mathlib-initiative/mathlib-tactics huggingface dataset:
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✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : ∀ (z : ℕ), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun a ↦ |k a ^ z * f a|
z : ℕ
m : β
hm : ∀ᶠ (x : β) in Filter.map (fun a ↦ |k a ^ (z + 1) * f a|) l, (fun x1 x2 ↦ x1 ≤ x2) x m
h1 : Tendsto (fun x ↦ 0) l (𝓝 0)
h2 : Tendsto (fun a ↦ |(k a)⁻¹| * m) l (𝓝 0)
x : α
hk0 : k x ≠ 0
hx : (fun x1 x2 ↦ x1 ≤ x2) |k x ^ (z + 1) * f x| m
⊢ |k x ^ z * f x| = |(k x)⁻¹ * (k x * k x ^ z) * f x|

Tactic:
← mul_assoc,

Goal state after:
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : ∀ (z : ℕ), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun a ↦ |k a ^ z * f a|
z : ℕ
m : β
hm : ∀ᶠ (x : β) in Filter.map (fun a ↦ |k a ^ (z + 1) * f a|) l, (fun x1 x2 ↦ x1 ≤ x2) x m
h1 : Tendsto (fun x ↦ 0) l (𝓝 0)
h2 : Tendsto (fun a ↦ |(k a)⁻¹| * m) l (𝓝 0)
x : α
hk0 : k x ≠ 0
hx : (fun x1 x2 ↦ x1 ≤ x2) |k x ^ (z + 1) * f x| m
⊢ |k x ^ z * f x| = |(k x)⁻¹ * k x * k x ^ z * f x|
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : 0 ≤ z
⊢ Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)

Tactic:
unfold Tendsto

Goal state after:
case pos
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : Field β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hz : 0 ≤ z
⊢ Filter.map (fun a ↦ k a ^ z * f a) l ≤ 𝓝 0
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
⊢ (expGrowthInf fun x ↦ b * v x) ≤ expGrowthInf v

Tactic:
rcases eq_zero_or_pos b with rfl | b_pos

Goal state after:
case inl
u v : ℕ → ℝ≥0∞
hb : 0 ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ 0 * v n
⊢ (expGrowthInf fun x ↦ 0 * v x) ≤ expGrowthInf v

case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ (expGrowthInf fun x ↦ b * v x) ≤ expGrowthInf v
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
⊢ (expGrowthSup fun x ↦ b * v x) ≤ expGrowthSup v

Tactic:
rcases eq_zero_or_pos b with rfl | b_pos

Goal state after:
case inl
u v : ℕ → ℝ≥0∞
hb : 0 ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ 0 * v n
⊢ (expGrowthSup fun x ↦ 0 * v x) ≤ expGrowthSup v

case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ (expGrowthSup fun x ↦ b * v x) ≤ expGrowthSup v
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: 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 = ⊥
u_1 : ∀ᶠ (n : ℕ) in atTop, u n ≤ 0
u_0' : linearGrowthSup u = 0
⊢ linearGrowthSup (u ∘ v) ≤ 0

Tactic:
apply (linearGrowthSup_eventually_monotone (v_top.eventually u_1)).trans_eq

Goal state after:
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 = ⊥
u_1 : ∀ᶠ (n : ℕ) in atTop, u n ≤ 0
u_0' : linearGrowthSup u = 0
⊢ (linearGrowthSup fun x ↦ 0) = 0
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsProbabilityMeasure μ
hf : Integrable f μ
⊢ 0 < μ {x | ∫ (a : α), f a ∂μ ≤ f x}

Tactic:
simpa only [average_eq_integral] using
    measure_average_le_pos (IsProbabilityMeasure.ne_zero μ) hf

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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
r'mem : r a ∈ r '' v
hr' : R0 / 2 < r a
⊢ ∃ a ∈ v, R0 / 2 < r a

Tactic:
exact ⟨a, hav, hr'⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_3
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set (Set α)
f g : α → ℝ≥0∞
h_eq : m0 = MeasurableSpace.generateFrom s
h_inter : IsPiSystem s
basic : ∀ t ∈ s, ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
h_univ : ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : α), g x ∂μ
hf_int : ∫⁻ (x : α), f x ∂μ ≠ ∞
t : ℕ → Set α
⊢ Pairwise (Function.onFun Disjoint t) →
    (∀ (i : ℕ), MeasurableSet (t i)) →
      (∀ (i : ℕ), ∫⁻ (x : α) in t i, f x ∂μ = ∫⁻ (x : α) in t i, g x ∂μ) →
        ∫⁻ (x : α) in ⋃ i, t i, f x ∂μ = ∫⁻ (x : α) in ⋃ i, t i, g x ∂μ

Tactic:
htd

Goal state after:
case refine_3
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set (Set α)
f g : α → ℝ≥0∞
h_eq : m0 = MeasurableSpace.generateFrom s
h_inter : IsPiSystem s
basic : ∀ t ∈ s, ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
h_univ : ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : α), g x ∂μ
hf_int : ∫⁻ (x : α), f x ∂μ ≠ ∞
t : ℕ → Set α
htd : Pairwise (Function.onFun Disjoint t)
⊢ (∀ (i : ℕ), MeasurableSet (t i)) →
    (∀ (i : ℕ), ∫⁻ (x : α) in t i, f x ∂μ = ∫⁻ (x : α) in t i, g x ∂μ) →
      ∫⁻ (x : α) in ⋃ i, t i, f x ∂μ = ∫⁻ (x : α) in ⋃ i, t i, g x ∂μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ MeasurableSet (e (n + 1))

Tactic:
· exact he₁ _

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

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hint : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
⊢ 0 < μ {x | x ∈ s ∧ ⨍⁻ (a : α) in s, f a ∂μ ≤ f x}

Tactic:
rw [hfg] at hint

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
⊢ 0 < μ {x | x ∈ s ∧ ⨍⁻ (a : α) in s, f a ∂μ ≤ f 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)
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
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
mu_ai : μ (a i) < ∞
⊢ ∫ (a : α), ‖g i a‖ * ‖f a - c‖ ∂μ ≤ ∫ (a_1 : α) in a i, K * (((μ.restrict (a i)).real univ)⁻¹ * ‖f a_1 - c‖) ∂μ

Tactic:
← mul_assoc,

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
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
mu_ai : μ (a i) < ∞
⊢ ∫ (a : α), ‖g i a‖ * ‖f a - c‖ ∂μ ≤ ∫ (a_1 : α) in a i, K * ((μ.restrict (a i)).real univ)⁻¹ * ‖f a_1 - c‖ ∂μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
heq : s.restrict (S ∩ T)ᶜ = (μ.singularPart ν).restrict (S ∩ T)ᶜ
A : Set α
hA : MeasurableSet A
hsinter : s (A ∩ (S ∩ T)) = 0
⊢ s A = s (A \ (S ∩ T))

Tactic:
AEDisjoint.measure_sdiff_left hsinter

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : s S = 0
hS₃ : ν Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : (μ.singularPart ν) T = 0
hT₃ : ν Tᶜ = 0
hνinter : ν (S ∩ T)ᶜ = 0
heq : s.restrict (S ∩ T)ᶜ = (μ.singularPart ν).restrict (S ∩ T)ᶜ
A : Set α
hA : MeasurableSet A
hsinter : s (A ∩ (S ∩ T)) = 0
⊢ s A = s A
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: 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
S : Set α
K : ℝ
x : α
hx : Tendsto (fun a ↦ μ (S ∩ a) / μ a) ((vitaliFamily μ K).filterAt x) (𝓝 1)
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
⊢ (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
    Tendsto (fun j ↦ μ (S ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)

Tactic:
xmem

Goal state after:
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
S : Set α
K : ℝ
x : α
hx : Tendsto (fun a ↦ μ (S ∩ a) / μ a) ((vitaliFamily μ K).filterAt x) (𝓝 1)
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)
⊢ Tendsto (fun j ↦ μ (S ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: α : Type u_1
inst✝⁶ : PseudoMetricSpace α
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : IsUnifLocDoublingMeasure μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
E : Type u_2
inst✝ : NormedAddCommGroup E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) ((vitaliFamily μ K).filterAt x) (𝓝 0)
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
⊢ (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
    Tendsto (fun j ↦ ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)

Tactic:
xmem

Goal state after:
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : IsUnifLocDoublingMeasure μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
E : Type u_2
inst✝ : NormedAddCommGroup E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) ((vitaliFamily μ K).filterAt x) (𝓝 0)
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)
⊢ Tendsto (fun j ↦ ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)
Module: Mathlib.MeasureTheory.Covering.DensityTheorem

Goal state before:
pp: α : Type u_1
inst✝⁸ : PseudoMetricSpace α
inst✝⁷ : MeasurableSpace α
μ : Measure α
inst✝⁶ : IsUnifLocDoublingMeasure μ
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, f y ∂μ) ((vitaliFamily μ K).filterAt x) (𝓝 (f x))
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
⊢ (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
    Tendsto (fun j ↦ ⨍ (y : α) in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x))

Tactic:
xmem

Goal state after:
α : Type u_1
inst✝⁸ : PseudoMetricSpace α
inst✝⁷ : MeasurableSpace α
μ : Measure α
inst✝⁶ : IsUnifLocDoublingMeasure μ
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : BorelSpace α
inst✝³ : IsLocallyFiniteMeasure μ
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, f y ∂μ) ((vitaliFamily μ K).filterAt x) (𝓝 (f x))
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)
⊢ Tendsto (fun j ↦ ⨍ (y : α) in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x))
Module: Mathlib.Analysis.Calculus.Monotone

Goal state before:
pp: f : ℝ → ℝ
s : Set ℝ
hf : MonotoneOn f s
a b : ℝ
as : a ∈ s
bs : b ∈ s
a✝ : a < b
g : ℝ → ℝ
hg : Monotone g
gf : EqOn f g (s ∩ Icc a b)
x : ℝ
hx : DifferentiableAt ℝ g x
h'x : x ∈ s ∩ Ioo a b
this : Ioo a b ∈ 𝓝[s] x
⊢ f =ᶠ[𝓝[s] x] g

Tactic:
filter_upwards [self_mem_nhdsWithin, this] with y hy h'y

Goal state after:
f : ℝ → ℝ
s : Set ℝ
hf : MonotoneOn f s
a b : ℝ
as : a ∈ s
bs : b ∈ s
a✝ : a < b
g : ℝ → ℝ
hg : Monotone g
gf : EqOn f g (s ∩ Icc a b)
x : ℝ
hx : DifferentiableAt ℝ g x
h'x : x ∈ s ∩ Ioo a b
this : Ioo a b ∈ 𝓝[s] x
y : ℝ
hy : y ∈ s
h'y : y ∈ Ioo a b
⊢ f y = g y
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
m n : ℕ
u : ℕ → α
hu : MonotoneOn u (Icc m n)
us : ∀ i ∈ Icc m n, u i ∈ s
hnm : n ≤ m
⊢ ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
m n : ℕ
u : ℕ → α
hu : MonotoneOn u (Icc m n)
us : ∀ i ∈ Icc m n, u i ∈ s
hmn : m ≤ n
⊢ ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

Tactic:
· simp [Finset.Ico_eq_empty_of_le hnm]

Goal state after:
case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
m n : ℕ
u : ℕ → α
hu : MonotoneOn u (Icc m n)
us : ∀ i ∈ Icc m n, u i ∈ s
hmn : m ≤ n
⊢ ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
m n : ℕ
u : ℕ → α
hu : MonotoneOn u (Icc m n)
us : ∀ i ∈ Icc m n, u i ∈ s
hmn : m ≤ n
⊢ ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s

Tactic:
let π := projIcc m n hmn

Goal state after:
case inr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
m n : ℕ
u : ℕ → α
hu : MonotoneOn u (Icc m n)
us : ∀ i ∈ Icc m n, u i ∈ s
hmn : m ≤ n
π : ℕ → ↑(Icc m n) := projIcc m n hmn
⊢ ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
h : ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
n : ℕ
u : ℕ → α
um : Monotone u
us : ∀ (i : ℕ), u i ∈ s
⊢ ∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i)) = 0

Tactic:
exact Finset.sum_eq_zero fun i _ => h _ (us i.succ) _ (us i)

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
s : Set α
f : α →ᵤ[singleton '' s] E
⊢ SemicontinuousAt (fun x1 x2 ↦ (fun f ↦ eVariationOn f s) x1 > x2) f

Tactic:
apply @lowerSemicontinuous_aux _ _ _ _ (UniformOnFun α E (s.image singleton)) id (𝓝 f) f s _
  simpa only [UniformOnFun.tendsto_iff_tendstoUniformlyOn, mem_image, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂, tendstoUniformlyOn_singleton_iff_tendsto] using! @tendsto_id _ (𝓝 f)

Goal state after:
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})
B : ν (s ∩ f ⁻¹' {∞}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {∞})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑t ^ (n + 1) * μ (s ∩ f ⁻¹' I) = ↑t ^ 2 * (↑t ^ (n - 1) * μ (s ∩ f ⁻¹' I))

Tactic:
← mul_assoc,

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
A : ν (s ∩ f ⁻¹' {0}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ν (s ∩ f ⁻¹' {∞}) ≤ (↑t ^ 2 • ρ) (s ∩ f ⁻¹' {∞})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑t ^ (n + 1) * μ (s ∩ f ⁻¹' I) = ↑t ^ 2 * ↑t ^ (n - 1) * μ (s ∩ f ⁻¹' I)
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
f g : C(α, R)₀
⊢ (∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p) → f ≤ g

Tactic:
rintro ⟨p, hp, rfl⟩

Goal state after:
case mpr
α : Type u_1
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : Zero α
R : Type u_2
inst✝⁸ : TopologicalSpace R
inst✝⁷ : CommSemiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : StarRing R
inst✝³ : StarOrderedRing R
inst✝² : IsTopologicalSemiring R
inst✝¹ : ContinuousStar R
inst✝ : StarOrderedRing C(α, R)
f p : C(α, R)₀
hp : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ f ≤ f + p
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 ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s

Tactic:
have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
    grw [measure_mono inter_subset_right, v.measure_limRatioMeas_zero hρ]
    exact zero_le

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
A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0})
⊢ ρ s ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s