mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteLattice β
hμ : ∀ (a : α), μ {a} ≠ 0
f : α → β
⊢ essSup f μ = ⨆ i, f i

Tactic:
by
  rw [essSup, ae_eq_top.2 hμ, limsup_top_eq_iSup]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f g : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ eLpNormEssSup (s.piecewise f g) μ = max (eLpNormEssSup f (μ.restrict s)) (eLpNormEssSup g (μ.restrict sᶜ))

Tactic:
simp only [eLpNormEssSup, ← ENNReal.essSup_piecewise hs]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f g : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ essSup (fun x ↦ ‖s.piecewise f g x‖ₑ) μ = essSup (s.piecewise (fun x ↦ ‖f x‖ₑ) fun x ↦ ‖g x‖ₑ) μ
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁵ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝³ : ∀ (i : ι), SigmaFinite (μ i)
inst✝² : (i : ι) → Group (α i)
inst✝¹ : ∀ (i : ι), MeasurableInv (α i)
inst✝ : ∀ (i : ι), (μ i).IsInvInvariant
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
A : Inv.inv ⁻¹' univ.pi s = univ.pi fun i ↦ Inv.inv ⁻¹' s i
⊢ (map Inv.inv (Measure.pi μ)) (univ.pi s) = ∏ i, (μ i) (s i)

Tactic:
Measure.map_apply measurable_inv (MeasurableSet.univ_pi hs),

Goal state after:
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁵ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝³ : ∀ (i : ι), SigmaFinite (μ i)
inst✝² : (i : ι) → Group (α i)
inst✝¹ : ∀ (i : ι), MeasurableInv (α i)
inst✝ : ∀ (i : ι), (μ i).IsInvInvariant
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
A : Inv.inv ⁻¹' univ.pi s = univ.pi fun i ↦ Inv.inv ⁻¹' s i
⊢ (Measure.pi μ) (Inv.inv ⁻¹' univ.pi s) = ∏ i, (μ i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: α : Type u
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
⊢ MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) (Measure.pi fun x ↦ μ) (μ.prod μ)

Tactic:
simpa only [Matrix.vec_single_eq_const, Matrix.vecCons_const] using
    measurePreserving_finTwoArrow_vec μ μ

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ ν : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hμν : ν ≤ μ
hp0 : p = 0
⊢ eLpNorm f p ν ≤ eLpNorm f p μ

Tactic:
simp [hp0]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
s : Set α
f : α → ε
hsf : Function.support f ⊆ s
hp0 : p = 0
⊢ eLpNorm f p (μ.restrict s) = eLpNorm f p μ

Tactic:
simp [hp0]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
c : ℝ≥0∞
hc : c ≠ 0
f : α → ε
p : ℝ≥0∞
μ : Measure α
hp0 : p = 0
⊢ eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ

Tactic:
simp [hp0]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
hp0 : p = 0
⊢ eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ

Tactic:
simp [hp0]

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ≥0∞
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ p.toNNReal.HolderTriple q.toNNReal r.toNNReal ↔ p.HolderTriple q r

Tactic:
← NNReal.holderTriple_coe_iff,

Goal state after:
p q r : ℝ≥0∞
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ (↑p.toNNReal).HolderTriple ↑q.toNNReal ↑r.toNNReal ↔ p.HolderTriple q r
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
hp : 1 < p.toNNReal
⊢ p.toNNReal.HolderConjugate q.toNNReal ↔ p.HolderConjugate q

Tactic:
← NNReal.holderTriple_coe_iff,

Goal state after:
p q : ℝ≥0∞
hp : 1 < p.toNNReal
⊢ (↑p.toNNReal).HolderTriple ↑q.toNNReal ↑1 ↔ p.HolderConjugate q
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < y - x
⊢ y ^ p - x ^ p < p * y ^ (p - 1) * (y - x)

Tactic:
← div_lt_div_iff_of_pos_right hy',

Goal state after:
p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < y - x
⊢ (y ^ p - x ^ p) / y ^ p < p * y ^ (p - 1) * (y - x) / y ^ p
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < z - y
⊢ p * y ^ (p - 1) * (z - y) < z ^ p - y ^ p

Tactic:
← div_lt_div_iff_of_pos_right hy',

Goal state after:
p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < z - y
⊢ p * y ^ (p - 1) * (z - y) / y ^ p < (z ^ p - y ^ p) / y ^ p
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
s : Set ι
⊢ MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i)

Tactic:
smul_nonneg_iff_pos_imp_nonneg,

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
s : Set ι
⊢ MonovaryOn f g s ↔
    ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → (0 < f j - f i → 0 ≤ g j - g i) ∧ (0 < g j - g i → 0 ≤ f j - f i)
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
⊢ (∀ ⦃i : ι⦄, i ∈ Set.univ → ∀ ⦃j : ι⦄, j ∈ Set.univ → 0 ≤ (f j - f i) • (g j - g i)) ↔
    ∀ (i j : ι), 0 ≤ (f j - f i) • (g j - g i)

Tactic:
by
    simp only [Set.mem_univ, forall_true_left]

Goal state after:
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
⊢ (∀ ⦃i : ι⦄, i ∈ Set.univ → ∀ ⦃j : ι⦄, j ∈ Set.univ → f i • g j + f j • g i ≤ f i • g i + f j • g j) ↔
    ∀ (i j : ι), f i • g j + f j • g i ≤ f i • g i + f j • g j

Tactic:
by
    simp only [Set.mem_univ, forall_true_left]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case e'_4
z₁ z₂ : ℝ≥0
p : ℝ
hp : 1 ≤ p
h'p : 1 < p
⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p

Tactic:
have A : p - 1 ≠ 0 := ne_of_gt (sub_pos.2 h'p)
    simp only [mul_rpow, rpow_sub' A, rpow_one]
    ring

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : (a ^ 1 + b ^ 1) ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
NNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a ^ 1 + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a ^ 1 + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
NNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
NNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
NNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case pos
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : p = 0
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ

Tactic:
simp [hp0]

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

Goal state before:
pp: α : Type u_2
inst✝ : MeasurableSpace α
p q r : ℝ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp0_ne : p ≠ 0
hp0 : 0 ≤ p
hq0_lt : 0 < q
hq0_ne : q ≠ 0
h_one_div_r : 1 / r = 1 / p - 1 / q
p2 : ℝ := q / p
q2 : ℝ := p2.conjExponent
hp2q2 : p2.HolderConjugate q2
⊢ 0 ≤ 1 / p

Tactic:
simp [hp0]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hr : r ≠ 0
⊢ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) =
    ((∑ i ∈ s, f i ^ p) ^ (r / p) * (∑ i ∈ s, g i ^ q) ^ (r / q)) ^ (1 / r)

Tactic:
simp only [← rpow_mul, mul_rpow]

Goal state after:
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hr : r ≠ 0
⊢ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) =
    (∑ i ∈ s, f i ^ p) ^ (r / p * (1 / r)) * (∑ i ∈ s, g i ^ q) ^ (r / q * (1 / r))
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
⊢ 0 < 1 / r.toReal

Tactic:
hpqr'

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
⊢ 0 < 1 / p.toReal + 1 / q.toReal
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
hr : r ≠ 0
⊢ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q) =
    ((∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)) ^ (1 / r)

Tactic:
simp only [← rpow_mul, mul_rpow]

Goal state after:
ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
hr : r ≠ 0
⊢ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q) =
    (∑' (i : ι), f i ^ p) ^ (r / p * (1 / r)) * (∑' (i : ι), g i ^ q) ^ (r / q * (1 / r))
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ p) (B ^ p)
⊢ ∃ C ≤ A + B, HasSum (fun i ↦ (f i + g i) ^ p) (C ^ p)

Tactic:
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
  obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
  have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
  have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
  refine ⟨(∑' i, (f i + g i) ^ p) ^ (1 / p), ?_, ?_⟩
  · simpa [hA, hB] using H₂
  · simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SeminormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
l : Filter ι
f : ι → α → E
g : α → E
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
⊢ Tendsto (fun i ↦ μ {x | ε ≤ ‖f i x - g x‖ₑ}) l (𝓝 0) ↔ Tendsto (fun i ↦ μ.real {x | ε ≤ ‖f i x - g x‖ₑ}) l (𝓝 0)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SeminormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
l : Filter ι
f : ι → α → E
g : α → E
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
⊢ Tendsto (fun i ↦ μ {x | ε ≤ ‖f i x - g x‖ₑ}) l (𝓝 0) ↔ Tendsto (fun i ↦ (μ {x | ε ≤ ‖f i x - g x‖ₑ}).toReal) l (𝓝 0)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetricSpace E
inst✝ : IsFiniteMeasure μ
f : ι → α → E
l : Filter ι
g : α → E
ε : ℝ
hε : 0 < ε
⊢ Tendsto (fun i ↦ μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0) ↔
    Tendsto (fun i ↦ μ.real {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetricSpace E
inst✝ : IsFiniteMeasure μ
f : ι → α → E
l : Filter ι
g : α → E
ε : ℝ
hε : 0 < ε
⊢ Tendsto (fun i ↦ μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0) ↔
    Tendsto (fun i ↦ (μ {x | ε ≤ dist (f i x) (g x)}).toReal) l (𝓝 0)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SeminormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
l : Filter ι
f : ι → α → E
g : α → E
ε : ℝ
hε : 0 < ε
⊢ Tendsto (fun i ↦ μ {x | ε ≤ ‖f i x - g x‖}) l (𝓝 0) ↔ Tendsto (fun i ↦ μ.real {x | ε ≤ ‖f i x - g x‖}) l (𝓝 0)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SeminormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
l : Filter ι
f : ι → α → E
g : α → E
ε : ℝ
hε : 0 < ε
⊢ Tendsto (fun i ↦ μ {x | ε ≤ ‖f i x - g x‖}) l (𝓝 0) ↔ Tendsto (fun i ↦ (μ {x | ε ≤ ‖f i x - g x‖}).toReal) l (𝓝 0)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
p : ℝ
w f : ι → ℝ≥0
hp : 1 ≤ p
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
exact NNReal.inner_le_weight_mul_Lp _ hp _ _

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
⊢ StrictMono (seqTendstoAeSeq hfg)

Tactic:
refine strictMono_nat_of_lt_succ fun n => ?_
  rw [seqTendstoAeSeq_succ]
  exact lt_of_lt_of_le (lt_add_one <| seqTendstoAeSeq hfg n) (le_max_right _ _)

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
⊢ (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i

Tactic:
have hp_pos : 0 < p := zero_lt_one.trans_le hp1
  suffices h_pow : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ≤ (∑' i, B i) ^ p by
      rwa [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv]
  have h_tsum_1 :
    ∀ g : ℕ → ℝ≥0∞, ∑' i, g i = atTop.liminf fun n => ∑ i ∈ Finset.range (n + 1), g i := by
    intro g
    rw [ENNReal.tsum_eq_liminf_sum_nat, ← liminf_nat_add _ 1]
  simp_rw [h_tsum_1 _]
  rw [← h_tsum_1]
  have h_liminf_pow :
    ∫⁻ a, (atTop.liminf fun n => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ =
      ∫⁻ a, atTop.liminf fun n => (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ := by
    refine lintegral_congr fun x => ?_
    have h_rpow_mono := ENNReal.strictMono_rpow_of_pos (zero_lt_one.trans_le hp1)
    have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2
    refine (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply ?_ ?_ ?_ ?_
    all_goals isBoundedDefault
  rw [h_liminf_pow]
  refine (lintegral_liminf_le' fun n ↦ ?_).trans <| liminf_le_of_frequently_le' <| .of_forall h
  exact ((Finset.range _).aemeasurable_fun_sum fun i _ ↦ ((hf _).sub (hf i)).enorm).pow_const _

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
n : ℕ
⊢ (fun t ↦ (↑x + t)⁻¹ʳ - (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 0] fun t ↦ ‖t‖ ^ n

Tactic:
refine EventuallyEq.trans_isBigO (.fun_sub (inverse_add_nth_order x n) (.refl _ _)) ?_

Goal state after:
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
n : ℕ
⊢ (fun i ↦
      (∑ i_1 ∈ range n, (-↑x⁻¹ * i) ^ i_1) * ↑x⁻¹ + (-↑x⁻¹ * i) ^ n * (↑x + i)⁻¹ʳ -
        (∑ i_1 ∈ range n, (-↑x⁻¹ * i) ^ i_1) * ↑x⁻¹) =O[𝓝 0]
    fun t ↦ ‖t‖ ^ n
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSpace β
f : α → β
hf : AEMeasurable f μ
s : Set β
hs : MeasurableSet s
⊢ (map f μ).real s = μ.real (f ⁻¹' s)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSpace β
f : α → β
hf : AEMeasurable f μ
s : Set β
hs : MeasurableSet s
⊢ ((map f μ) s).toReal = (μ (f ⁻¹' s)).toReal
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
⊢ (μ.restrict s).real t = μ.real (t ∩ s)

Tactic:
by
  simp only [measureReal_def, restrict_apply' hs]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : Fintype β
f : β → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : β), MeasurableSet (f i)
h' : ∀ (i : β), μ (f i) ≠ ∞
⊢ μ.real (⋃ b, f b) = ∑ p, μ.real (f p)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : Fintype β
f : β → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : β), MeasurableSet (f i)
h' : ∀ (i : β), μ (f i) ≠ ∞
⊢ (μ (⋃ b, f b)).toReal = ∑ p, (μ (f p)).toReal
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ.real s + μ.real (t \ s) = μ.real (s ∪ t)

Tactic:
rw [← measureReal_union' (@disjoint_sdiff_right _ s t) hs h₁
    (measure_ne_top_of_subset sdiff_subset h₂), union_sdiff_self]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ ‖c‖ₑ.toReal * (μ s ^ (1 / p.toReal)).toReal = ‖c‖ * μ.real s ^ (1 / p.toReal)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ ‖c‖ₑ.toReal * (μ s ^ (1 / p.toReal)).toReal = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) = ‖c‖ₑ * ENNReal.ofReal (μ.real s ^ (1 / p.toReal))

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) = ‖c‖ₑ * ENNReal.ofReal ((μ s).toReal ^ (1 / p.toReal))
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : IsFiniteMeasure μ
c : E
inst✝ : NeZero μ
hp_zero : p ≠ 0
this : μ ≠ 0
⊢ (‖c‖ₑ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * μ.real Set.univ ^ (1 / p.toReal)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : IsFiniteMeasure μ
c : E
inst✝ : NeZero μ
hp_zero : p ≠ 0
this : μ ≠ 0
⊢ (‖c‖ₑ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ∞
⊢ (‖c‖ₑ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * μ.real Set.univ ^ (1 / p.toReal)

Tactic:
measureReal_def,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ∞
⊢ (‖c‖ₑ * μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
hf : AEStronglyMeasurable f μ
p q : ℝ
hp✝ : 0 ≤ p
hq✝ : 0 ≤ q
hpq : p ≤ q
hint : Integrable (fun x ↦ ‖f x‖ ^ (ENNReal.ofReal q).toReal) μ
hp : 0 < p
hq : 0 < q
⊢ ENNReal.ofReal q ≠ 0

Tactic:
simp [hq]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ HasFiniteIntegral f μ ∧ HasFiniteIntegral g μ

Tactic:
exact ⟨hf.hasFiniteIntegral, hg.hasFiniteIntegral⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : TopologicalSpace β
inst✝ : BorelSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : NullMeasurableSet s μ
t : Set α
ts : t ⊆ s
ht : MeasurableSet t
t_eq_s : t =ᵐ[μ] s
⊢ AEMeasurable f (μ.restrict t)

Tactic:
exact ContinuousOn.aemeasurable (hf.mono ts) ht

Goal state after: