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

Goal state before:
pp: case neg.refine_2
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
a' : EReal
a_a' : a < a'
a_top' : a' ≠ ⊤
a_0' : 0 < a'
a_a_inv' : a'⁻¹ < a⁻¹
a_v' : ∀ᶠ (n : ℕ) in atTop, a' * ↑n ≤ ↑(v n)
b_u : ∃ᶠ (x : ℕ) in atTop, (b * ↑x ≤ u x ∧ 0 < x) ∧ ∃ m, a'⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a⁻¹ * ↑x
M M' : ℕ
aM_M' : a' * ↑M ≤ ↑M'
n : ℕ
n_M' : M' ≤ n
bn_un : b * ↑n ≤ u n
right✝ : 0 < n
k : ℕ
an_k' : ↑n / a' ≤ ↑k
k_an : ↑k ≤ a⁻¹ * ↑n
ak_vk' : a' * ↑k ≤ ↑(v k)
⊢ b * (a * ↑k) ≤ u (v k)

Tactic:
div_le_iff_le_mul a_0' a_top'

Goal state after:
case neg.refine_2
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv : (linearGrowthInf fun n ↦ ↑(v n)) ≠ 0
v_0 : 0 < linearGrowthInf fun n ↦ ↑(v n)
u_0 : ¬u = ⊥
a : EReal
a_0 : 0 < a
a_v : a < linearGrowthInf fun n ↦ ↑(v n)
b : EReal
b_0 : 0 < b
a' : EReal
a_a' : a < a'
a_top' : a' ≠ ⊤
a_0' : 0 < a'
a_a_inv' : a'⁻¹ < a⁻¹
a_v' : ∀ᶠ (n : ℕ) in atTop, a' * ↑n ≤ ↑(v n)
b_u : ∃ᶠ (x : ℕ) in atTop, (b * ↑x ≤ u x ∧ 0 < x) ∧ ∃ m, a'⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a⁻¹ * ↑x
M M' : ℕ
aM_M' : a' * ↑M ≤ ↑M'
n : ℕ
n_M' : M' ≤ n
bn_un : b * ↑n ≤ u n
right✝ : 0 < n
k : ℕ
an_k' : ↑n ≤ a' * ↑k
k_an : ↑k ≤ a⁻¹ * ↑n
ak_vk' : a' * ↑k ≤ ↑(v k)
⊢ b * (a * ↑k) ≤ u (v k)
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
⊢ ⨍⁻ (_x : α), 0 ∂μ = 0

Tactic:
laverage,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
⊢ ∫⁻ (x : α), 0 ∂(μ univ)⁻¹ • μ = 0
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → ℝ≥0∞
⊢ ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ

Tactic:
laverage,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → ℝ≥0∞
⊢ ∫⁻ (x : α), f x ∂(μ univ)⁻¹ • μ = ∫⁻ (x : α), 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
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
c : ℝ≥0∞
⊢ ⨍⁻ (_x : α) in s, c ∂μ = c

Tactic:
simp only [setLAverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ,
    univ_inter, div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one]

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

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : IsFiniteMeasure (μ.restrict s)
hμ0 : μ s = 0
⊢ ⨍⁻ (x : α) in s, f x ∂μ ≤ essSup f μ

Tactic:
laverage,

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : IsFiniteMeasure (μ.restrict s)
hμ0 : μ s = 0
⊢ ∫⁻ (x : α), f x ∂((μ.restrict s) univ)⁻¹ • μ.restrict s ≤ essSup f μ
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hgi : ∫⁻ (x : α), g x ∂μ ≠ ∞
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
⊢ f =ᵐ[μ] g

Tactic:
suffices f =ᵐ[μ.restrict (s ∪ t)] g by
    refine ae_of_ae_restrict_of_ae_restrict_compl _ this ?_
    simp only [Set.compl_union]
    have h1 : f =ᵐ[μ.restrict sᶜ] 0 := hf'.ae_eq_zero_compl
    have h2 : g =ᵐ[μ.restrict tᶜ] 0 := hg'.ae_eq_zero_compl
    rw [ae_restrict_iff' (hf'.measurableSet.compl.inter hg'.measurableSet.compl)]
    rw [EventuallyEq, ae_restrict_iff' hf'.measurableSet.compl] at h1
    rw [EventuallyEq, ae_restrict_iff' hg'.measurableSet.compl] at h2
    filter_upwards [h1, h2] with x h1 h2 hx
    rw [h1 (Set.inter_subset_left hx), h2 (Set.inter_subset_right hx)]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hgi : ∫⁻ (x : α), g x ∂μ ≠ ∞
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
⊢ f =ᵐ[μ.restrict (s ∪ t)] g
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_2
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 : Set α
ht : MeasurableSet t
h_eq : ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
⊢ ∫⁻ (x : α) in t, g x ∂μ ≠ ∞

Tactic:
refine ne_of_lt ?_

Goal state after:
case refine_2
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 : Set α
ht : MeasurableSet t
h_eq : ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
⊢ ∫⁻ (x : α) in t, g x ∂μ < ∞
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_2
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 : Set α
ht : MeasurableSet t
h_eq : ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
⊢ ∫⁻ (x : α) in t, f x ∂μ ≠ ∞

Tactic:
refine ne_of_lt ?_

Goal state after:
case refine_2
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 : Set α
ht : MeasurableSet t
h_eq : ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
⊢ ∫⁻ (x : α) in t, f x ∂μ < ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
hl : ¬μ.HaveLebesgueDecomposition ν
⊢ ν.withDensity (μ.rnDeriv ν) ≤ μ

Tactic:
· rw [rnDeriv, dif_neg hl, withDensity_zero]
    exact Measure.zero_le μ

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
⊢ AEMeasurable ((r • ν).rnDeriv μ) μ

pp: case hg
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
⊢ AEMeasurable (r • ν.rnDeriv μ) μ

pp: case hfi
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
⊢ ∫⁻ (x : α), (r • ν).rnDeriv μ x ∂μ ≠ ∞

Tactic:
· exact (measurable_rnDeriv _ _).aemeasurable

Goal state after:
case hg
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
⊢ AEMeasurable (r • ν.rnDeriv μ) μ

case hfi
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
⊢ ∫⁻ (x : α), (r • ν).rnDeriv μ x ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case hf
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
⊢ AEMeasurable (ν.rnDeriv (r • μ)) (r • μ)

pp: case hg
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
⊢ AEMeasurable (r⁻¹ • ν.rnDeriv μ) (r • μ)

pp: case hfi
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
⊢ ∫⁻ (x : α), ν.rnDeriv (r • μ) x ∂r • μ ≠ ∞

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
⊢ (r • μ).withDensity (ν.rnDeriv (r • μ)) = (r • μ).withDensity (r⁻¹ • ν.rnDeriv μ)

Tactic:
· exact (measurable_rnDeriv _ _).aemeasurable

Goal state after:
case hg
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
⊢ AEMeasurable (r⁻¹ • ν.rnDeriv μ) (r • μ)

case hfi
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
⊢ ∫⁻ (x : α), ν.rnDeriv (r • μ) x ∂r • μ ≠ ∞

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
⊢ (r • μ).withDensity (ν.rnDeriv (r • μ)) = (r • μ).withDensity (r⁻¹ • ν.rnDeriv μ)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝⁴ : IsFiniteMeasure ν₁
inst✝³ : IsFiniteMeasure ν₂
inst✝² : ν₁.HaveLebesgueDecomposition μ
inst✝¹ : ν₂.HaveLebesgueDecomposition μ
inst✝ : (ν₁ + ν₂).HaveLebesgueDecomposition μ
⊢ AEMeasurable ((ν₁ + ν₂).rnDeriv μ) μ

pp: case hg
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝⁴ : IsFiniteMeasure ν₁
inst✝³ : IsFiniteMeasure ν₂
inst✝² : ν₁.HaveLebesgueDecomposition μ
inst✝¹ : ν₂.HaveLebesgueDecomposition μ
inst✝ : (ν₁ + ν₂).HaveLebesgueDecomposition μ
⊢ AEMeasurable (ν₁.rnDeriv μ + ν₂.rnDeriv μ) μ

pp: case hfi
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝⁴ : IsFiniteMeasure ν₁
inst✝³ : IsFiniteMeasure ν₂
inst✝² : ν₁.HaveLebesgueDecomposition μ
inst✝¹ : ν₂.HaveLebesgueDecomposition μ
inst✝ : (ν₁ + ν₂).HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), (ν₁ + ν₂).rnDeriv μ x ∂μ ≠ ∞

Tactic:
· exact (measurable_rnDeriv _ _).aemeasurable

Goal state after:
case hg
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝⁴ : IsFiniteMeasure ν₁
inst✝³ : IsFiniteMeasure ν₂
inst✝² : ν₁.HaveLebesgueDecomposition μ
inst✝¹ : ν₂.HaveLebesgueDecomposition μ
inst✝ : (ν₁ + ν₂).HaveLebesgueDecomposition μ
⊢ AEMeasurable (ν₁.rnDeriv μ + ν₂.rnDeriv μ) μ

case hfi
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝⁴ : IsFiniteMeasure ν₁
inst✝³ : IsFiniteMeasure ν₂
inst✝² : ν₁.HaveLebesgueDecomposition μ
inst✝¹ : ν₂.HaveLebesgueDecomposition μ
inst✝ : (ν₁ + ν₂).HaveLebesgueDecomposition μ
⊢ ∫⁻ (x : α), (ν₁ + ν₂).rnDeriv μ x ∂μ ≠ ∞
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
⊢ ∀ᵐ (x : α) ∂μ,
    ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
      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:
by
  filter_upwards [(vitaliFamily μ K).ae_tendsto_average_norm_sub hf] with x hx ι l w δ δlim
    xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})
⊢ ∫⁻ (x : α) in A, (fun a ↦ max (f a) (g a)) x ∂μ ≤ ν A

Tactic:
rw [setLIntegral_max hf.1 hg.1]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : f ∈ measurableLE μ ν
hg : g ∈ measurableLE μ ν
A : Set α
hA : MeasurableSet A
h₁ : MeasurableSet (A ∩ {a | f a ≤ g a})
h₂ : MeasurableSet (A ∩ {a | g a < f a})
⊢ ∫⁻ (x : α) in A ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in A ∩ {x | g x < f x}, f x ∂μ ≤ ν 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
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 c)}
⊢ ∃ a b,
    MeasurableSet a ∧ MeasurableSet b ∧ {x | v.limRatio ρ x < ↑p} ⊆ a ∧ {x | ↑q < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0

Tactic:
let o : ℕ → Set α := spanningSets (ρ + μ)

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
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)
⊢ ∃ a b,
    MeasurableSet a ∧ MeasurableSet b ∧ {x | v.limRatio ρ x < ↑p} ⊆ a ∧ {x | ↑q < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
⊢ AEMeasurable ((r • ν).rnDeriv μ) μ

pp: case hg
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
⊢ AEMeasurable (r • ν.rnDeriv μ) μ

Tactic:
· exact (measurable_rnDeriv _ _).aemeasurable

Goal state after:
case hg
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
⊢ AEMeasurable (r • ν.rnDeriv μ) μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (ν.rnDeriv (r • μ)) (r • μ)

pp: case hg
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (r⁻¹ • ν.rnDeriv μ) (r • μ)

Tactic:
· exact (measurable_rnDeriv _ _).aemeasurable

Goal state after:
case hg
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (r⁻¹ • ν.rnDeriv μ) (r • μ)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

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

Tactic:
have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right' ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
  rwa [ENNReal.smul_def, ENNReal.coe_toNNReal hr_ne_top,
    ← ENNReal.toNNReal_inv, ENNReal.smul_def, ENNReal.coe_toNNReal (ENNReal.inv_ne_top.mpr hr)] at h

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable ((ν₁ + ν₂).rnDeriv μ) μ

pp: case hg
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable (ν₁.rnDeriv μ + ν₂.rnDeriv μ) μ

Tactic:
· exact (measurable_rnDeriv _ _).aemeasurable

Goal state after:
case hg
α : Type u_1
m : MeasurableSpace α
ν₁ ν₂ μ : Measure α
inst✝² : SigmaFinite ν₁
inst✝¹ : SigmaFinite ν₂
inst✝ : SigmaFinite μ
⊢ AEMeasurable (ν₁.rnDeriv μ + ν₂.rnDeriv μ) μ
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:
pp: case mp
α : Type u_1
inst✝⁸ : TopologicalSpace α
R : Type u_2
inst✝⁷ : PartialOrder R
inst✝⁶ : NonUnitalSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : StarOrderedRing R
inst✝³ : TopologicalSpace R
inst✝² : ContinuousStar R
inst✝¹ : IsTopologicalSemiring R
inst✝ : ContinuousSqrt R
f g : C(α, R)
⊢ f ≤ g → ∃ s, g = f + star s * s

Tactic:
rw [ContinuousMap.le_def]
    intro h
    use (mk _ ContinuousSqrt.continuousOn_sqrt.domRestrict).comp
      ⟨_, map_continuous (f.prodMk g) |>.codRestrict (s := {x | x.1 ≤ x.2}) (by exact h)⟩
    ext x
    simpa [IsSelfAdjoint.star_eq <| .of_nonneg (ContinuousSqrt.sqrt_nonneg (f x, g x) (h x))]
      using ContinuousSqrt.sqrt_mul_sqrt (f x, g x) (h x)

Goal state after:
Module: Mathlib.Topology.ContinuousMap.StarOrdered

Goal state before:

Tactic:
f ≤ f + g₁ := h₁ f
          _ ≤ (f + g₁) + g₂ := h₂ (f + g₁)
          _ = f + (g₁ + g₂) := add_assoc _ _ _

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
A : ρ.rnDeriv μ =ᵐ[μ] v.limRatioMeas hρ
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))

Tactic:
filter_upwards [v.ae_tendsto_limRatioMeas hρ, A] with _ _ h'x

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
A : ρ.rnDeriv μ =ᵐ[μ] v.limRatioMeas hρ
a✝¹ : α
a✝ : Tendsto (fun a ↦ ρ a / μ a) (v.filterAt a✝¹) (𝓝 (v.limRatioMeas hρ a✝¹))
h'x : ρ.rnDeriv μ a✝¹ = v.limRatioMeas hρ a✝¹
⊢ Tendsto (fun a ↦ ρ a / μ a) (v.filterAt a✝¹) (𝓝 (ρ.rnDeriv μ a✝¹))
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case e'_3.a
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
x y : β
hx : x ∈ t
hy : y ∈ t
h : x ≤ y
u : β
us : u ∈ t
vφx : φ x ≤ φ u
vφy : φ u ≤ φ y
⊢ φ u ∈ φ '' (t ∩ Icc x y)

Tactic:
rcases le_total x u with (xu | ux)

Goal state after:
case e'_3.a.inl
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
x y : β
hx : x ∈ t
hy : y ∈ t
h : x ≤ y
u : β
us : u ∈ t
vφx : φ x ≤ φ u
vφy : φ u ≤ φ y
xu : x ≤ u
⊢ φ u ∈ φ '' (t ∩ Icc x y)

case e'_3.a.inr
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
β : Type u_3
inst✝ : LinearOrder β
f : α → E
t : Set β
φ : β → α
hφ : MonotoneOn φ t
x y : β
hx : x ∈ t
hy : y ∈ t
h : x ≤ y
u : β
us : u ∈ t
vφx : φ x ≤ φ u
vφy : φ u ≤ φ y
ux : u ≤ x
⊢ φ u ∈ φ '' (t ∩ Icc x y)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case h
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
N : 𝓝[s ∩ Ici y ∩ Iio x] x = 𝓝[s ∩ Iio x] x
⊢ (𝓝[s ∩ Ici y ∩ Iio x] x).NeBot

pp: case ha
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
N : 𝓝[s ∩ Ici y ∩ Iio x] x = 𝓝[s ∩ Iio x] x
⊢ x ∈ s ∩ Ici y

pp: case h'f
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
N : 𝓝[s ∩ Ici y ∩ Iio x] x = 𝓝[s ∩ Iio x] x
⊢ Tendsto f (𝓝[s ∩ Ici y ∩ Iio x] x) (𝓝 l)

Tactic:
· convert h using 1

Goal state after:
case ha
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
N : 𝓝[s ∩ Ici y ∩ Iio x] x = 𝓝[s ∩ Iio x] x
⊢ x ∈ s ∩ Ici y

case h'f
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
H : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x) + edist (f x) l) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
y : α
hy : y ∈ s ∩ Iio x
N : 𝓝[s ∩ Ici y ∩ Iio x] x = 𝓝[s ∩ Iio x] x
⊢ Tendsto f (𝓝[s ∩ Ici y ∩ Iio x] x) (𝓝 l)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : CommRing R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
inst✝ : UniqueHom R A
hf : ContinuousOn f ((fun x ↦ -x) '' spectrum R a)
ha : p a
⊢ cfc (fun x ↦ f (-x)) a = cfc f (-a)

Tactic:
rw [cfc_comp' .., cfc_neg_id _]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a

Tactic:
simp only [cfc_apply_of_not_continuousOn _ hf, le_rfl]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0

Tactic:
simp only [cfc_apply_of_not_continuousOn _ hf, le_rfl]

Goal state after: