mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℝ
a : ℝ
hf : CircleIntegrable f c R
h₂f : ∀ x ∈ sphere c |R|, f x ≤ a
⊢ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, a

Tactic:
smul_eq_mul,

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℝ
a : ℝ
hf : CircleIntegrable f c R
h₂f : ∀ x ∈ sphere c |R|, f x ≤ a
⊢ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) ≤ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, a
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ |(2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)

Tactic:
smul_eq_mul,

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ |(2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ |(2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)

Tactic:
smul_eq_mul,

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ |(2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: case pos
c : ℂ
R : ℝ
f : ℂ → ℝ
h₂f : ∀ x ∈ sphere c |R|, 0 ≤ f x
hf : CircleIntegrable f c R
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, 0 ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)

Tactic:
smul_eq_mul,

Goal state after:
case pos
c : ℂ
R : ℝ
f : ℂ → ℝ
h₂f : ∀ x ∈ sphere c |R|, 0 ≤ f x
hf : CircleIntegrable f c R
⊢ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, 0 ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: case pos
c : ℂ
R : ℝ
f : ℂ → ℝ
h₂f : ∀ x ∈ sphere c |R|, 0 ≤ f x
hf : CircleIntegrable f c R
⊢ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, 0 ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)

Tactic:
smul_eq_mul,

Goal state after:
case pos
c : ℂ
R : ℝ
f : ℂ → ℝ
h₂f : ∀ x ∈ sphere c |R|, 0 ≤ f x
hf : CircleIntegrable f c R
⊢ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, 0 ≤ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)
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
w : ℂ
hw : w ∈ Ioi 0 ×ℂ Ioi 0
⊢ (fun x ↦ x * I) w ∈ Iio 0 ×ℂ Ioi 0

Tactic:
by
    simpa only [mem_reProdIm, mul_I_re, mul_I_im, neg_lt_zero, mem_Iio] using! hw.symm

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

Goal state before:
pp: case inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) (-1) = -(-1)⁻¹ / log (-1) ^ 2

Tactic:
deriv_zero_of_not_differentiableAt ‹_›

Goal state after:
case inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ 0 = -(-1)⁻¹ / log (-1) ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: case inr.inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) 0 = -0⁻¹ / log 0 ^ 2

Tactic:
deriv_zero_of_not_differentiableAt ‹_›

Goal state after:
case inr.inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ 0 = -0⁻¹ / log 0 ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: case inr.inr.inr
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) 1 = -1⁻¹ / log 1 ^ 2

Tactic:
deriv_zero_of_not_differentiableAt ‹_›

Goal state after:
case inr.inr.inr
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ 0 = -1⁻¹ / log 1 ^ 2
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 ∨ 0 ∉ [[a, b]]
⊢ IntervalIntegrable (fun x ↦ ↑x ^ r) μ a b

Tactic:
by_cases h2 : (0 : ℝ) ∉ [[a, b]]
  · -- Easy case #1: 0 ∉ [a, b] -- use continuity.
    refine (continuousOn_of_forall_continuousAt fun x hx => ?_).intervalIntegrable
    exact Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_mem_of_not_mem hx h2)
  rw [eq_false h2, or_false] at h
  rcases lt_or_eq_of_le h with (h' | h')
  · -- Easy case #2: 0 < re r -- again use continuity
    exact (Complex.continuous_ofReal_cpow_const h').intervalIntegrable _ _
  -- Now the hard case: re r = 0 and 0 is in the interval.
  refine (IntervalIntegrable.intervalIntegrable_norm_iff ?_).mp ?_
  · refine (measurable_of_continuousOn_compl_singleton (0 : ℝ) ?_).aestronglyMeasurable
    exact continuousOn_of_forall_continuousAt fun x hx =>
      Complex.continuousAt_ofReal_cpow_const x r (Or.inr hx)
  -- reduce to case of integral over `[0, c]`
  suffices ∀ c : ℝ, IntervalIntegrable (fun x : ℝ => ‖(x : ℂ) ^ r‖) μ 0 c from
    (this a).symm.trans (this b)
  intro c
  rcases le_or_gt 0 c with (hc | hc)
  · -- case `0 ≤ c`: integrand is identically 1
    have : IntervalIntegrable (fun _ => 1 : ℝ → ℝ) μ 0 c := intervalIntegrable_const
    rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hc] at this ⊢
    refine IntegrableOn.congr_fun this (fun x hx => ?_) measurableSet_Ioc
    rw [Complex.norm_cpow_eq_rpow_re_of_pos hx.1, ← h', rpow_zero]
  · -- case `c < 0`: integrand is identically constant, *except* at `x = 0` if `r ≠ 0`.
    apply IntervalIntegrable.symm
    rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hc.le]
    rw [← Ioo_union_right hc, integrableOn_union, and_comm]; constructor
    · exact integrableOn_singleton (by simp) <|
        isFiniteMeasureOnCompacts_of_isLocallyFiniteMeasure.lt_top_of_isCompact isCompact_singleton
    · have : ∀ x : ℝ, x ∈ Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖(x : ℂ) ^ r‖ := by
        intro x hx
        rw [Complex.ofReal_cpow_of_nonpos hx.2.le, norm_mul, ← Complex.ofReal_neg,
          Complex.norm_cpow_eq_rpow_re_of_pos (neg_pos.mpr hx.2), ← h',
          rpow_zero, one_mul]
      refine IntegrableOn.congr_fun ?_ this measurableSet_Ioo
      rw [integrableOn_const_iff]
      right
      refine (measure_mono Set.Ioo_subset_Icc_self).trans_lt ?_
      exact isFiniteMeasureOnCompacts_of_isLocallyFiniteMeasure.lt_top_of_isCompact isCompact_Icc

