mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case h₀
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
y : E
hy : y ∈ Metric.eball 0 r
i : ℕ
⊢ i ∈ Finset.range (n + 1) → i ≠ n → ((q i) fun x ↦ y) = 0

Tactic:
hi

Goal state after:
case h₀
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := fun k ↦ if h : n = k then h ▸ p n else 0
y : E
hy : y ∈ Metric.eball 0 r
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ i ≠ n → ((q i) fun x ↦ y) = 0
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: r : ℝ
hr : r < 1
hr' : r ≠ 0
h : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
y : ℝ := deriv (fun x ↦ x ^ r) 0
x : ℝ
hx : 0 < x
⊢ x ^ r = x * x ^ (r - 1)

Tactic:
← add_sub_cancel 1 r,

Goal state after:
r : ℝ
hr : r < 1
hr' : r ≠ 0
h : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
y : ℝ := deriv (fun x ↦ x ^ r) 0
x : ℝ
hx : 0 < x
⊢ x ^ (1 + (r - 1)) = x * x ^ (r - 1)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x : E
p : ℝ
m : ℕ
hf : ContDiffAt ℝ (↑m) f x
h : ↑m ≤ p
⊢ ContDiffAt ℝ (↑m) (fun x ↦ f x ^ p) x

Tactic:
rw [← contDiffWithinAt_univ] at *

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x : E
p : ℝ
m : ℕ
hf : ContDiffWithinAt ℝ (↑m) f Set.univ x
h : ↑m ≤ p
⊢ ContDiffWithinAt ℝ (↑m) (fun x ↦ f x ^ p) Set.univ x
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: x y : ℝ≥0∞
h : x < y
⊢ (if x = 0 then ⊥ else if x = ∞ then ⊤ else ↑(Real.log x.toReal)) <
    if y = 0 then ⊥ else if y = ∞ then ⊤ else ↑(Real.log y.toReal)

Tactic:
split_ifs <;> simp_all [Real.log_lt_log, toReal_pos_iff, pos_iff_ne_zero, lt_top_iff_ne_top]

Goal state after:
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : X ≃ᵢ Y
D : DeloneSet X
⊢ (fun D ↦
        (mapBilipschitz f.symm.toEquiv 1 1 ⋯ ⋯ ⋯ ⋯ D).copy (⇑f.symm '' D.carrier) D.packingRadius D.coveringRadius ⋯ ⋯
          ⋯)
      ((fun D ↦ (mapBilipschitz f.toEquiv 1 1 ⋯ ⋯ ⋯ ⋯ D).copy (⇑f '' D.carrier) D.packingRadius D.coveringRadius ⋯ ⋯ ⋯)
        D) =
    D

Tactic:
ext <;> simp [copy_eq]

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

Goal state before:
pp: x y : EReal
⊢ (x + y).exp = x.exp * y.exp

Tactic:
by
  induction x
  · simp
  · induction y
    · simp
    · simp only [← EReal.coe_add, exp_coe]
      rw [← ENNReal.ofReal_mul (Real.exp_nonneg _), Real.exp_add]
    · simp only [EReal.coe_add_top, exp_top, exp_coe]
      rw [ENNReal.mul_top]
      simp [Real.exp_pos]
  · induction y
    · simp
    · simp only [EReal.top_add_coe, exp_top, exp_coe]
      rw [ENNReal.top_mul]
      simp [Real.exp_pos]
    · simp

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

Goal state before:
pp: case pos
x : ℝ≥0∞
hx_top : x = ∞
⊢ x.log.exp = x

Tactic:
simp [hx_top]

Goal state after:
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : X ≃ᵢ Y
D : DeloneSet Y
⊢ (fun D ↦ (mapBilipschitz f.toEquiv 1 1 ⋯ ⋯ ⋯ ⋯ D).copy (⇑f '' D.carrier) D.packingRadius D.coveringRadius ⋯ ⋯ ⋯)
      ((fun D ↦
          (mapBilipschitz f.symm.toEquiv 1 1 ⋯ ⋯ ⋯ ⋯ D).copy (⇑f.symm '' D.carrier) D.packingRadius D.coveringRadius ⋯ ⋯
            ⋯)
        D) =
    D

Tactic:
ext <;> simp [copy_eq]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : TopologicalSpace β
inst✝² : CommSemiring β
inst✝¹ : ContinuousAdd β
inst✝ : ContinuousMul β
hf : SuperpolynomialDecay l k f
p✝ p q : β[X]
hp : SuperpolynomialDecay l k fun x ↦ eval (k x) p * f x
hq : SuperpolynomialDecay l k fun x ↦ eval (k x) q * f x
⊢ SuperpolynomialDecay l k fun x ↦ eval (k x) (p + q) * f x

Tactic:
simpa [add_mul] using! hp.add hq

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f g : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hf : SuperpolynomialDecay l k f
hfg : abs ∘ g ≤ᶠ[l] abs ∘ f
⊢ SuperpolynomialDecay l k g

Tactic:
rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢

Goal state after:
α : Type u_1
β : Type u_2
l : Filter α
k f g : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hf : ∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)
hfg : abs ∘ g ≤ᶠ[l] abs ∘ f
⊢ ∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * g a|) l (𝓝 0)
Module: Mathlib.MeasureTheory.Covering.VitaliFamily

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
⊢ ∀ (x : X),
    ∀ ε > 0, ∃ s ∈ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}, s ⊆ closedBall x ε

Tactic:
intro x ε εpos
    rcases v.nontrivial x ε εpos with ⟨s, hs, h's⟩
    exact ⟨s, mem_union_left _ hs, h's⟩

Goal state after:
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: x✝ : ℕ
⊢ (0 x✝).log = ⊥ x✝

Tactic:
Pi.zero_apply,

Goal state after:
x✝ : ℕ
⊢ log 0 = ⊥ x✝
Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x ↦ ⨅ i, (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x
⊢ AEMeasurable f μ

Tactic:
have f'_meas : Measurable f' := by fun_prop (disch := simp_all)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x ↦ ⨅ i, (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x
f'_meas : Measurable f'
⊢ AEMeasurable f μ
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u : ℕ → ℝ≥0∞
v : ℕ → ℕ
hu : 1 ≤ᶠ[atTop] u
hv : Tendsto v atTop atTop
n : ℕ
h : 1 ≤ u n
⊢ 0 n ≤ (u n).log

Tactic:
Pi.zero_apply,

Goal state after:
u : ℕ → ℝ≥0∞
v : ℕ → ℕ
hu : 1 ≤ᶠ[atTop] u
hv : Tendsto v atTop atTop
n : ℕ
h : 1 ≤ u n
⊢ 0 ≤ (u n).log
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg.a
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
h' : ∀ᶠ (x : ℕ) in atTop, 0 ≤ u x
⊢ linearGrowthInf (u ∘ v) ≤ a * linearGrowthInf u

Tactic:
rw [← hv.limsup_eq] at ha ha' ⊢
      exact h.linearGrowthInf_comp_le ha ha'

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
x : α
⊢ ∃ R, 0 < R ∧ R ≤ 1 ∧ μ (closedBall x (20 * R)) < ∞

Tactic:
by
    refine ((eventually_le_nhds one_pos).and ?_).exists_gt
    refine (tendsto_closedBall_smallSets x).comp ?_ (μ.finiteAt_nhds x).eventually
    exact Continuous.tendsto' (by fun_prop) _ _ (mul_zero _)

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
a : ι
⊢ a ∈ v → dist (c a) x ≤ r a + R x

Tactic:
hav

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ dist (c a) x ≤ r a + R x
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
i : ι
⊢ 0 < ∫ (y : α), g i y ∂μ → Integrable (g i) μ

Tactic:
hi

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
i : ι
hi : 0 < ∫ (y : α), g i y ∂μ
⊢ Integrable (g i) μ
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ ν' + ⊤.restrict sᶜ ∈ {d | μ ≤ d + ν}

pp: case a.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ (fun μ ↦ μ.restrict s) (ν' + ⊤.restrict sᶜ) = ν'.restrict s

Tactic:
· rw [mem_ofPred_eq, add_right_comm, Measure.le_iff]
      intro t h_meas_t
      repeat rw [← measure_inter_add_sdiff t h_meas_s]
      refine add_le_add ?_ ?_
      · rw [add_apply, add_apply]
        apply le_add_right _
        rw [← restrict_eq_self μ inter_subset_right,
          ← restrict_eq_self ν inter_subset_right]
        apply h_ν'_in
      · rw [add_apply, restrict_apply (h_meas_t.diff h_meas_s), sdiff_eq, inter_assoc, inter_self,
          ← add_apply]
        have h_mu_le_add_top : μ ≤ ν' + ν + ⊤ := by simp only [add_top, le_top]
        exact Measure.le_iff'.1 h_mu_le_add_top _

Goal state after:
case a.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ (fun μ ↦ μ.restrict s) (ν' + ⊤.restrict sᶜ) = ν'.restrict s
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
⊢ support (g i) ⊆ a i →
    (∀ (x : α), |g i x| ≤ K / μ.real (a i)) →
      IntegrableOn f (a i) μ →
        0 < ∫ (y : α), g i y ∂μ → ‖∫ (y : α), g i y • (f y - c) ∂μ‖ ≤ K * ⨍ (y : α) in a i, ‖f y - c‖ ∂μ

Tactic:
hi

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
⊢ (∀ (x : α), |g i x| ≤ K / μ.real (a i)) →
    IntegrableOn f (a i) μ →
      0 < ∫ (y : α), g i y ∂μ → ‖∫ (y : α), g i y • (f y - c) ∂μ‖ ≤ K * ⨍ (y : α) in a i, ‖f y - c‖ ∂μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
h : IsHahnDecomposition μ ν s
⊢ μ.restrict sᶜᶜ ≤ ν.restrict sᶜᶜ

Tactic:
simpa using h.le_on

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case h₂
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ (ε • μ) (s ∩ o) ≤ ρ (s ∩ o)

Tactic:
refine v.measure_le_of_frequently_le ρ smul_absolutelyContinuous _ ?_

Goal state after:
case h₂
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ρ o = 0
μo : μ oᶜ = 0
⊢ ∀ x ∈ s ∩ o, ∃ᶠ (a : Set α) in v.filterAt x, (↑ε • μ) a ≤ ρ a
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: case pos.inl.left
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := ⋯
H : closedBall y r ⊆ closedBall x (R / 4)
hr : r ≤ R
⊢ closedBall y r ⊆ closedBall x ((K + 1) * r)

Tactic:
apply closedBall_subset_closedBall'
      rw [dist_comm]
      linarith

Goal state after:
Module: Mathlib.MeasureTheory.Covering.OneDim

Goal state before:
pp: x : ℝ
⊢ Tendsto (fun y ↦ Icc y x) (𝓝[<] x) ((vitaliFamily volume 1).filterAt x)

Tactic:
by
  refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
  · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
  · intro ε εpos
    filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy
    rw [closedBall_eq_Icc]
    exact Icc_subset_Icc hy.1 (by linarith)

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case refine_5
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n ↦ s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n ↦ s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
⊢ ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0

Tactic:
intro m n

Goal state after:
case refine_5
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n ↦ s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n ↦ s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
Module: Mathlib.Topology.EMetricSpace.VariationOnFromTo

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
⊢ variationOnFromTo f s a b = -variationOnFromTo f s b a

Tactic:
rcases lt_trichotomy a b with (ab | rfl | ba)

Goal state after:
case inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
ab : a < b
⊢ variationOnFromTo f s a b = -variationOnFromTo f s b a

case inr.inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a : α
⊢ variationOnFromTo f s a a = -variationOnFromTo f s a a

case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
ba : b < a
⊢ variationOnFromTo f s a b = -variationOnFromTo f s b a
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ 0 ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {∞})

pp: case a
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ μ (s ∩ f ⁻¹' {∞}) = 0

Tactic:
· exact zero_le

Goal state after:
case a
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ μ (s ∩ f ⁻¹' {∞}) = 0
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
⊢ ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0})

Tactic:
grw [measure_mono inter_subset_right, v.measure_limRatioMeas_zero hρ]

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : Measure α := μ.withDensity (v.limRatioMeas hρ)
f : α → ℝ≥0∞ := v.limRatioMeas hρ
f_meas : Measurable f
⊢ 0 ≤ (t • ν) (s ∩ f ⁻¹' {0})