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

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz1 : z.re ∈ {1}
⊢ (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) * ‖f z‖ ≤ 1

Tactic:
mem_singleton_iff

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz1 : z.re = 1
⊢ (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) * ‖f z‖ ≤ 1
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case neg
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
z : ℂ
⊢ DifferentiableAt ℂ (fun z ↦ interpStrip (fun a ↦ f a) z) z

Tactic:
simp_rw [interpStrip_eq_of_pos f _ (lt_of_le_of_ne (sSupNormIm_nonneg f 0) h0)
      (lt_of_le_of_ne (sSupNormIm_nonneg f 1) h1)]

Goal state after:
case neg
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
z : ℂ
⊢ DifferentiableAt ℂ (fun z ↦ ↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z) z
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)
I₂ : E := I • ∫ (y : ℝ) in c.im..w.im, f (↑w.re + ↑y * I)
I₃ : E := ∫ (x : ℝ) in c.re..z.re, f (↑x + ↑c.im * I)
I₄ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑z.re + ↑y * I)
I₅ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₆ : E := I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
I₇ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑c.im * I)
I₈ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑w.re + ↑y * I)
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
⊢ c.wedgeIntegral w f - c.wedgeIntegral z f = z.wedgeIntegral w f

Tactic:
have integrableVert (a b₁ b₂ : ℝ) (hb₁ : a + b₁ * I ∈ ball c r) (hb₂ : a + b₂ * I ∈ ball c r) :
      IntervalIntegrable (fun y ↦ f (a + y * I)) volume b₁ b₂ :=
    ((f_cont.mono (mem_ball_of_map_im_aux₁ hb₁ hb₂)).comp (by fun_prop)
      (mapsTo_image _ _)).intervalIntegrable

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)
I₂ : E := I • ∫ (y : ℝ) in c.im..w.im, f (↑w.re + ↑y * I)
I₃ : E := ∫ (x : ℝ) in c.re..z.re, f (↑x + ↑c.im * I)
I₄ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑z.re + ↑y * I)
I₅ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₆ : E := I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
I₇ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑c.im * I)
I₈ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑w.re + ↑y * I)
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
⊢ c.wedgeIntegral w f - c.wedgeIntegral z f = z.wedgeIntegral w f
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case hf.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      ‖↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z‖
⊢ 0 ≤ 1 - z.re

Tactic:
simp only [sub_nonneg, le_of_lt hz.2]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
⊢ ‖f z‖ ≤ ‖interpStrip f z‖

Tactic:
apply le_on_closure (fun w hw ↦ norm_le_interpStrip_of_mem_verticalStrip_zero f w hd hB hw)
    (Continuous.comp_continuousOn' continuous_norm hd.2)
    (Continuous.comp_continuousOn' continuous_norm (diffContOnCl_interpStrip f).2)
  rwa [verticalClosedStrip, ← closure_Ioo zero_ne_one, ← closure_preimage_re] at hz

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

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
hR : R < 0
⊢ circleAverage f c |R| = circleAverage f c R

Tactic:
rw [abs_of_neg hR, circleAverage_neg_radius]

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

Goal state before:
pp: φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
h_subst : (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)) ≤ (R + r) / (R - r)
this :
  R * R * normSq (cexp (↑θ * I)) + r * r * normSq (cexp (↑φ * I)) -
      2 * (R * Real.cos θ * (r * Real.cos φ) + R * Real.sin θ * (r * Real.sin φ)) =
    R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)
⊢ (‖↑R * cexp (↑θ * I)‖ ^ 2 - ‖↑r * cexp (↑φ * I)‖ ^ 2) / ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2 =
    (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ))

Tactic:
rw [← this]; simp [← normSq_eq_norm_sq, Complex.normSq_sub]

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

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
h : 0 ≤ z.im
⊢ ‖f z‖ ≤ max C C'

Tactic:
· refine quadrant_I (hd.mono fun _ => And.left) ⟨c, hc, B, ?_⟩
          (fun x hx => (hC' x hx).trans <| le_max_right _ _)
          (fun x _ => (him x).trans (le_max_left _ _)) hz h
      exact hO.mono (inf_le_inf_left _ <| principal_mono.2 fun _ => And.left)

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

Goal state before:
pp: case inr.inl
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) / (z - 0)).re • f z = f z

Tactic:
simp [div_self (a := z) (by aesop)]

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

Goal state before:
pp: case h
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntegrableOn ?f (Ι 0 c) volume

Tactic:
← intervalIntegrable_iff

Goal state after:
case h
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable ?m.108 volume 0 c

a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ ℝ → ℝ
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: ⊢ log⁺ 0 = 0

Tactic:
by simp [posLog]

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

Goal state before:
pp: ⊢ log⁺ 1 = 0

Tactic:
by simp [posLog]

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

Goal state before:
pp: x : ℝ
⊢ 0 ≤ log⁺ x

Tactic:
by simp [posLog]

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

Goal state before:
pp: x : ℝ
⊢ log⁺ (-x) = log⁺ x

Tactic:
by simp [posLog]

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

Goal state before:
pp: x : ℝ
⊢ log⁺ |x| = log⁺ x

Tactic:
by simp [posLog]

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

Goal state before:
pp: x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
⊢ log⁺ x ≤ log⁺ y

Tactic:
simp only [posLog, le_sup_iff, sup_le_iff, le_refl, true_and]

Goal state after:
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: case pos.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
t₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤
g : ℝ → E
h₁g : AnalyticOnNhd ℝ g [[a, b]]
h₂g : ∀ (u : ↑[[a, b]]), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin [[a, b]]] (∏ᶠ (u : ℝ), (fun x ↦ x - u) ^ (divisor f [[a, b]]) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin [[a, b]]]
    (∑ᶠ (u : ℝ), fun x ↦ ↑((divisor f [[a, b]]) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
i : ℝ
⊢ IntervalIntegrable (fun x ↦ ↑((divisor f [[a, b]]) i) * log ‖x - i‖) volume a b

Tactic:
apply IntervalIntegrable.const_mul

Goal state after:
case pos.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
t₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤
g : ℝ → E
h₁g : AnalyticOnNhd ℝ g [[a, b]]
h₂g : ∀ (u : ↑[[a, b]]), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin [[a, b]]] (∏ᶠ (u : ℝ), (fun x ↦ x - u) ^ (divisor f [[a, b]]) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin [[a, b]]]
    (∑ᶠ (u : ℝ), fun x ↦ ↑((divisor f [[a, b]]) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
i : ℝ
⊢ IntervalIntegrable (fun x ↦ log ‖x - i‖) volume a b
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

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

Tactic:
rw [sub_ne_zero]

Goal state after:
a : ℂ
h : ‖a‖ < 1
x : ℂ
hx : x ∈ closedBall 0 |1|
this : ‖x * a‖ < 1
⊢ 1 ≠ x * a
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: case pos.hf
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‖
⊢ CircleIntegrable (∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f (sphere c |R|)) u) * log ‖x - u‖) c R

Tactic:
apply CircleIntegrable.finsum

Goal state after:
case pos.hf
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), CircleIntegrable (fun x ↦ ↑((divisor f (sphere c |R|)) i) * log ‖x - i‖) c R
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: case pos.hf
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
⊢ CircleIntegrable (fun x ↦ ↑((divisor f (sphere c |R|)) i) * log ‖x - i‖) c R

Tactic:
apply IntervalIntegrable.const_mul

Goal state after:
case pos.hf
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
⊢ IntervalIntegrable (fun x ↦ log ‖circleMap c R x - i‖) volume 0 (2 * π)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
⊢ ∫ (θ : ℝ) in 0..2 * π, log ‖circleMap 0 1 θ - a‖ = 0

Tactic:
obtain ⟨ζ, hζ⟩ : ∃ ζ, a⁻¹ = circleMap 0 1 ζ := by simp [Set.exists_range_iff.1, h]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
⊢ ∫ (θ : ℝ) in 0..2 * π, log ‖circleMap 0 1 θ - a‖ = 0
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 4 - 4 * cos (x / 2) ^ 2 = 4 * sin (x / 2) ^ 2

Tactic:
nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 4 * (sin (x / 2) ^ 2 + cos (x / 2) ^ 2) - 4 * cos (x / 2) ^ 2 = 4 * sin (x / 2) ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

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

Tactic:
rw [sub_ne_zero]

Goal state after:
a : ℂ
h : 1 < ‖a‖
x : ℂ
hx : x ∈ closedBall 0 |1|
⊢ x ≠ a
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
ha : 1 < a
hb : 1 < b
⊢ ∫ (t : ℝ) in a..b, t⁻¹ / log t = log (log b) - log (log a)

Tactic:
rw [← intervalIntegral.integral_congr (fun _ _ ↦ deriv_log_log_apply)]
  refine integral_deriv_eq_sub (fun _ _ ↦ ?_) ?_
  · exact differentiableOn_log_log.differentiableAt (Ioi_mem_nhds (by grind [Set.uIcc]))
  refine (?_ : ContinuousOn _ _).congr (fun _ _ ↦ deriv_log_log_apply) |>.intervalIntegrable
  fun_prop (disch := grind [log_pos, Set.uIcc])

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
⊢ (circleMap 0 r₁ θ * (starRingEnd ℂ) ρ).re = r₁ * ‖ρ‖ * Real.cos (θ - ρ.arg)

Tactic:
conv_lhs => rw [← norm_mul_exp_arg_mul_I ρ, ← circleMap_zero, conj_circleMap_zero,
      circleMap_zero_mul, circleMap_zero_re, ← sub_eq_add_neg]

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.GaussLucas

Goal state before:
pp: P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
x : ℂ
hx : x ∈ s
⊢ z - x ≠ 0

Tactic:
rw [sub_ne_zero]

Goal state after:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z (derivative P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
x : ℂ
hx : x ∈ s
⊢ z ≠ x
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
m n : ℕ
hc : Continuous fun u ↦ u ^ m * (1 - u ^ 2) ^ n
x✝ : ℝ
⊢ cos x✝ * cos x✝ = 1 - sin x✝ * sin x✝

Tactic:
rw [← sq, ← sq, cos_sq']

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

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

Tactic:
apply IntervalIntegrable.const_mul

Goal state after:
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
u : ℂ
hu : u ∈ h.toFinset
⊢ IntervalIntegrable (fun x ↦ log ‖circleMap c R x - u‖) volume 0 (2 * π)
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ ∑ i ∈ h.toFinset, circleAverage (fun x ↦ ↑(D i) * log ‖x - i‖) c R = ∑ u ∈ h.toFinset, ↑(D u) * log R

Tactic:
by
    apply Finset.sum_congr rfl
    intro u hu
    simp_rw [← smul_eq_mul, circleAverage_fun_smul]
    congr
    rw [circleAverage_log_norm_sub_const_of_mem_closedBall]
    apply D.supportWithinDomain
    simp_all

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: case inr
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
h3 : 0 < 3
h23 : 0 < 2 / 3
hf : 0 < ‖f‖
⊢ ∃ g, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖

Tactic:
have hf3 : -‖f‖ / 3 < ‖f‖ / 3 := (div_lt_div_iff_of_pos_right h3).2 (Left.neg_lt_self hf)

Goal state after:
case inr
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
h3 : 0 < 3
h23 : 0 < 2 / 3
hf : 0 < ‖f‖
hf3 : -‖f‖ / 3 < ‖f‖ / 3
⊢ ∃ g, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖
Module: Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints

Goal state before:
pp: g✝ : GL (Fin 2) ℝ
z : ℍ
g : GL (Fin 2) ℝ
hpos : 0 < (↑g).det
hell : g.IsElliptic
hc : 0 < ↑g 1 0
hd : discrim (↑(↑g 1 0)) (↑(↑g 1 1) - ↑(↑g 0 0)) (-↑(↑g 0 1)) = (Complex.I * ↑√(-(↑g).discr)) ^ 2
⊢ ↑z = (-(↑(↑g 1 1) - ↑(↑g 0 0)) + Complex.I * ↑√(-(↑g).discr)) / (2 * ↑(↑g 1 0)) ↔ z = fixedPt g hell

Tactic:
simp [fixedPt, UpperHalfPlane.ext_iff, abs_of_pos hc, field]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Preorder R
m : Matrix (Fin 2) (Fin 2) R
⊢ (-m).IsElliptic ↔ m.IsElliptic

Tactic:
simp [IsElliptic, discr_fin_two, det_neg]

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

Goal state before:
pp: τ : ℍ
U : Set ℝ
hU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)
s : Set ℍ
hs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U
ε : ℝ
hεpos : ε > 0
hεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s
r : ℝ
hr : |r - ‖↑τ‖| < ε
hr' : r < 0
⊢ False

Tactic:
abs_lt

Goal state after:
τ : ℍ
U : Set ℝ
hU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)
s : Set ℍ
hs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U
ε : ℝ
hεpos : ε > 0
hεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s
r : ℝ
hr : -ε < r - ‖↑τ‖ ∧ r - ‖↑τ‖ < ε
hr' : r < 0
⊢ False
Module: Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Algebra R ℝ
g : SL(2, R)
z : ℍ
⊢ g • z =
    {
      coe :=
        (↑((algebraMap R ℝ) (↑g 0 0)) * ↑z + ↑((algebraMap R ℝ) (↑g 0 1))) /
          (↑((algebraMap R ℝ) (↑g 1 0)) * ↑z + ↑((algebraMap R ℝ) (↑g 1 1))),
      coe_im_pos := ⋯ }

Tactic:
ext; simp [coe_specialLinearGroup_apply]

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
⊢ c - a = b - c

Tactic:
simp [c]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
⊢ (a + b) / 2 - a = b - (a + b) / 2
Module: Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction

Goal state before:
pp: g : GL (Fin 2) ℝ
τ : ℂ
⊢ denom (J * g) τ = denom g τ

Tactic:
by
  simp [denom, vecMul, vecHead, vecTail]

Goal state after:
Module: Mathlib.RingTheory.Norm.Transitivity

Goal state before:
pp: S : Type u_2
m : Type u_5
inst✝² : CommRing S
M : Matrix m m S
inst✝¹ : DecidableEq m
k : m
inst✝ : Fintype m
i j : m
lt : i = k ∧ ¬j = k
⊢ (∑ x, if x = k then -(M i x * M k j) else if x = j then M i x * M k k else 0) = 0

Tactic:
Finset.sum_ite,

Goal state after:
S : Type u_2
m : Type u_5
inst✝² : CommRing S
M : Matrix m m S
inst✝¹ : DecidableEq m
k : m
inst✝ : Fintype m
i j : m
lt : i = k ∧ ¬j = k
⊢ (∑ x with x = k, -(M i x * M k j) + ∑ x with ¬x = k, if x = j then M i x * M k k else 0) = 0
Module: Mathlib.Geometry.Euclidean.Inversion.Basic

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c x y : P
R : ℝ
hx : x ≠ c
hy : y ≠ c
hR : R ≠ 0
hy' : inversion c R y ≠ c
| dist (inversion c R x) y

Tactic:
← inversion_inversion c hR y

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c x y : P
R : ℝ
hx : x ≠ c
hy : y ≠ c
hR : R ≠ 0
hy' : inversion c R y ≠ c
| dist (inversion c R x) (inversion c R (inversion c R y))
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: case left
R : Type u_4
inst✝² : Ring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
⊢ Continuous[inst✝¹, instTopologicalSpaceMatrix] fun x ↦ !![1, x; 0, 1]

Tactic:
refine continuous_matrix fun i j ↦ ?_

Goal state after:
case left
R : Type u_4
inst✝² : Ring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
i j : Fin 2
⊢ Continuous[inst✝¹, inst✝¹] fun a ↦ !![1, a; 0, 1] i j
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: case right
R : Type u_4
inst✝² : Ring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
⊢ Continuous[inst✝¹, instTopologicalSpaceMatrix] fun x ↦ !![1, -x; 0, 1]

Tactic:
refine continuous_matrix fun i j ↦ ?_

Goal state after:
case right
R : Type u_4
inst✝² : Ring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
i j : Fin 2
⊢ Continuous[inst✝¹, inst✝¹] fun a ↦ !![1, -a; 0, 1] i j
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z : ℍ
r : ℝ
w : ℂ
h : dist w ↑(z.center r) ≤ z.im * Real.sinh r
⊢ dist (↑(z.center r)) w ≤ z.im * Real.sinh r

Tactic:
rwa [dist_comm]

Goal state after:
Module: Mathlib.Topology.Category.CompHausLike.Basic

Goal state before:
pp: P : TopCat → Prop
X Y : CompHausLike P
f : X ⟶ Y
bij : Function.Bijective ⇑(ConcreteCategory.hom f)
E : (fun X ↦ ↑X.toTop) X ≃ (fun X ↦ ↑X.toTop) Y := Equiv.ofBijective (⇑(ConcreteCategory.hom f)) bij
⊢ IsIso f

Tactic:
have hE : Continuous E.symm := by
    rw [continuous_iff_isClosed]
    intro S hS
    rw [← E.image_eq_preimage_symm]
    exact isClosedMap f S hS

Goal state after:
P : TopCat → Prop
X Y : CompHausLike P
f : X ⟶ Y
bij : Function.Bijective ⇑(ConcreteCategory.hom f)
E : (fun X ↦ ↑X.toTop) X ≃ (fun X ↦ ↑X.toTop) Y := Equiv.ofBijective (⇑(ConcreteCategory.hom f)) bij
hE : Continuous ⇑E.symm
⊢ IsIso f
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
u : Ultrafilter (Ultrafilter α)
x : Ultrafilter α
⊢ ↑u ≤ 𝓝 x ↔ joinM u = x

Tactic:
← Ultrafilter.coe_le_coe

Goal state after:
α : Type u
u : Ultrafilter (Ultrafilter α)
x : Ultrafilter α
⊢ ↑u ≤ 𝓝 x ↔ ↑(joinM u) ≤ ↑x
Module: Mathlib.Topology.Compactification.StoneCech

Goal state before:
pp: α : Type u
A B : Ultrafilter α
hB : B ∈ connectedComponent A
⊢ B = A

Tactic:
← Ultrafilter.coe_le_coe

Goal state after:
α : Type u
A B : Ultrafilter α
hB : B ∈ connectedComponent A
⊢ ↑B ≤ ↑A
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
x₁ x₂ : X
⊢ (↑x₁ = ↑x₂) = (x₁ = x₂)

Tactic:
exact Option.some.injEq x₁ x₂

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
x : OnePoint X
⊢ x ∈ {∞} ↔ x = ∞

Tactic:
mem_singleton_iff

Goal state after:
X : Type u_1
x : OnePoint X
⊢ x = ∞ ↔ x = ∞