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

Goal state before:
pp: 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

Tactic:
grw [thickening_singleton] at h_disj

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 (ball w d) (thickening d (ball c R)ᶜ)
⊢ ball w d ⊆ ball c R
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: 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‖

Tactic:
grw [thickening_singleton] at h_disj

Goal state after:
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 (ball w d) (thickening d (ball c R)ᶜ)
⊢ ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap c R θ - x‖
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)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log ‖f x‖) c R

Tactic:
← abs_of_nonneg hR

Goal state after:
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|)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log ‖f x‖) c R
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)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log⁺ ‖f x‖) c R

Tactic:
← abs_of_nonneg hR

Goal state after:
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|)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log⁺ ‖f x‖) c R
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a c : ℂ
R : ℝ
hR : R ≠ 0
⊢ circleAverage (fun x ↦ log ‖x - a‖) c R = circleAverage (fun z ↦ log ‖↑R * (z + ↑R⁻¹ * (c - a))‖) 0 1

Tactic:
by
    rw [circleAverage_eq_circleAverage_zero_one]
    congr
    ext z
    congr
    rw [Complex.ofReal_inv R]
    field [Complex.ofReal_ne_zero.mpr hR]

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

Goal state before:
pp: q : ℤ[X]
this : ¬0 < q.natDegree
p : ℤ[X]
h : ∀ (z : ℂ), ¬((aeval z) (q * p) = 0 ∧ (aeval z) (q * p).mirror = 0)
hq' : q ∣ (q * p).mirror
hp : IsUnit (q * p).leadingCoeff
⊢ IsUnit q

Tactic:
rw [leadingCoeff_mul] at hp

Goal state after:
q : ℤ[X]
this : ¬0 < q.natDegree
p : ℤ[X]
h : ∀ (z : ℂ), ¬((aeval z) (q * p) = 0 ∧ (aeval z) (q * p).mirror = 0)
hq' : q ∣ (q * p).mirror
hp : IsUnit (q.leadingCoeff * p.leadingCoeff)
⊢ IsUnit q
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: case e'_5
c : ℂ
r : ℝ
z : ℂ
hz : z ∈ Metric.ball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.ball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c

Tactic:
smul_eq_mul,

Goal state after:
case e'_5
c : ℂ
r : ℝ
z : ℂ
hz : z ∈ Metric.ball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.ball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ * (z - c) ^ n • iteratedDeriv n f c
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: case e'_5
c : ℂ
r : ℝ
z : ℂ
hz : z ∈ Metric.ball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.ball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ * (z - c) ^ n • iteratedDeriv n f c

Tactic:
smul_eq_mul,

Goal state after:
case e'_5
c : ℂ
r : ℝ
z : ℂ
hz : z ∈ Metric.ball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.ball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c)
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: case e'_5
c : ℂ
r : ENNReal
z : ℂ
hz : z ∈ Metric.eball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.eball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c

Tactic:
smul_eq_mul,

Goal state after:
case e'_5
c : ℂ
r : ENNReal
z : ℂ
hz : z ∈ Metric.eball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.eball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ * (z - c) ^ n • iteratedDeriv n f c
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: case e'_5
c : ℂ
r : ENNReal
z : ℂ
hz : z ∈ Metric.eball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.eball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ * (z - c) ^ n • iteratedDeriv n f c

Tactic:
smul_eq_mul,

Goal state after:
case e'_5
c : ℂ
r : ENNReal
z : ℂ
hz : z ∈ Metric.eball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.eball c r)
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c)
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: case e'_5
c z : ℂ
f : ℂ → ℂ
hf : Differentiable ℂ f
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c

Tactic:
smul_eq_mul,

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

Goal state before:
pp: case e'_5
c z : ℂ
f : ℂ → ℂ
hf : Differentiable ℂ f
n : ℕ
⊢ (↑n !)⁻¹ * (z - c) ^ n * iteratedDeriv n f c = (↑n !)⁻¹ * (z - c) ^ n • iteratedDeriv n f c

Tactic:
smul_eq_mul,

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

Goal state before:
pp: 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
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n ↦ (fun g ↦ g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
⊢ ∃ g, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f

Tactic:
have :
    Tendsto (fun n => (g n).compContinuous e) atTop
      (𝓝 <| (limUnder atTop g).compContinuous e) :=
    ((continuous_compContinuous e).tendsto _).comp hg_cau.tendsto_limUnder

Goal state after:
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
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n ↦ (fun g ↦ g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
this : Tendsto (fun n ↦ (g n).compContinuous e) atTop (𝓝 ((atTop.limUnder g).compContinuous e))
⊢ ∃ g, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: case refine_1
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
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n ↦ (fun g ↦ g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
this : Tendsto (fun n ↦ (g n).compContinuous e) atTop (𝓝 ((atTop.limUnder g).compContinuous e))
hge : (atTop.limUnder g).compContinuous e = f
⊢ dist 0 (atTop.limUnder g) ≤ ‖f‖

Tactic:
← g0

Goal state after:
case refine_1
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
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n ↦ (fun g ↦ g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
this : Tendsto (fun n ↦ (g n).compContinuous e) atTop (𝓝 ((atTop.limUnder g).compContinuous e))
hge : (atTop.limUnder g).compContinuous e = f
⊢ dist (g 0) (atTop.limUnder g) ≤ ‖f‖
Module: Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction

Goal state before:
pp: s : SL(2, ℤ)
g : ↥GL(2, ℝ)⁺
z : ℍ
⊢ (toGLPos ((SpecialLinearGroup.map (Int.castRingHom ℝ)) s) * g) • z = s • g • z

Tactic:
apply mul_smul'

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
⊢ M k k • 1 * ((M * auxMat M k).toSquareBlock (fun x ↦ x = k) False).det =
    M k k * ((M * auxMat M k).toSquareBlock (fun x ↦ x = k) False).det

Tactic:
smul_eq_mul,

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
⊢ M k k * 1 * ((M * auxMat M k).toSquareBlock (fun x ↦ x = k) False).det =
    M k k * ((M * auxMat M k).toSquareBlock (fun x ↦ x = k) False).det
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Manifold

Goal state before:
pp: case right
n : ℕ∞ω
z : ℂ
hz : 0 < z.im
⊢ ContDiffWithinAt ℂ n (↑(extChartAt 𝓘(ℂ, ℂ) (↑ofComplex z)) ∘ ↑ofComplex ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z).symm)
    (Set.range ↑𝓘(ℂ, ℂ)) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)

Tactic:
· -- smoothness in local chart
    simpa using! contDiffAt_id.congr_of_eventuallyEq (eventuallyEq_coe_comp_ofComplex hz)

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

Goal state before:
pp: x✝ : ℝ
⊢ (deriv cosh / cosh) x✝ = tanh x✝

Tactic:
Real.deriv_cosh,

Goal state after:
x✝ : ℝ
⊢ (sinh / cosh) x✝ = tanh x✝
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: case e_a
a : { x // 0 < x }
y₁ y₂ : ℍ
⊢ ‖↑a‖ * dist ↑y₁ ↑y₂ / (2 * √(↑a * ↑a * (y₁.im * y₂.im))) = dist ↑y₁ ↑y₂ / (2 * √(y₁.im * y₂.im))

Tactic:
Real.sqrt_mul (mul_self_nonneg _),

Goal state after:
case e_a
a : { x // 0 < x }
y₁ y₂ : ℍ
⊢ ‖↑a‖ * dist ↑y₁ ↑y₂ / (2 * (√(↑a * ↑a) * √(y₁.im * y₂.im))) = dist ↑y₁ ↑y₂ / (2 * √(y₁.im * y₂.im))
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)
⊢ (∞ ∈ ⋃₀ S → IsCompact (some ⁻¹' ⋃₀ S)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' ⋃₀ S)

Tactic:
suffices IsOpen ((↑) ⁻¹' ⋃₀ S : Set X) by
      refine ⟨?_, this⟩
      rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩
      refine IsCompact.of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl ?_
      exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)
    rw [preimage_sUnion]
    exact isOpen_biUnion fun s hs => (ho s hs).2

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

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
s : Set (OnePoint X✝)
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x : OnePoint X
x✝¹ : x ∈ univ
y : OnePoint X
x✝ : y ∈ univ
hxy : x ≠ y
⊢ (fun x y ↦
      ∃ u v,
        IsOpen[instTopologicalSpace] u ∧ IsOpen[instTopologicalSpace] v ∧ x ∈ u ∧ y ∈ v ∧ univ ⊆ u ∪ v ∧ Disjoint u v)
    x y

Tactic:
by
    cases x with
    | infty =>
      refine ⟨{y}ᶜ, {y}, isOpen_compl_singleton, ?_, hxy, rfl, (compl_union_self _).symm.subset,
        disjoint_compl_left⟩
      rw [OnePoint.isOpen_iff_of_notMem]
      exacts [isOpen_discrete _, hxy]
    | coe val =>
      refine ⟨{some val}, {some val}ᶜ, ?_, isOpen_compl_singleton, rfl, hxy.symm, by simp,
        disjoint_compl_right⟩
      rw [OnePoint.isOpen_iff_of_notMem]
      exacts [isOpen_discrete _, (Option.some_ne_none val).symm]

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction

Goal state before:
pp: case h₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
α : Type u_2
s : Finset α
f : α → ℂ → E
r : ℝ
hf : ∀ a ∈ s, Meromorphic (f a)
hr : 1 ≤ r
⊢ logCounting (∑ a ∈ s, f a) ⊤ r ≤ (∑ a ∈ s, logCounting (f a) ⊤) r

Tactic:
apply logCounting_sum_top_le s f hf hr

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation

Goal state before:
pp: f : ℂ → ℂ
R : ℝ
h : Meromorphic f
μ : Measure ℝ := volume.restrict (Ioc 0 (2 * π))
h_meas_K : Measurable fun a ↦ (cartanKernel f R a.1 a.2)⁺
h_int_posLog : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖) μ
h_int_Bound : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖ + log 2) μ
β : ℝ
h_int_nonneg : 0 ≤ ∫ (α : ℝ), (cartanKernel f R α β)⁺ ∂μ
h_bound_nonneg : 0 ≤ 2 * π * (log⁺ ‖f (circleMap 0 R β)‖ + log 2)
this :
  ∫ (α : ℝ) in Ioc 0 (2 * π), (cartanKernel f R α β)⁺ ≤
    (volume.restrict (Ioc 0 (2 * π))).real univ • (log⁺ ‖f (circleMap 0 R β)‖ + log 2)
⊢ ∫ (α : ℝ), (cartanKernel f R α β)⁺ ∂μ ≤ 2 * π * (log⁺ ‖f (circleMap 0 R β)‖ + log 2)

Tactic:
smul_eq_mul,

Goal state after:
f : ℂ → ℂ
R : ℝ
h : Meromorphic f
μ : Measure ℝ := volume.restrict (Ioc 0 (2 * π))
h_meas_K : Measurable fun a ↦ (cartanKernel f R a.1 a.2)⁺
h_int_posLog : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖) μ
h_int_Bound : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖ + log 2) μ
β : ℝ
h_int_nonneg : 0 ≤ ∫ (α : ℝ), (cartanKernel f R α β)⁺ ∂μ
h_bound_nonneg : 0 ≤ 2 * π * (log⁺ ‖f (circleMap 0 R β)‖ + log 2)
this :
  ∫ (α : ℝ) in Ioc 0 (2 * π), (cartanKernel f R α β)⁺ ≤
    (volume.restrict (Ioc 0 (2 * π))).real univ * (log⁺ ‖f (circleMap 0 R β)‖ + log 2)
⊢ ∫ (α : ℝ), (cartanKernel f R α β)⁺ ∂μ ≤ 2 * π * (log⁺ ‖f (circleMap 0 R β)‖ + log 2)
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: case inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
h : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
xs : x ∈ s
y : ℝ
ys : y ∈ s
yx : y ≤ x
⊢ ↑l * (y - x) ≤ 0

Tactic:
exact mul_nonpos_of_nonneg_of_nonpos l.prop (sub_nonpos_of_le yx)

Goal state after:
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: case mp.inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
h : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
xs : x ∈ s
y : ℝ
ys : y ∈ s
yx : y ≤ x
⊢ variationOnFromTo f s x y = ↑l * (y - x)

Tactic:
rw [variationOnFromTo.eq_of_ge f s yx, h ys xs yx]

Goal state after:
case mp.inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
h : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
xs : x ∈ s
y : ℝ
ys : y ∈ s
yx : y ≤ x
⊢ -(ENNReal.ofReal (↑l * (x - y))).toReal = ↑l * (y - x)