mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ -1 + 1 ≤ z / y

Tactic:
neg_add_cancel,

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
⊢ 0 ≤ z / y
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : ∀ i ∈ s, f i ^ p = 0
i : ι
his : i ∈ s
⊢ f i = 0

Tactic:
exact (rpow_eq_zero_iff.mp (hf i his)).left

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✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_pow_liminf : liminf (fun n ↦ eLpNorm' (f n) p μ) atTop ^ p = liminf (fun n ↦ eLpNorm' (f n) p μ ^ p) atTop
⊢ liminf (fun i ↦ ∫⁻ (a : α), ‖f i a‖ₑ ^ p ∂μ) atTop = liminf (fun n ↦ eLpNorm' (f n) p μ ^ p) atTop

Tactic:
simp_rw [eLpNorm'_eq_lintegral_enorm, ← ENNReal.rpow_mul, one_div,
    inv_mul_cancel₀ hp_pos.ne.symm, ENNReal.rpow_one]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
β : Type u_7
inst✝ : MeasurableSpace β
μb : Measure β
g : β →ₘ[μb] E
hg : g ∈ Lp E p μb
f : α → β
hf : MeasurePreserving f μ μb
⊢ g.compMeasurePreserving f hf ∈ Lp E p μ

Tactic:
Lp.mem_Lp_iff_eLpNorm_lt_top

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
β : Type u_7
inst✝ : MeasurableSpace β
μb : Measure β
g : β →ₘ[μb] E
hg : eLpNorm (↑g) p μb < ∞
f : α → β
hf : MeasurePreserving f μ μb
⊢ eLpNorm (↑(g.compMeasurePreserving f hf)) p μ < ∞
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⁻¹
⊢ ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞

Tactic:
ENNReal.top_rpow_of_pos (by simp [hp_pos] : 0 < p⁻¹)

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 < ∞
⊢ ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞
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✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
p : ℝ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < B N
h1 : ∀ (n : ℕ), eLpNorm' (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i
⊢ ∀ᵐ (x : α) ∂μ, Summable fun i ↦ f (i + 1) x - f i x

Tactic:
have h2 n :
        ∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' i, B i) ^ p :=
      lintegral_rpow_sum_enorm_sub_le_rpow_tsum hp1 n (h1 n)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
p : ℝ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < B N
h1 : ∀ (n : ℕ), eLpNorm' (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i
h2 : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
⊢ ∀ᵐ (x : α) ∂μ, Summable fun i ↦ f (i + 1) x - f i x
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h
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
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
⊢ StrictMono ns ∧ μ {a | ¬Tendsto (fun i ↦ f (ns i) a) atTop (𝓝 (g a))} = 0

Tactic:
refine ⟨ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono hfg, measure_mono_null (fun x => ?_) hμs⟩

Goal state after:
case h
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
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∈ {a | ¬Tendsto (fun i ↦ f (ns i) a) atTop (𝓝 (g a))} → x ∈ s
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
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : SeminormedAddCommGroup E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
l : Filter ι
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) l (𝓝 0)
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)

Tactic:
replace hfg := ENNReal.Tendsto.const_mul (a := 1 / ε ^ p.toReal)
    (Tendsto.ennrpow_const p.toReal hfg) (Or.inr <| by simp [hε.ne'])

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : SeminormedAddCommGroup E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
l : Filter ι
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
hfg : Tendsto (fun b ↦ 1 / ε ^ p.toReal * eLpNorm (f b - g) p μ ^ p.toReal) l (𝓝 (1 / ε ^ p.toReal * 0 ^ p.toReal))
⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: case mpr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → TopologicalSpace (E i)
inst✝⁵ : (i : ι) → AddCommGroup (E i)
inst✝⁴ : (i : ι) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
⊢ f ∈
      {f |
        Continuous[Pi.topologicalSpace, inst✝] ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
          (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
                (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
            ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
                c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)} →
    f ∈ range toUniformOnFun

Tactic:
· rintro ⟨hcont, hadd, hsmul⟩
    exact ⟨⟨⟨f, by intro; convert! hadd, by intro; convert! hsmul⟩, hcont⟩, rfl⟩

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ IsTopCompl q p ↔ Continuous[inst✝³, instTopologicalSpaceSubtype] (⇑(p.quotientEquivOfIsCompl q h) ∘ ⇑p.mkQL)

Tactic:
exact h.symm.isTopCompl_iff_projectionOnto

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
⊢ MemLp f 1 μ ↔ Integrable f μ

Tactic:
Integrable,

Goal state after:
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
⊢ MemLp f 1 μ ↔ AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
this : AEStronglyMeasurable (fun x ↦ c) μ
⊢ Integrable (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ

Tactic:
Integrable,

Goal state after:
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
this : AEStronglyMeasurable (fun x ↦ c) μ
⊢ AEStronglyMeasurable (fun x ↦ c) μ ∧ HasFiniteIntegral (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case pos
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
i : ι
hi : ‖m i‖ = 0
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
hf₀ hi,

Goal state after:
case pos
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
i : ι
hi : ‖m i‖ = 0
⊢ 0 ≤ C * ∏ i, ‖m i‖
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
a : α
f : α → ε
hf : StronglyMeasurable f
hfa : ‖f a‖ₑ < ∞
⊢ HasFiniteIntegral f (Measure.dirac a)

Tactic:
simpa [HasFiniteIntegral, lintegral_dirac' _ hf.enorm]

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
⊢ IntegrableOn f {x} μ ↔ ‖f x‖ₑ = 0 ∨ μ {x} < ∞

Tactic:
have : f =ᵐ[μ.restrict {x}] fun _ => f x := by
    filter_upwards [ae_restrict_mem (measurableSet_singleton x)] with _ ha
    simp only [mem_singleton_iff.1 ha]
  rw [IntegrableOn, integrable_congr this, integrable_const_iff_enorm, isFiniteMeasure_restrict,
    lt_top_iff_ne_top]
  exact hfx

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hs : MeasurableSet s
⊢ Integrable (s.indicator f) μ ↔ Integrable f (μ.restrict s)

Tactic:
Integrable,

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hs : MeasurableSet s
⊢ AEStronglyMeasurable (s.indicator f) μ ∧ HasFiniteIntegral (s.indicator f) μ ↔
    AEStronglyMeasurable f (μ.restrict s) ∧ HasFiniteIntegral f (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : AEStronglyMeasurable f μ
⊢ Integrable (fun x ↦ ‖f x‖ₑ) μ ↔ Integrable f μ

Tactic:
Integrable,

Goal state after:
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable (fun x ↦ ‖f x‖ₑ) μ ∧ HasFiniteIntegral (fun x ↦ ‖f x‖ₑ) μ ↔
    AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ
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 : α → β
hf : AEStronglyMeasurable f μ
⊢ Integrable (fun a ↦ ‖f a‖) μ ↔ Integrable f μ

Tactic:
Integrable,

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable (fun a ↦ ‖f a‖) μ ∧ HasFiniteIntegral (fun a ↦ ‖f a‖) μ ↔
    AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
⊢ Integrable f Measure.count ↔ Summable fun x ↦ ‖f x‖

Tactic:
Integrable,

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
⊢ AEStronglyMeasurable f Measure.count ∧ HasFiniteIntegral f Measure.count ↔ Summable fun x ↦ ‖f x‖
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 : MeasurableSet s
a✝ : Nontrivial α
inhabited_h : Inhabited α
this : (s.piecewise f fun x ↦ f default) =ᵐ[μ.restrict s] f
⊢ AEMeasurable f (μ.restrict s)

Tactic:
refine ⟨Set.piecewise s f fun _ => f default, ?_, this.symm⟩

Goal state after:
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 : MeasurableSet s
a✝ : Nontrivial α
inhabited_h : Inhabited α
this : (s.piecewise f fun x ↦ f default) =ᵐ[μ.restrict s] f
⊢ Measurable (s.piecewise f fun x ↦ f default)
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 s)

Tactic:
rw [← Measure.restrict_congr_set t_eq_s]

Goal state after:
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)
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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
⊢ 𝓤 (ContinuousMultilinearMap 𝕜 E G) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖-f.1 + f.2‖ < r}

Tactic:
simp only [A]

Goal state after:
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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
⊢ 𝓤 (ContinuousMultilinearMap 𝕜 E G) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖f.1 - f.2‖ < r}
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
⊢ Integrable (fun x ↦ (f x).toReal) μ ↔ ∫⁻ (x : α), f x ∂μ ≠ ∞

Tactic:
Integrable,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
⊢ AEStronglyMeasurable (fun x ↦ (f x).toReal) μ ∧ HasFiniteIntegral (fun x ↦ (f x).toReal) μ ↔ ∫⁻ (x : α), f x ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hfm : AEStronglyMeasurable f μ
hf : 0 ≤ᵐ[μ] f
⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ ≠ ∞ ↔ Integrable f μ

Tactic:
Integrable,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hfm : AEStronglyMeasurable f μ
hf : 0 ≤ᵐ[μ] f
⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ ≠ ∞ ↔ AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case h
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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
c : 𝕜
hc : 1 < ‖c‖
hc₀ : 0 < ‖c‖
⊢ ∀ (f : ContinuousMultilinearMap 𝕜 E G), (∀ (x : (i : ι) → E i), ‖x‖ ≤ ‖c‖ → ‖f x‖ ≤ 1) → ‖f‖ ≤ 1

Tactic:
intro f hf

Goal state after:
case h
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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
c : 𝕜
hc : 1 < ‖c‖
hc₀ : 0 < ‖c‖
f : ContinuousMultilinearMap 𝕜 E G
hf : ∀ (x : (i : ι) → E i), ‖x‖ ≤ ‖c‖ → ‖f x‖ ≤ 1
⊢ ‖f‖ ≤ 1
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
𝕜 : Type u_8
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : CompleteSpace 𝕜
E : Type u_9
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
c : E
hc : c ≠ 0
⊢ Integrable (fun x ↦ f x • c) μ ↔ Integrable f μ

Tactic:
Integrable,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
𝕜 : Type u_8
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : CompleteSpace 𝕜
E : Type u_9
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
c : E
hc : c ≠ 0
⊢ AEStronglyMeasurable (fun x ↦ f x • c) μ ∧ HasFiniteIntegral (fun x ↦ f x • c) μ ↔
    AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ ENNReal.ofReal ‖f - g‖ = ∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ

Tactic:
simp_rw [ofReal_norm_eq_lintegral, ← edist_zero_right]
  apply lintegral_congr_ae
  filter_upwards [Lp.coeFn_sub f g] with _ ha
  simp only [ha, Pi.sub_apply]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : UniformSpace E
inst✝¹⁵ : IsUniformAddGroup E
inst✝¹⁴ : AddCommGroup F
inst✝¹³ : UniformSpace F
inst✝¹² : IsUniformAddGroup F
inst✝¹¹ : T0Space F
inst✝¹⁰ : AddCommMonoid Eₗ
inst✝⁹ : UniformSpace Eₗ
inst✝⁸ : ContinuousAdd Eₗ
inst✝⁷ : Semiring 𝕜
inst✝⁶ : Semiring 𝕜₂
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : Module 𝕜 Eₗ
inst✝² : ContinuousConstSMul 𝕜 Eₗ
inst✝¹ : ContinuousConstSMul 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h_dense : DenseRange ⇑e
h_e : IsUniformInducing ⇑e
x : E
⊢ ⋯.extend (⇑f) (e x) = f x

Tactic:
exact IsDenseInducing.extend_eq (h_e.isDenseInducing h_dense) f.cont _

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

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : DecidablePred fun x ↦ x ≠ 0
m : MeasurableSpace α
T : Set α → F →L[ℝ] F'
f : α →ₛ F
⊢ setToSimpleFunc T f = ∑ x ∈ f.range with x ≠ 0, (T (⇑f ⁻¹' {x})) x

Tactic:
by
  symm
  refine sum_filter_of_ne fun x _ => mt fun hx0 => ?_
  rw [hx0]
  exact map_zero _

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : ∀ (y : E), y ≠ 0 → μ (⇑g ⁻¹' {y}) < ∞
x : E
hx_ne : x ≠ 0
⊢ μ (⇑(-g) ⁻¹' {x}) < ∞

Tactic:
SimpleFunc.coe_neg,

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : ∀ (y : E), y ≠ 0 → μ (⇑g ⁻¹' {y}) < ∞
x : E
hx_ne : x ≠ 0
⊢ μ ((-⇑g) ⁻¹' {x}) < ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ↥(Lp E 1 μ)
⊢ integral (-f) = -integral f

Tactic:
simp only [integral]
  exact map_neg integralCLM f

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
ν : Measure β
f g : α → β → G
h : ∀ᵐ (a : α) ∂μ, f a =ᵐ[ν] g a
⊢ ∫ (a : α), ∫ (b : β), f a b ∂ν ∂μ = ∫ (a : α), ∫ (b : β), g a b ∂ν ∂μ

Tactic:
apply integral_congr_ae

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
ν : Measure β
f g : α → β → G
h : ∀ᵐ (a : α) ∂μ, f a =ᵐ[ν] g a
⊢ (fun a ↦ ∫ (b : β), f a b ∂ν) =ᵐ[μ] fun a ↦ ∫ (b : β), g a b ∂ν