mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : z.re ≤ 0
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Iio 0 ×ℂ Ioi 0) (Ioi 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖(f ∘ Neg.neg) (↑x * I)‖ ≤ C

Tactic:
rw [comp_apply, ← neg_mul, ← ofReal_neg]
    exact him (-x) (neg_nonpos.2 hx)

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

Goal state before:
pp: case pos
w c z : ℂ
η₂ : z - c ≠ 0
hz : z ∈ sphere c ‖z - c‖
hw : w ∈ ball c ‖z - c‖
η₀ : 0 < ‖z - c‖
h₁w : ‖w - c‖ = 0
⊢ (‖z - c‖ - ‖w - c‖) / (‖z - c‖ + ‖w - c‖) ≤ ((z - c + (w - c)) / (z - c - (w - c))).re

Tactic:
aesop

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

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
⊢ ContinuousOn (herglotzRieszKernel c w) (sphere c |R|)

Tactic:
by
  apply ContinuousOn.div (by fun_prop) (by fun_prop)
  grind [mem_sphere, mem_ball, le_abs_self R]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: f : ℂ → ℂ
z : ℂ
h₁f : AnalyticAt ℂ f z
h₂f : f z ≠ 0
this : (fun x ↦ Real.log ‖f x‖) = 2⁻¹ • Real.log ∘ ⇑normSq ∘ f
h₃f : f z ∉ slitPlane
⊢ ⇑normSq ∘ f = ⇑normSq ∘ (-f)

Tactic:
aesop

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hR : 0 < R
hw : 0 ∈ ball 0 R
z : ℂ
hz : z ∈ sphere 0 R
⊢ z ≠ 0

Tactic:
aesop

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
z : ℂ
hz : z ∈ sphere 0 R
⊢ z ≠ 0

Tactic:
aesop

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

Goal state before:
pp: case funProp.discharger
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
⊢ ∀ x ∈ closedBall 0 R, ↑q * x - W ≠ 0

Tactic:
aesop

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

Goal state before:
pp: ⊢ Tendsto (fun x ↦ log (log x)) (nhdsWithin 1 {1}ᶜ) (cobounded ℝ)

Tactic:
exact (tendsto_log_nhdsNE_zero.mono_right atBot_le_cobounded).comp <| log_one ▸
    HasDerivAt.tendsto_nhdsNE (by simpa using hasDerivAt_log one_ne_zero) one_ne_zero

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

Goal state before:
pp: case neg.inr.refine_2.inl
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 : 0 ≤ c
this : IntegrableOn (fun x ↦ 1) (Ioc 0 c) μ
x : ℝ
hx : x ∈ Ioc 0 c
⊢ 1 = ‖↑x ^ r‖

Tactic:
rw [Complex.norm_cpow_eq_rpow_re_of_pos hx.1, ← h', rpow_zero]

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

Goal state before:
pp: case funProp.discharger
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
h₁ : ∀ z ∈ sphere 0 R, z - w ≠ 0
h₃ : ContinuousOn f (sphere 0 R)
⊢ ∀ x ∈ sphere 0 R, q • x - W ≠ 0

Tactic:
aesop

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
c : ℂ
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : 0 < R
x✝ : ℂ
⊢ x✝ ∈ ball 0 R → (fun z ↦ z + c) x✝ ∈ ball c R

Tactic:
aesop

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

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
this : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (thickening d {w}) (thickening d (ball c R)ᶜ)
⊢ ∃ d > 0, ball w d ⊆ ball c R ∧ ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap c R θ - x‖

Tactic:
refine ⟨d, hd, ?_, ?_⟩

Goal state after:
case refine_1
R : ℝ
w c : ℂ
hw : w ∈ ball c R
this : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (thickening d {w}) (thickening d (ball c R)ᶜ)
⊢ ball w d ⊆ ball c R

case refine_2
R : ℝ
w c : ℂ
hw : w ∈ ball c R
this : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (thickening d {w}) (thickening d (ball c R)ᶜ)
⊢ ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap c R θ - x‖
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case inr
a b : ℝ
r : ℂ
h : ¬r + 1 = 0 ∧ 0 ∉ [[a, b]]
⊢ r + 1 ≠ 0

Tactic:
exact h.1

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

Goal state before:
pp: a b : ℝ
f : ℝ → ℝ
hf : MeromorphicOn f [[a, b]]
⊢ log ∘ f = fun x ↦ log ‖f x‖

Tactic:
aesop

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‖ ≤ 1

Tactic:
aesop

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

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
⊢ a ≠ 0

Tactic:
aesop

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

Goal state before:
pp: case neg
a b c : ℝ
hc : ¬c = 0
⊢ c * ∫ (x : ℝ) in a..b, (c ^ 2 + x ^ 2)⁻¹ = arctan (b / c) - arctan (a / c)

Tactic:
integral_inv_sq_add_sq hc

Goal state after:
case neg
a b c : ℝ
hc : ¬c = 0
⊢ c * (c⁻¹ * (arctan (b / c) - arctan (a / c))) = arctan (b / c) - arctan (a / c)
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|)
⊢ CircleIntegrable (fun x ↦ log⁺ ‖f x‖) c R

Tactic:
simp_rw [← half_mul_log_add_log_abs, mul_add]
  apply CircleIntegrable.add
  · apply hf.circleIntegrable_log_norm.const_mul
  · apply hf.circleIntegrable_log_norm.abs.const_mul

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

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
n : ℕ
⊢ Tendsto (fun x ↦ g n ↑x) atTop (𝓝 0)

Tactic:
rw [tendsto_zero_iff_norm_tendsto_zero]

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
n : ℕ
⊢ Tendsto (fun x ↦ ‖g n ↑x‖) atTop (𝓝 0)
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
⊢ Continuous (𝕢 h)

Tactic:
unfold qParam

Goal state after:
h : ℝ
⊢ Continuous fun z ↦ cexp (2 * ↑π * I * z / ↑h)
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
z₀ : ℂ
hf : AnalyticAt ℂ f z₀
⊢ (∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀) ∨ 𝓝 (f z₀) ≤ map f (𝓝 z₀)

Tactic:
by
  /- The function `f` is analytic in a neighborhood of `z₀`; by the isolated zeros principle, if `f`
    is not constant in a neighborhood of `z₀`, then it is nonzero, and therefore bounded below, on
    every small enough circle around `z₀` and then `DiffContOnCl.ball_subset_image_closedBall`
    provides an explicit ball centered at `f z₀` contained in the range of `f`. -/
  refine or_iff_not_imp_left.mpr fun h => ?_
  refine (nhds_basis_ball.le_basis_iff (nhds_basis_closedBall.map f)).mpr fun R hR => ?_
  have h1 := (hf.eventually_eq_or_eventually_ne analyticAt_const).resolve_left h
  have h2 : ∀ᶠ z in 𝓝 z₀, AnalyticAt ℂ f z := (isOpen_analyticAt ℂ f).eventually_mem hf
  obtain ⟨ρ, hρ, h3, h4⟩ :
    ∃ ρ > 0, AnalyticOnNhd ℂ f (closedBall z₀ ρ) ∧ ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀ := by
    simpa only [ofPred_and, subset_inter_iff] using!
      nhds_basis_closedBall.mem_iff.mp (h2.and (eventually_nhdsWithin_iff.mp h1))
  replace h3 : DiffContOnCl ℂ f (ball z₀ ρ) :=
    ⟨h3.differentiableOn.mono ball_subset_closedBall,
      (closure_ball z₀ hρ.lt.ne.symm).symm ▸ h3.continuousOn⟩
  let r := ρ ⊓ R
  have hr : 0 < r := lt_inf_iff.mpr ⟨hρ, hR⟩
  have h5 : closedBall z₀ r ⊆ closedBall z₀ ρ := closedBall_subset_closedBall inf_le_left
  have h6 : DiffContOnCl ℂ f (ball z₀ r) := h3.mono (ball_subset_ball inf_le_left)
  have h7 : ∀ z ∈ sphere z₀ r, f z ≠ f z₀ := fun z hz =>
    h4 z (h5 (sphere_subset_closedBall hz)) (ne_of_mem_sphere hz hr.ne.symm)
  have h8 : (sphere z₀ r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
  have h9 : ContinuousOn (fun x => ‖f x - f z₀‖) (sphere z₀ r) := continuous_norm.comp_continuousOn
    ((h6.sub_const (f z₀)).continuousOn_ball.mono sphere_subset_closedBall)
  obtain ⟨x, hx, hfx⟩ := (isCompact_sphere z₀ r).exists_isMinOn h8 h9
  refine ⟨‖f x - f z₀‖ / 2, half_pos (norm_sub_pos_iff.mpr (h7 x hx)), ?_⟩
  exact (h6.ball_subset_image_closedBall hr (fun z hz => hfx hz) (not_eventually.mp h)).trans
    (by gcongr; exact inf_le_right)

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

Goal state before:
pp: h : ℝ
⊢ Differentiable ℂ (𝕢 h)

Tactic:
unfold qParam

Goal state after:
h : ℝ
⊢ Differentiable ℂ fun z ↦ cexp (2 * ↑π * I * z / ↑h)
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
m : WithTop ℕ∞
⊢ ContDiff ℂ m (𝕢 h)

Tactic:
unfold qParam

Goal state after:
h : ℝ
m : WithTop ℕ∞
⊢ ContDiff ℂ m fun z ↦ cexp (2 * ↑π * I * z / ↑h)
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
hh : 0 < h
⊢ Tendsto (𝕢 h) I∞ (𝓝 0)

Tactic:
rw [tendsto_zero_iff_norm_tendsto_zero]

Goal state after:
h : ℝ
hh : 0 < h
⊢ Tendsto (fun x ↦ ‖𝕢 h x‖) I∞ (𝓝 0)
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
hh : 0 < h
⊢ Tendsto (fun y ↦ rexp (-2 * π * y / h)) atTop (𝓝 0)

Tactic:
refine Real.tendsto_exp_atBot.comp (.atBot_div_const hh (tendsto_id.const_mul_atTop_of_neg ?_))

Goal state after:
h : ℝ
hh : 0 < h
⊢ -2 * π < 0
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
g : E → ℂ
z₀ : E
hg : AnalyticAt ℂ g z₀
ray : E → ℂ → E := fun z t ↦ z₀ + t • z
gray : E → ℂ → ℂ := fun z ↦ g ∘ ray z
r : ℝ
hr : r > 0
hgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}
z : E
h1 : AnalyticAt ℂ (gray z) 0
hz : z ∈ sphere 0 1
hrz : ¬∀ᶠ (t : ℂ) in 𝓝 0, gray z t = gray z 0
h7 : 𝓝 (gray z 0) ≤ map (gray z) (𝓝 0)
⊢ 𝓝 (g z₀) ≤ map g (𝓝 z₀)

Tactic:
show gray z 0 = g z₀ by simp [gray, ray],

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
g : E → ℂ
z₀ : E
hg : AnalyticAt ℂ g z₀
ray : E → ℂ → E := fun z t ↦ z₀ + t • z
gray : E → ℂ → ℂ := fun z ↦ g ∘ ray z
r : ℝ
hr : r > 0
hgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}
z : E
h1 : AnalyticAt ℂ (gray z) 0
hz : z ∈ sphere 0 1
hrz : ¬∀ᶠ (t : ℂ) in 𝓝 0, gray z t = gray z 0
h7 : 𝓝 (g z₀) ≤ map (gray z) (𝓝 0)
⊢ 𝓝 (g z₀) ≤ map g (𝓝 z₀)
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).re = c₀
h₁U : IsOpen U
h₂U : IsConnected U
z₀ : ℂ
h✝ : z₀ ∈ U
h₅ : ¬∃ c, ∀ x ∈ U, f x = c
⊢ re '' f '' U = {c₀}

Tactic:
aesop

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

Goal state before:
pp: case h
f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).re = c₀
h₁U : IsOpen U
h₂U : IsConnected U
cc : ℂ
hcc : ∀ x ∈ U, f x = cc
⊢ ∀ x ∈ U, (f x).re = (↑c₀ + ↑cc.im * I).re ∧ (f x).im = (↑c₀ + ↑cc.im * I).im

Tactic:
aesop

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

Goal state before:
pp: f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).im = c₀
h₁U : IsOpen U
h₂U : IsConnected U
z₀ : ℂ
h✝ : z₀ ∈ U
h₅ : ¬∃ c, ∀ x ∈ U, f x = c
⊢ im '' f '' U = {c₀}

Tactic:
aesop

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

Goal state before:
pp: case h
f : ℂ → ℂ
U : Set ℂ
c₀ : ℝ
h₁f : AnalyticOnNhd ℂ f U
h₂f : ∀ x ∈ U, (f x).im = c₀
h₁U : IsOpen U
h₂U : IsConnected U
cc : ℂ
hcc : ∀ x ∈ U, f x = cc
⊢ ∀ x ∈ U, (f x).re = (↑cc.re + ↑c₀ * I).re ∧ (f x).im = (↑cc.re + ↑c₀ * I).im

Tactic:
aesop

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

Goal state before:
pp: case inr
p : ℂ[X]
h : ¬∃ x, C x = p
hp : IsOpenMap fun x ↦ eval x p
⊢ IsOpenQuotientMap fun x ↦ eval x p

Tactic:
exact ⟨IsAlgClosed.eval_surjective <| natDegree_eq_zero.not.mpr h, p.continuous_aeval, hp⟩

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

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
⊢ Tendsto f I∞ (𝓝 (cuspFunction h f 0))

Tactic:
suffices Tendsto (cuspFunction h f) (𝓝[≠] 0) (𝓝 <| cuspFunction h f 0) by
    simpa only [Function.comp_def, eq_cuspFunction hh.ne' hf] using this.comp (qParam_tendsto hh)
  exact tendsto_nhdsWithin_of_tendsto_nhds
    (differentiableAt_cuspFunction_zero hh hf h_hol h_bd).continuousAt.tendsto

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

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3

Tactic:
have := @integral_sin_pow_mul_cos_pow_odd a b 2 0

Goal state after:
a b : ℝ
this : ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x ^ (2 * 0 + 1) = ∫ (u : ℝ) in sin a..sin b, u ^ 2 * (1 - u ^ 2) ^ 0
⊢ ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3
Module: Mathlib.Analysis.Complex.Positivity

Goal state before:
pp: f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ ↑(z - c).re
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * ↑(z - c).re ^ n = f z
hz' : z - c = ↑(z - c).re
n : ℕ
⊢ 0 ≤ (↑n.factorial)⁻¹ * iteratedDeriv n f c * ↑(z - c).re ^ n

Tactic:
rw [← ofReal_natCast, ← ofReal_pow, ← ofReal_inv, eq_re_of_ofReal_le (h n), ← ofReal_mul,
    ← ofReal_mul]

Goal state after:
f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ ↑(z - c).re
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * ↑(z - c).re ^ n = f z
hz' : z - c = ↑(z - c).re
n : ℕ
⊢ 0 ≤ ↑((↑n.factorial)⁻¹ * (iteratedDeriv n f c).re * (z - c).re ^ n)
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
n : ℕ
hr_lt : ∀ (n : ℕ), 0 < ↑n + 2
⊢ R - ‖w‖ ≤ R * 1

Tactic:
aesop

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

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
this : ∀ {n : ℕ}, circleAverage (herglotzLogIntegrand w ρ) 0 (r n) = log ‖w - ρ‖
⊢ circleAverage (Complex.re ∘ herglotzRieszKernel 0 w • fun x ↦ log ‖x - ρ‖) 0 R = log ‖w - ρ‖

Tactic:
aesop

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

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
⊢ ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0

Tactic:
by
  -- WLOG, `0 ∉ U`, otherwise we choose `a ∉ U` and replace `U` with `-a +ᵥ U`
  wlog hU₀ : 0 ∉ U
  · rw [ne_univ_iff_exists_notMem] at hU
    rcases hU with ⟨a, ha⟩
    specialize this (hUo.vadd (-a)) (by simpa) (by simp [hU])
      (by simpa [mem_vadd_set_iff_neg_vadd_mem])
    rcases this with ⟨f, hf_inj, hf_dense, hdf⟩
    refine ⟨f ∘ (-a + ·), hf_inj.comp (add_right_injective (-a)), ?_, fun z hz ↦ ?_⟩
    · simpa only [← image_vadd, Set.image_image] using! hf_dense
    · simpa [Function.comp_def, deriv_comp_const_add] using hdf (-a + z) (mapsTo_image _ _ hz)
  -- Choose a continuous branch of `√z` on `U`.
  -- This is the function we're looking for.
  rcases exists_continuousOn_pow_eq hUc hUo continuousOn_id (by rwa [image_id]) two_ne_zero
    with ⟨f, hfc, hf_inv⟩
  replace hf_inv : LeftInverse (· ^ 2) f := hf_inv
  -- Then `0 ∉ f '' U`
  have hf₀ : ∀ z ∈ U, f z ≠ 0 := by
    intro z hz hfz
    simpa [hfz, (ne_of_mem_of_not_mem hz hU₀).symm] using hf_inv z
  -- Note that `f` is strictly differentiable at every point of `U`
  -- with derivative `1 / (2 * f z)`.
  have hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z := by
    intro z hz
    apply HasStrictDerivAt.of_local_left_inverse
    · exact hfc.continuousAt <| hUo.mem_nhds hz
    · simpa using hasStrictDerivAt_pow 2 (f z)
    · simpa using hf₀ z hz
    · exact .of_forall hf_inv
  -- `f` has a left inverse, so it's injective. Let's show that `f '' U` isn't dense in `ℂ`.
  refine ⟨f, hf_inv.injective, ?_, fun z hz ↦ ?_⟩
  · -- Take a point `x ∈ U`.
    simp only [Dense, not_forall, mem_closure_iff_frequently, not_frequently]
    rcases hUc.nonempty with ⟨x, hx⟩
    -- Show that `-f x` has a neighborhood disjoint with `f '' U`.
    use -f x
    -- Since `f` is strictly differentiable at `x` with nonzero derivative,
    -- `f '' U` is a neighborhood of `f x`.
    have : f '' U ∈ 𝓝 (f x) := by
      rw [← (hdf x hx).map_nhds_eq (by simpa using hf₀ x hx)]
      exact Filter.image_mem_map <| hUo.mem_nhds hx
    -- Then `-f '' U` is a neighborhood of `-f x`.
    rw [nhds_neg, eventually_neg]
    -- This neighborhood has to be disjoint with `f '' U`,
    -- because `f a = - f b` implies `a = (f a) ^ 2 = (- f b) ^ 2 = b`, hence `f a = f b = 0`,
    -- which is impossible.
    filter_upwards [this]
    rintro _ ⟨a, ha, rfl⟩ ⟨b, hb, hab⟩
    obtain rfl : a = b := by
      rw [← hf_inv b, hab]
      simp [hf_inv a]
    refine hf₀ a ha ?_
    linear_combination hab / 2
  · simpa [(hdf z hz).hasDerivAt.deriv] using hf₀ z hz

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

Goal state before:
pp: case h
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
u : ℂ
⊢ u ∉ ↑h.toFinset → u ∉ Function.support fun u x ↦ ↑(D u) * log ‖x - u‖

Tactic:
aesop

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

Goal state before:
pp: case h
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
u : ℂ
⊢ (u ∈ Function.support fun u ↦ ↑(D u) * log R) → u ∈ ↑h.toFinset

Tactic:
aesop

Goal state after:
Module: Mathlib.Order.Monotone.Union

Goal state before:
pp: case inl.inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : StrictMonoOn f s
h₂ : StrictMonoOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
B : ∀ x ∈ s ∪ t, c ≤ x → x ∈ t
x : α
hx : x ∈ s ∪ t
y : α
hy : y ∈ s ∪ t
hxy : x < y
hxc : x < c
xs : x ∈ s
hyc : y < c
⊢ f x < f y

Tactic:
exact h₁ xs (A _ hy hyc.le) hxy

Goal state after:
Module: Mathlib.Order.Monotone.Union

Goal state before:
pp: case inl.inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : MonotoneOn f s
h₂ : MonotoneOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
B : ∀ x ∈ s ∪ t, c ≤ x → x ∈ t
x : α
hx : x ∈ s ∪ t
y : α
hy : y ∈ s ∪ t
hxy : x ≤ y
hxc : x < c
xs : x ∈ s
hyc : y < c
⊢ f x ≤ f y

Tactic:
exact h₁ xs (A _ hy hyc.le) hxy

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

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall c |R|)
CB : Set ℂ := closedBall c |R|
h₂f : ¬∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤
⊢ ¬∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u ≠ ⊤

Tactic:
aesop

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

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall c |R|)
CB : Set ℂ := closedBall c |R|
h₂f✝ : ¬∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤
h₂f : ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u = ⊤
this : divisor f CB = 0
⊢ meromorphicOrderAt f c = ⊤

Tactic:
aesop

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

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a : 𝕜
h : im I = 1
⊢ (map ℂ 𝕜)
      (↑√((‖(map 𝕜 ℂ) a‖ + ((map 𝕜 ℂ) a).re) / 2) +
        (if 0 ≤ ((map 𝕜 ℂ) a).im then 1 else -1) * ↑√((‖(map 𝕜 ℂ) a‖ - ((map 𝕜 ℂ) a).re) / 2) * Complex.I) =
    ↑√((‖a‖ + re a) / 2) + (if 0 ≤ im a then 1 else -1) * ↑√((‖a‖ - re a) / 2) * I

Tactic:
aesop

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

Goal state before:
pp: X : Type u
inst✝⁵ : TopologicalSpace X
inst✝⁴ : NormalSpace X
s : Set X
hs : IsClosed s
𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
f : ↑s →ᵇ E
hf : ‖f‖ = 0
⊢ ‖0‖ = ‖f‖ ∧ domRestrict 0 s = f

Tactic:
aesop

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
u : Rˣ
⊢ mk ((GeneralLinearGroup.scalar n) u) = 1

Tactic:
rw [← MonoidHom.mem_ker, ker_mk, GeneralLinearGroup.center_eq_range_scalar]
  simp

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Nonempty n
F : Type u_3
inst✝¹ : Field F
inst✝ : IsAlgClosed F
r : Fˣ
x : F
hx : (X ^ Fintype.card n - C ↑r).IsRoot x
⊢ x ≠ 0

Tactic:
aesop

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: case left
R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
h : m.IsParabolic
⊢ m ∉ (scalar (Fin 2)).range

Tactic:
exact h.1

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

Goal state before:
pp: g g' : GL (Fin 2) ℝ
z : ℂ
⊢ (if 0 < ↑(GeneralLinearGroup.det g) * ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ
      else Complex.conjCAE)
      z =
    (if 0 < ↑(GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE)
      ((if 0 < ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE) z)

Tactic:
rcases g.det_ne_zero.lt_or_gt with (h | h)

Goal state after:
case inl
g g' : GL (Fin 2) ℝ
z : ℂ
h : (↑g).det < 0
⊢ (if 0 < ↑(GeneralLinearGroup.det g) * ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ
      else Complex.conjCAE)
      z =
    (if 0 < ↑(GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE)
      ((if 0 < ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE) z)

case inr
g g' : GL (Fin 2) ℝ
z : ℂ
h : 0 < (↑g).det
⊢ (if 0 < ↑(GeneralLinearGroup.det g) * ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ
      else Complex.conjCAE)
      z =
    (if 0 < ↑(GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE)
      ((if 0 < ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE) z)