by simp)).filter_mono (nhdsWithin_mono _ (Set.subset_insert ..)),
    eventually_mem_nhdsWithin] with a h₁a using laplacianWithin_smul v h₁a hs

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : IsScalarTower ℝ 𝕜 F
f : E → F
x : E
v : 𝕜
h : ContDiffAt ℝ 2 f x
⊢ 2 ≠ ↑⊤

Tactic:
by simp

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

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f : E → F
x : E
s : Set E
l : F →L[ℝ] G
h : ContDiffWithinAt ℝ 2 f s x
hs : UniqueDiffOn ℝ s
⊢ 2 ≠ ↑⊤

Tactic:
by simp

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

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f : E → F
x : E
l : F →L[ℝ] G
h : ContDiffAt ℝ 2 f x
⊢ 2 ≠ ↑⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
h : IsConservativeOn f (ball c r)
w : ℂ
I₁ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₂ : E := ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
⊢ I₁ + I • I₂ - (↑(w - z).re + ↑(w - z).im * I) • f z = I₁ + I • I₂ - (↑w.re - ↑z.re + (↑w.im - ↑z.im) * I) • f z

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: case calc_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
h : IsConservativeOn f (ball c r)
w : ℂ
I₁ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₂ : E := ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
⊢ I₁ + I • I₂ - ((↑w.re - ↑z.re) • f z + ((↑w.im - ↑z.im) * I) • f z) =
    I₁ - (↑w.re - ↑z.re) • f z + (I • I₂ - (I * (↑w.im - ↑z.im)) • f z)

Tactic:
mul_comm I

Goal state after:
case calc_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
h : IsConservativeOn f (ball c r)
w : ℂ
I₁ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₂ : E := ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
⊢ I₁ + I • I₂ - ((↑w.re - ↑z.re) • f z + ((↑w.im - ↑z.im) * I) • f z) =
    I₁ - (↑w.re - ↑z.re) • f z + (I • I₂ - ((↑w.im - ↑z.im) * I) • f z)
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
h₁f : ContDiffAt ℝ 2 f x
⊢ minSmoothness ℝ 2 ≤ 2

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: case e_a
f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
h₁f : ContDiffAt ℝ 2 f x
⊢ -(I * ↑(((fderiv ℝ (fderiv ℝ f) x) I) I)) = I * ↑(((fderiv ℝ (fderiv ℝ f) x) 1) 1)

Tactic:
· have h₂f := hf.2.eq_of_nhds
    simp only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply, Fin.isValue,
      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, Pi.zero_apply,
      add_eq_zero_iff_eq_neg] at h₂f
    simp [h₂f]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalAverage

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hab : a ≠ b
hf : ContinuousOn f [[a, b]]
⊢ volume (Ι a b) ≠ ⊤

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: f : ℂ → ℝ
z : ℂ
R : ℝ
hf : HarmonicOnNhd f (ball z R)
hR : ¬R ≤ 0
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : DifferentiableOn ℂ g (ball z R)
F : ℂ → ℂ
hF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x
h₁F : DifferentiableOn ℂ F (ball z R)
h₂F : DifferentiableOn ℝ F (ball z R)
x : ℂ
hx : x ∈ ball z R
y : ℂ
hy : y ∈ ball z R
h₄F : DifferentiableAt ℂ F y
h₅F : DifferentiableAt ℝ F y
a : ℂ
⊢ a = a.re • 1 + a.im • I

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ Complex.reCLM (circleAverage F c R) = g (f c)

pp: F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ CircleIntegrable F c R

Tactic:
h₃F.diffContOnCl.circleAverage

Goal state after:
F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ Complex.reCLM (F c) = g (f c)

F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ CircleIntegrable F c R
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.im
hz_im : 0 ≤ z.re
⊢ ‖f (z * I)‖ ≤ C

Tactic:
change ‖(f ∘ (· * I)) z‖ ≤ C

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.im
hz_im : 0 ≤ z.re
⊢ ‖(f ∘ fun x ↦ x * I) z‖ ≤ C
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Basic

Goal state before:
pp: ⊢ 0 < Complex.I.im

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: case inl
x : ℝ
hx : x ≠ 0
⊢ ContinuousAt (fun x ↦ x * log x) x

Tactic:
exact (continuous_id'.continuousAt).mul (continuousAt_log hx)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: ⊢ Disjoint (𝓝 (deriv (fun x ↦ x * log x) 0)) Filter.atBot

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: ⊢ StrictConcaveOn ℝ (Set.Ici 0) negMulLog

Tactic:
simpa only [negMulLog_eq_neg] using! strictConvexOn_mul_log.neg

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

Goal state before:
pp: case neg
x : ℝ
hx : ¬x = 0
⊢ |sin x| ≤ 1 * |x|

Tactic:
rw [one_mul]

Goal state after:
case neg
x : ℝ
hx : ¬x = 0
⊢ |sin x| ≤ |x|
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: H : Tendsto (fun t ↦ (log t * t)⁻¹) (nhdsWithin 0 {0}ᶜ) (nhds (deriv (fun x ↦ (log x)⁻¹) 0))
⊢ Disjoint (nhds (deriv (fun x ↦ (log x)⁻¹) 0)) atTop

Tactic:
by simp

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

Goal state before:
pp: a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
⊢ ‖‖↑0 ^ r‖‖ₑ ≠ ⊤

Tactic:
by simp

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

Goal state before:
pp: case hbc
a b x : ℝ
hx : 0 < x
⊢ ContinuousOn log [[1, x]]

Tactic:
apply Real.continuousOn_log.mono

Goal state after:
case hbc
a b x : ℝ
hx : 0 < x
⊢ [[1, x]] ⊆ {0}ᶜ
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: α : Type u_1
s : Finset α
f : α → ℝ
hs : s.Nonempty
t_max : α
ht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|
⊢ |∑ t ∈ s, f t| ∈ Set.Ici 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
z : ℂ
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z.re
this : ∀ᶠ (ε : ℝ) in 𝓝[<] 0, ‖cexp (↑ε * z) • f z‖ ≤ C
⊢ ‖cexp (↑0 * z) • f z‖ = ‖f z‖

Tactic:
by simp

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

Goal state before:
pp: a b r : ℝ
h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
h' : -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]
this : (∫ (x : ℝ) in a..b, ↑x ^ ↑r).re = ((↑b ^ (↑r + 1) - ↑a ^ (↑r + 1)) / (↑r + 1)).re
⊢ ↑r + 1 = ↑(r + 1)

Tactic:
by push_cast; rfl

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

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..a + 1, b = b

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
x : ℂ
hx : x ∈ closedBall 0 |1|
⊢ ‖x * a‖ = ‖x‖ * ‖a‖

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: this✝ : AnalyticOnNhd ℝ (fun x ↦ 4 * sin x ^ 2) Set.univ
this : ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ ∈ codiscrete ℝ
a : ℝ
ha : ¬sin a = 0
⊢ 4 ≠ 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
t₀ : ∀ (u : ↑(sphere c |R|)), meromorphicOrderAt f ↑u ≠ ⊤
g : ℂ → E
h₁g : AnalyticOnNhd ℂ g (sphere c |R|)
h₂g : ∀ (u : ↑(sphere c |R|)), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin (sphere c |R|)] (∏ᶠ (u : ℂ), (fun x ↦ x - u) ^ (divisor f (sphere c |R|)) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin (sphere c |R|)]
    (∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f (sphere c |R|)) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
i : ℂ
⊢ AnalyticOnNhd ℝ (circleMap c R) [[0, 2 * π]]

Tactic:
apply (analyticOnNhd_circleMap c R).mono (by tauto)

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

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, sin x = cos a - cos b

Tactic:
rw [integral_deriv_eq_sub' fun x => -cos x]

Goal state after:
a b : ℝ
⊢ -cos b - -cos a = cos a - cos b

case hderiv
a b : ℝ
⊢ (deriv fun x ↦ -cos x) = sin

case hdiff
a b : ℝ
⊢ ∀ x ∈ [[a, b]], DifferentiableAt ℝ (fun x ↦ -cos x) x

case hcont
a b : ℝ
⊢ ContinuousOn sin [[a, b]]
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
t₀ : ¬∀ (u : ↑(sphere c |R|)), meromorphicOrderAt f ↑u ≠ ⊤
⊢ 1 < Module.rank ℝ ℂ

Tactic:
by simp

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

Goal state before:
pp: case hderiv
a b : ℝ
t : ℂ
ht : t ≠ -1
this : t + 1 ≠ 0
x : ℝ
a✝ : x ∈ [[a, b]]
f : HasDerivAt (fun y ↦ 1 + y ^ 2) (2 * ↑x) ↑x
g : ∀ {z : ℂ}, 0 < z.re → HasDerivAt (fun z ↦ z ^ (t + 1) / (2 * (t + 1))) (z ^ t / 2) z
⊢ 0 < (1 + ↑x ^ 2).re

Tactic:
· exact mod_cast add_pos_of_pos_of_nonneg zero_lt_one (sq_nonneg x)

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: r₀ r R : ℝ
ρ : ℂ
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hR : 0 < R
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ r₁ : ℝ
⊢ ↑(normSq (circleMap 0 r₁ θ) + normSq ρ - 2 * (circleMap 0 r₁ θ * (starRingEnd ℂ) ρ).re) =
    ↑(r₁ ^ 2 + R ^ 2 - 2 * r₁ * R * Real.cos (θ - ρ.arg))

Tactic:
suffices (circleMap 0 r₁ θ * (conj) ρ).re = r₁ * ‖ρ‖ * Real.cos (θ - ρ.arg) by
      simp [normSq_eq_norm_sq, hρ, -mul_re, this, mul_assoc]

Goal state after:
r₀ r R : ℝ
ρ : ℂ
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hR : 0 < R
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ r₁ : ℝ
⊢ (circleMap 0 r₁ θ * (starRingEnd ℂ) ρ).re = r₁ * ‖ρ‖ * Real.cos (θ - ρ.arg)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case succ
n k : ℕ
ih : ∫ (x : ℝ) in 0..π, sin x ^ (2 * k + 1) = 2 * ∏ i ∈ Finset.range k, (2 * ↑i + 2) / (2 * ↑i + 3)
⊢ ∫ (x : ℝ) in 0..π, sin x ^ (2 * (k + 1) + 1) = 2 * ∏ i ∈ Finset.range (k + 1), (2 * ↑i + 2) / (2 * ↑i + 3)

Tactic:
| succ k ih =>

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

Goal state before:
pp: case succ
n k : ℕ
ih : ∫ (x : ℝ) in 0..π, sin x ^ (2 * k) = π * ∏ i ∈ Finset.range k, (2 * ↑i + 1) / (2 * ↑i + 2)
⊢ ∫ (x : ℝ) in 0..π, sin x ^ (2 * (k + 1)) = π * ∏ i ∈ Finset.range (k + 1), (2 * ↑i + 1) / (2 * ↑i + 2)

Tactic:
| succ k ih =>

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
h_bd : I∞.BoundedAtFilter f
⊢ 1 ∘ invQParam h =ᶠ[𝓝[≠] 0] 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
⊢ IntervalIntegrable (fun x ↦ |log (2 * R)|) volume 0 (2 * π)

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: ρ w : ℂ
R : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
r : ℕ → ℝ
hr_lt : ∀ (n : ℕ), r n < R
hr_tendsto : Tendsto r atTop (𝓝 R)
bound : ℝ → ℝ :=
  fun θ ↦ (R + ‖w‖) / ((R + ‖w‖) / 2 - ‖w‖) * (|log (2 * R)| + |log √((R + ‖w‖) / 2 / R)| + |log ‖circleMap 0 R θ - ρ‖|)
⊢ IntervalIntegrable (fun x ↦ |log √((R + ‖w‖) / 2 / R)|) volume 0 (2 * π)

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: c z : ℂ
f : ℂ → ℂ
hf : Differentiable ℂ f
⊢ ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z

Tactic:
convert! taylorSeries_eq_of_entire hf c z using 3 with n

Goal state after:
case e'_5
c z : ℂ
f : ℂ → ℂ
hf : Differentiable ℂ f
n : ℕ
⊢ (↑n !)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: ⊢ ∫ (x : ℝ) in -(π / 2)..π / 2, cos x ^ 2 = π / 2

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.RCLike.Sqrt

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a : 𝕜
ha : 0 ≤ a
h : I = 0
⊢ sqrt (-a) = I * sqrt a

Tactic:
simp [h, sqrt_eq_ite, Real.sqrt_eq_zero', nonneg_iff.mp ha]

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
⊢ c ∈ closedBall c |R|

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.RCLike.Sqrt

Goal state before:
pp: ⊢ 0 ≤ 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.RCLike.Sqrt

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
⊢ 0 ≤ 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.RCLike.Sqrt

Goal state before:
pp: ⊢ 0 ≤ I.im

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.RCLike.Sqrt

Goal state before:
pp: ⊢ (-I).im < 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Exp

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
τ : ℍ
⊢ -2 * π * (↑τ).im / ↑n < 0

Tactic:
neg_mul,

Goal state after:
n : ℕ
inst✝ : NeZero n
τ : ℍ
⊢ -(2 * π) * (↑τ).im / ↑n < 0
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Exp

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
τ : ℍ
⊢ -(2 * π) * τ.im / ↑n < 0

Tactic:
neg_mul,

Goal state after:
n : ℕ
inst✝ : NeZero n
τ : ℍ
⊢ -(2 * π * τ.im) / ↑n < 0
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
integral_bound : circleAverage (fun x ↦ Real.log ‖f x‖) c R ≤ Real.log M
u : ℂ
h1 : u ∈ closedBall c |R|
h2 : u ∈ closedBall c |r|
h3 : u = c
⊢ c ∈ closedBall c |R|

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
s : Set X
hs : IsClosed[inst✝²] s
Y : Type v
inst✝ : TopologicalSpace Y
f : C(↑s, Y)
t : Set Y
hf : ∀ (x : ↑s), f x ∈ t
ht : TietzeExtension ↑t
g : C(X, ↑t)
hg : restrict s g = { toFun := Set.codRestrict (⇑f) t hf, continuous_toFun := ⋯ }
⊢ ∀ (x : X), ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp g) x ∈ t

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X₁ : Type u₁
inst✝³ : TopologicalSpace X₁
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
e : X₁ → X
he : IsClosedEmbedding e
Y : Type v
inst✝ : TopologicalSpace Y
f : C(X₁, Y)
t : Set Y
hf : ∀ (x : X₁), f x ∈ t
ht : TietzeExtension ↑t
g : C(X, ↑t)
hg : g.comp { toFun := e, continuous_toFun := ⋯ } = { toFun := Set.codRestrict (⇑f) t hf, continuous_toFun := ⋯ }
⊢ ∀ (x : X), ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp g) x ∈ t

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: Y : Type v
Z : Type w
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TietzeExtension Z
e : Y ≃ₜ Z
⊢ (↑e.symm).comp ↑e = ContinuousMap.id Y

Tactic:
by simp

Goal state after: