mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
X : ι → Type u_4
inst✝ : Unique ι
m : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
e : ((i : ι) → X i) ≃ᵐ X default := MeasurableEquiv.piUnique X
⊢ Measure.map (⇑e) (Measure.pi μ) = μ default

Tactic:
have : (piPremeasure fun i => (μ i).toOuterMeasure) = Measure.map e.symm (μ default) := by
      ext1 s
      rw [piPremeasure, Fintype.prod_unique, e.symm.map_apply, coe_toOuterMeasure]
      congr 1; exact e.toEquiv.image_eq_preimage_symm s

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
X : ι → Type u_4
inst✝ : Unique ι
m : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
e : ((i : ι) → X i) ≃ᵐ X default := MeasurableEquiv.piUnique X
this : (piPremeasure fun i ↦ (μ i).toOuterMeasure) = ⇑(Measure.map (⇑e.symm) (μ default))
⊢ Measure.map (⇑e) (Measure.pi μ) = μ default
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : 0 < y - x
hyz : 0 < z - y
hxz : 0 < z - x
hxz' : x ≠ z
ha : 0 < (z - y) / (z - x)
hb : 0 < (y - x) / (z - x)
⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1

Tactic:
field

Goal state after:
Module: Mathlib.Data.ENNReal.Holder

Goal state before:
pp: p q : ℝ≥0∞
⊢ p.HolderConjugate q ↔ p⁻¹ + q⁻¹ = 1

Tactic:
simp [holderTriple_iff]

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
h : 0 < z - y
hyz' : 0 < z / y
hyz'' : z / y ≠ 1
⊢ z / y - 1 = y⁻¹ * (z - y)

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
h : 0 < y - x
hxy' : 0 < x / y
hxy'' : x / y ≠ 1
⊢ y⁻¹ * (y - x) = 1 - x / y

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 ≤ (z - y) / (z - x)
hb : 0 ≤ (y - x) / (z - x)
⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 < (z - y) / (z - x)
hb : 0 < (y - x) / (z - x)
⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1

Tactic:
field

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

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

Tactic:
have : 0 ≤ p.toReal⁻¹ := by positivity
  refine (eLpNorm_smul_measure_of_ne_top hp ..).trans ?_
  simp [ENNReal.smul_def, ENNReal.coe_rpow_of_nonneg, this]

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

Goal state before:
pp: p q : ℝ≥0∞
h : p.toReal.HolderConjugate q.toReal
⊢ p.HolderConjugate q

Tactic:
rw [Real.HolderConjugate] at h
  exact HolderTriple.of_toReal (toReal_one ▸ h)

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  rw [one_div_one, one_div] at h
  repeat' rw [ENNReal.rpow_one] at h
  exact (ENNReal.le_rpow_inv_iff hp_pos).mp h

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ≥0∞
h : p ∈ Set.Ioo 0 1
⊢ 1 ≤ p⁻¹.toReal

Tactic:
← ENNReal.toReal_one

Goal state after:
p : ℝ≥0∞
h : p ∈ Set.Ioo 0 1
⊢ ENNReal.toReal 1 ≤ p⁻¹.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq✝ : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
r : ℝ := p * q / (q - p)
hpqr : 1 / p = 1 / q + 1 / r
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) =
    (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
rw [hpqr]

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq✝ : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
r : ℝ := p * q / (q - p)
hpqr : 1 / p = 1 / q + 1 / r
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) =
    (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / q + 1 / r - 1 / q)
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
⊢ 1 / r = 1 / p - 1 / q

Tactic:
rw [hpqr]

Goal state after:
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
⊢ 1 / r = 1 / q + 1 / r - 1 / q
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:

Tactic:
(∫⁻ a : α, (f * g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ a : α, f a ^ p * g a ^ p ∂μ) ^ (1 / p) := by
      simp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0]
    _ ≤ ((∫⁻ a, f a ^ (p * p2) ∂μ) ^ (1 / p2) *
        (∫⁻ a, g a ^ (p * q2) ∂μ) ^ (1 / q2)) ^ (1 / p) := by
      gcongr
      simp_rw [ENNReal.rpow_mul]
      exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hp2q2 (hf.pow_const _) (hg.pow_const _)
    _ = (∫⁻ a : α, f a ^ q ∂μ) ^ (1 / q) * (∫⁻ a : α, g a ^ r ∂μ) ^ (1 / r) := by
      rw [@ENNReal.mul_rpow_of_nonneg _ _ (1 / p) (by simp [hp0]), ← ENNReal.rpow_mul, ←
        ENNReal.rpow_mul]
      have hpp2 : p * p2 = q := by
        symm
        rw [mul_comm, ← div_eq_iff hp0_ne]
      have hpq2 : p * q2 = r := by
        rw [← inv_inv r, ← one_div, ← one_div, h_one_div_r]
        simp [field, p2, q2, Real.conjExponent]
      simp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case right
ι : Type u
s : Finset ι
f : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
⊢ (∑ i ∈ s, f i ^ p) ^ (1 / p) ∈ upperBounds ((fun g ↦ ∑ i ∈ s, f i * g i) '' {g | ∑ i ∈ s, g i ^ q ≤ 1})

Tactic:
· rintro _ ⟨g, hg, rfl⟩
    apply le_trans (inner_le_Lp_mul_Lq s f g hpq)
    simpa only [mul_one] using
      mul_le_mul_right (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : SeminormedAddGroup E
ι : Type u_3
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : LinearOrder ι
f : ι → α → E
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ eLpNorm f_lim ∞ μ ≤ liminf (fun n ↦ eLpNorm (f n) ∞ μ) atTop

Tactic:
rw [eLpNorm_exponent_top_lim_eq_essSup_liminf h_lim]
  simp_rw [eLpNorm_exponent_top, eLpNormEssSup]
  exact ENNReal.essSup_liminf_le _

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
h_eq : (fun n ↦ eLpNorm (↑↑(f n) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) p μ) = fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ
⊢ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) p μ) fi (𝓝 0) ↔
    Tendsto (fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)

Tactic:
rw [h_eq]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

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

Tactic:
by
  lift w to ι → ℝ≥0 using hw
  lift f to ι → ℝ≥0 using hf
  beta_reduce at *
  norm_cast at *
  exact NNReal.inner_le_weight_mul_Lp _ hp _ _

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : 0 < ε
δ : ℝ≥0∞
hδ : δ > 0
hδi : δ = ∞
⊢ ∃ N, ∀ n ≥ N, μ {x | ε ≤ edist (f n x) (g x)} ≤ δ

pp: case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : 0 < ε
δ : ℝ≥0∞
hδ : δ > 0
hδi : ¬δ = ∞
⊢ ∃ N, ∀ n ≥ N, μ {x | ε ≤ edist (f n x) (g x)} ≤ δ

Tactic:
· simp only [hδi, imp_true_iff, le_top, exists_const]

Goal state after:
case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : 0 < ε
δ : ℝ≥0∞
hδ : δ > 0
hδi : ¬δ = ∞
⊢ ∃ N, ∀ n ≥ N, μ {x | ε ≤ edist (f n x) (g x)} ≤ δ
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_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < B N
n : ℕ
⊢ eLpNorm' (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i

Tactic:
by
  let f_norm_diff i x := ‖f (i + 1) x - f i x‖
  have hgf_norm_diff :
    ∀ n,
      (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) =
        ∑ i ∈ Finset.range (n + 1), f_norm_diff i :=
    fun n => funext fun x => by simp [f_norm_diff]
  rw [hgf_norm_diff]
  refine (eLpNorm'_sum_le (fun i _ => ((hf (i + 1)).sub (hf i)).norm) hp1).trans ?_
  simp_rw [eLpNorm'_norm]
  refine (Finset.sum_le_sum ?_).trans <| ENNReal.sum_le_tsum _
  exact fun m _ => (h_cau m (m + 1) m (Nat.le_succ m) (le_refl m)).le

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∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞
rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞
x : α
hx : (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞
⊢ ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞

Tactic:
← ENNReal.lt_rpow_inv_iff hp_pos,

Goal state after:
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∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞
rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞
x : α
hx : ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞ ^ p⁻¹
⊢ ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞
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✝² : PseudoEMetricSpace E
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → E
g : α → E
hfg : TendstoInMeasure μ f u g
⊢ ∃ ns, Tendsto ns atTop u ∧ TendstoInMeasure μ (fun n ↦ f (ns n)) atTop g

Tactic:
by
  obtain ⟨ns, h_tendsto_ns⟩ : ∃ ns : ℕ → ι, Tendsto ns atTop u := exists_seq_tendsto u
  exact ⟨ns, h_tendsto_ns, fun ε hε => (hfg ε hε).comp h_tendsto_ns⟩

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ∀ (c : 𝕜), ‖c‖ ≤ 1
⊢ Absorbs 𝕜 V (⇑f '' s)

Tactic:
then

Goal state after:
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ∀ (c : 𝕜), ‖c‖ ≤ 1
⊢ Absorbs 𝕜 V (⇑f '' s)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
a : α
ha : ↑(f a) ≠ 0 → g a = g' a
h'a : f a ≠ 0
⊢ ↑(f a) ≠ 0

Tactic:
simpa only [Ne, ENNReal.coe_eq_zero] using h'a

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 (⋃ b, f b)).toReal = ∑ p, (μ (f p)).toReal

Tactic:
measure_iUnion hn h,

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) ≠ ∞
⊢ (∑' (i : β), μ (f i)).toReal = ∑ p, (μ (f p)).toReal
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
ι : Type u_3
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (μ univ).toReal < ∑ i ∈ s, (μ (t i)).toReal
⊢ (∑ i ∈ s, μ (t i)).toReal = ∑ i ∈ s, (μ (t i)).toReal

Tactic:
rw [ENNReal.toReal_sum (by finiteness)]

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

Goal state before:
pp: case refine_1
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : α → E
h : Disjoint (support f) (support g)
hf : StronglyMeasurable f
hg : StronglyMeasurable g
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
hfg : MemLp (f + g) p μ
⊢ MemLp f p μ

pp: case refine_2
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : α → E
h : Disjoint (support f) (support g)
hf : StronglyMeasurable f
hg : StronglyMeasurable g
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
hfg : MemLp (f + g) p μ
⊢ MemLp g p μ

Tactic:
· rw [← Set.indicator_add_eq_left h]; exact hfg.indicator (measurableSet_support hf.measurable)

Goal state after:
case refine_2
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : α → E
h : Disjoint (support f) (support g)
hf : StronglyMeasurable f
hg : StronglyMeasurable g
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
hfg : MemLp (f + g) p μ
⊢ MemLp g p μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case neg
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0
ε : ι → ℝ
C : ℝ
hε : ∀ (i : ι), 0 < ε i
c : ι → 𝕜
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
hm : ∀ (i : ι), ‖m i‖ ≠ 0
δ : ι → 𝕜
hδ0 : ∀ (i : ι), δ i ≠ 0
hδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i
hle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖
a✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
have hδ0 : 0 < ∏ i, ‖δ i‖ := prod_pos fun i _ => norm_pos_iff.2 (hδ0 i)

Goal state after:
case neg
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0
ε : ι → ℝ
C : ℝ
hε : ∀ (i : ι), 0 < ε i
c : ι → 𝕜
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
hm : ∀ (i : ι), ‖m i‖ ≠ 0
δ : ι → 𝕜
hδ0✝ : ∀ (i : ι), δ i ≠ 0
hδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i
hle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖
a✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖
hδ0 : 0 < ∏ i, ‖δ i‖
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
⊢ IntegrableOn f univ μ ↔ Integrable f μ

Tactic:
rw [IntegrableOn, Measure.restrict_univ]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case succ
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝¹ : Fintype ι
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
⊢ Continuous[inst✝⁴, Pi.topologicalSpace] ⇑ξ.equivFun

Tactic:
have : FiniteDimensional 𝕜 E := ξ.finiteDimensional_of_finite

Goal state after:
case succ
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝² : Fintype ι
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this : FiniteDimensional 𝕜 E
⊢ Continuous[inst✝⁴, Pi.topologicalSpace] ⇑ξ.equivFun
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
⊢ (∃ s ∈ map e l, IntegrableOn f s μ) ↔ ∃ s ∈ map e l, IntegrableOn f s (μ.restrict (range e))

Tactic:
constructor <;> rintro ⟨s, hs, int⟩

Goal state after:
case mp
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ∈ map e l
int : IntegrableOn f s μ
⊢ ∃ s ∈ map e l, IntegrableOn f s (μ.restrict (range e))

case mpr
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
s : Set β
hs : s ∈ map e l
int : IntegrableOn f s (μ.restrict (range e))
⊢ ∃ s ∈ map e l, IntegrableOn f s μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f g : α → E
h : Disjoint (support f) (support g)
hf : StronglyMeasurable f
hg : StronglyMeasurable g
⊢ Integrable (f + g) μ ↔ Integrable f μ ∧ Integrable g μ

Tactic:
by
  refine ⟨fun hfg => ⟨?_, ?_⟩, fun h => h.1.add h.2⟩
  · rw [← indicator_add_eq_left h]; exact hfg.indicator hf.measurableSet_support
  · rw [← indicator_add_eq_right h]; exact hfg.indicator hg.measurableSet_support

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
b : ℝ
hm : ‖m‖ ≤ b
⊢ ‖f m‖ ≤ ‖f‖ * b ^ Fintype.card ι

Tactic:
simpa only [prod_const] using! f.le_opNorm_mul_prod_of_le fun i => (norm_le_pi_norm m i).trans hm

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_9
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : SigmaFinite μ
C : ℝ≥0∞
hC : C < ∞
f : α → ε
hf_meas : AEStronglyMeasurable f μ
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → ∫⁻ (x : α) in s, ‖f x‖ₑ ∂μ ≤ C
⊢ SigmaFinite (μ.trim ⋯)

Tactic:
by rwa [@trim_eq_self _ m]

Goal state after: