mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case pos
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 (n : ℕ) in atTop, u n ≤ 0
⊢ linearGrowthInf (u ∘ v) = a * linearGrowthInf u

Tactic:
replace h' (n : ℕ) : u n ≤ 0 := by
      obtain ⟨m, n_m, um_1⟩ := (frequently_atTop.1 h') n
      exact (h n_m).trans um_1
    have u_0' : linearGrowthInf u = 0 := by
      apply le_antisymm _ (h.linearGrowthInf_nonneg u_0)
      exact (linearGrowthInf_monotone h').trans_eq (linearGrowthInf_const zero_ne_bot zero_ne_top)
    rw [u_0', mul_zero]
    apply le_antisymm _ (linearGrowthInf_comp_nonneg h u_0 v_top)
    apply (linearGrowthInf_monotone fun n ↦ h' (v n)).trans_eq
    exact linearGrowthInf_const zero_ne_bot zero_ne_top

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

Goal state before:
pp: case pos
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
⊢ linearGrowthSup (u ∘ v) = a * linearGrowthSup u

Tactic:
have u_0' : linearGrowthSup u = 0 := by
      apply le_antisymm _ (h.linearGrowthSup_nonneg u_0)
      apply (linearGrowthSup_eventually_monotone u_1).trans_eq
      exact (linearGrowthSup_const zero_ne_bot zero_ne_top)

Goal state after:
case pos
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) = a * linearGrowthSup u
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case neg.inr
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a < 0
b : ι
rb : b ∈ t ∧ 0 ≤ r b
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

Tactic:
rcases A b ⟨rb.1, rb.2⟩ with ⟨c, cu, _⟩

Goal state after:
case neg.inr
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a < 0
b : ι
rb : b ∈ t ∧ 0 ≤ r b
c : ι
cu : c ∈ u
right✝ : closedBall (x b) (r b) ⊆ closedBall (x c) (τ * r c)
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case neg.inr
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
t' : Set ι := {a | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a ≤ 0
b : ι
rb : b ∈ t ∧ 0 < r b
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases A b ⟨rb.1, rb.2⟩ with ⟨c, cu, _⟩

Goal state after:
case neg.inr
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h✝ : t.Nonempty
t' : Set ι := {a | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a ≤ 0
b : ι
rb : b ∈ t ∧ 0 < r b
c : ι
cu : c ∈ u
right✝ : ball (x b) (r b) ⊆ ball (x c) (τ * r c)
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → E
⊢ ⨍ (x : α), f x ∂0 = 0

Tactic:
rw [average, smul_zero, integral_zero_measure]

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

Goal state before:
pp: case inl
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
hμ : μ = 0
⊢ μ.real univ • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ

pp: case inr
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
hμ : μ ≠ 0
⊢ μ.real univ • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ

Tactic:
· rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]

Goal state after:
case inr
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
hμ : μ ≠ 0
⊢ μ.real univ • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s t : Set α
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hfs : IntegrableOn f s μ
hft : IntegrableOn f t μ
hse : μ s = 0
⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ -[ℝ] ⨍ (x : α) in t, f x ∂μ]

pp: case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s t : Set α
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hfs : IntegrableOn f s μ
hft : IntegrableOn f t μ
hse : ¬μ s = 0
⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ -[ℝ] ⨍ (x : α) in t, f x ∂μ]

Tactic:
· rw [← ae_eq_empty] at hse
    rw [restrict_congr_set (hse.union EventuallyEq.rfl), empty_union]
    exact right_mem_segment _ _ _

Goal state after:
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s t : Set α
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hfs : IntegrableOn f s μ
hft : IntegrableOn f t μ
hse : ¬μ s = 0
⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ -[ℝ] ⨍ (x : α) in t, f x ∂μ]
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
⊢ 0 < ∫ (x : α) in s, f x - ⨍ (a : α) in s, f a ∂μ ∂μ

Tactic:
refine (setIntegral_pos_iff_support_of_nonneg_ae ?_ ?_).2 ?_

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
⊢ 0 ≤ᵐ[μ.restrict s] fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ

case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
⊢ IntegrableOn (fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ) s μ

case refine_3
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
this : Fact (μ s < ∞)
⊢ 0 < μ ((support fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ) ∩ s)
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
⊢ ∫⁻ (x : α), g x ∂μ ≠ ∞

Tactic:
rwa [← setLIntegral_univ, ← withDensity_apply g MeasurableSet.univ, ← hfg,
        withDensity_apply f MeasurableSet.univ, setLIntegral_univ]

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
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
k : Set α := ⋃ a ∈ w, B ↑a
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
b : ι
hbv : b ∈ v
⊢ ⟨b, ⋯⟩ ∈ w → z ∉ B b

Tactic:
_

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
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
k : Set α := ⋃ a ∈ w, B ↑a
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
b : ι
hbv : b ∈ v
h✝ : ⟨b, ⋯⟩ ∈ w
⊢ z ∉ B b
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ

Tactic:
refine (absolutelyContinuous_smul <| ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · simp_rw [ENNReal.smul_def]
    rw [withDensity_smul _ (measurable_rnDeriv _ _)]
    suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ))
        = ν.singularPart (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ) by
      rwa [add_right_inj] at this
    rw [← ν.haveLebesgueDecomposition_add (r • μ), singularPart_smul_right _ _ _ hr,
      ENNReal.smul_def r, withDensity_smul_measure, ← ENNReal.smul_def, ← smul_assoc,
      smul_eq_mul, inv_mul_cancel₀ hr, one_smul]
    exact ν.haveLebesgueDecomposition_add μ

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

Goal state before:
pp: case pos.inr
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 := scalingScaleOf μ (max (4 * K + 3) 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
⊢ μ (closedBall x (3 * (R / 4))) ≤ ↑(max (Classical.choose ⋯) 1) * μ (closedBall y r)

Tactic:
← le_max_right,

Goal state after:
case pos.inr
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 := scalingScaleOf μ (max (4 * K + 3) 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
⊢ μ (closedBall x (3 * (R / 4))) ≤ ↑1 * μ (closedBall y r)
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case hc
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
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
h1x : ∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c
⊢ (∃ᶠ (x : Set α) in v.filterAt x, ↑d < ρ x / μ x) → ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ a

Tactic:
_

Goal state after:
case hc
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
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
h1x : ∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c
a✝ : ∃ᶠ (x : Set α) in v.filterAt x, ↑d < ρ x / μ x
⊢ ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ a
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case hd
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
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
⊢ (∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c) →
    (∃ᶠ (x : Set α) in v.filterAt x, ↑d < ρ x / μ x) → ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a

Tactic:
_

Goal state after:
case hd
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
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
a✝ : ∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c
⊢ (∃ᶠ (x : Set α) in v.filterAt x, ↑d < ρ x / μ x) → ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
A : Set α
hE₃ : ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hA : MeasurableSet A
⊢ ↑ε * ν (A ∩ E) + ∫⁻ (a : α) in A ∩ E, ξ a ∂ν ≤ μ (A ∩ E)

Tactic:
rw [Measure.sub_apply (hA.inter hE₁) hle, withDensity_apply _ (hA.inter hE₁)] at hE₃

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ∂ν) (f n) = g n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
A : Set α
hE₃ : ↑ε * ν (A ∩ E) ≤ μ (A ∩ E) - ∫⁻ (a : α) in A ∩ E, ξ a ∂ν
hA : MeasurableSet A
⊢ ↑ε * ν (A ∩ E) + ∫⁻ (a : α) in A ∩ E, ξ a ∂ν ≤ μ (A ∩ E)
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 < b → ∀ᵐ (x : ℝ), x ∈ s ∩ Ioo a b → DifferentiableWithinAt ℝ f s x

Tactic:
_

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
⊢ ∀ᵐ (x : ℝ), x ∈ s ∩ Ioo a b → DifferentiableWithinAt ℝ f s x
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
h : b ≤ a
⊢ -variationOnFromTo f s b a = -(eVariationOn f (s ∩ Icc b a)).toReal

Tactic:
neg_inj,

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
h : b ≤ a
⊢ variationOnFromTo f s b a = (eVariationOn f (s ∩ Icc b a)).toReal
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:

Tactic:
∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))
        = ∑ i ∈ Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) :=
      Finset.sum_congr rfl fun i hi ↦ by
        rw [Finset.mem_Ico] at hi
        simp only [v, π, projIcc_of_mem hmn ⟨hi.1, hi.2.le⟩,
          projIcc_of_mem hmn ⟨hi.1.trans i.le_succ, hi.2⟩]
    _ ≤ ∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i)) :=
      Finset.sum_mono_set _ (Nat.Iio_eq_range n ▸ Finset.Ico_subset_Iio_self)
    _ ≤ eVariationOn f s :=
      sum_le (fun i j h ↦ hu (π i).2 (π j).2 (monotone_projIcc hmn h)) fun i ↦ us _ (π i).2

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)
⊢ ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ = ↑t ^ (n + 1) * μ (s ∩ f ⁻¹' I)

Tactic:
simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]

Goal state after:
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 p : C(α, R)₀
hp : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ f ≤ f + p

Tactic:
induction hp using AddSubmonoid.closure_induction generalizing f with

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 : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0})
B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (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) = ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ

Tactic:
simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]

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
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ ∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i)) +
      ∑ i ∈ Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i)) ≤
    ∑ i ∈ Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i))

Tactic:
rw [← Finset.sum_union]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ ∑ x ∈ Finset.range n ∪ Finset.Ico (n + 1) (n + 1 + m), edist (f (w (x + 1))) (f (w x)) ≤
    ∑ i ∈ Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i))

α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i ↦ if i ≤ n then u i else v (i - (n + 1))
⊢ Disjoint (Finset.range n) (Finset.Ico (n + 1) (n + 1 + m))