mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁵ : Semifield α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Semifield β
inst✝¹ : LinearOrder β
inst✝ : IsStrictOrderedRing β
f : ι → α
g : ι → β
hf : StrongLT 0 f
hg : StrongLT 0 g
⊢ Monovary f⁻¹ g⁻¹ ↔ Monovary f g

Tactic:
rw [monovary_inv_left₀ hf, antivary_inv_right₀ hg]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw' : ∑ i ∈ s, w i = 1
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
⊢ ∑ i ∈ s, w i * z i = ∑ i ∈ s, w i * x

Tactic:
refine sum_congr rfl fun i hi => ?_
      rcases eq_or_ne (w i) 0 with hwi | hwi
      · rw [hwi, zero_mul, zero_mul]
      · rw [hx i hi hwi]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: w₁ w₂ p₁ p₂ : ℝ≥0
⊢ w₁ + w₂ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ ≤ w₁ * p₁ + w₂ * p₂

Tactic:
by
  simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,
    Finset.univ_eq_empty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using!
    geom_mean_le_arith_mean_weighted univ ![w₁, w₂] ![p₁, p₂]

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

Goal state before:
pp: case e'_4
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
p : ℝ≥0∞
hp : p ≠ 0
h'p : p < 1
⊢ p.LpAddConst = 2 ^ (1 / p.toReal - 1)

Tactic:
have : p ∈ Set.Ioo (0 : ℝ≥0∞) 1 := ⟨hp.bot_lt, h'p⟩
      simp only [LpAddConst, if_pos this]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos
a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h : a = ∞ ∨ b = ∞
⊢ ∞ = a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q

Tactic:
repeat rw [div_eq_mul_inv]

Goal state after:
case pos
a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h : a = ∞ ∨ b = ∞
⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹
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
hpp2 : p * p2 = q
hpq2 : p * q2 = r
⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 * 1 / (p2 * p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 * 1 / (q2 * p)) =
    (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)

Tactic:
mul_one,

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
h_one_div_r : 1 / r = 1 / p - 1 / q
p2 : ℝ := q / p
q2 : ℝ := p2.conjExponent
hp2q2 : p2.HolderConjugate q2
hpp2 : p * p2 = q
hpq2 : p * q2 = r
⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / (p2 * p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / (q2 * p)) =
    (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)
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))
⊢ eLpNorm' f_lim p μ ≤ liminf (fun n ↦ eLpNorm' (f n) p μ) atTop

Tactic:
rw [eLpNorm'_lim_eq_lintegral_liminf h_lim]
  rw [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv]
  refine (lintegral_liminf_le' fun m => (hf m).enorm.pow_const _).trans_eq ?_
  have h_pow_liminf :
    atTop.liminf (fun n ↦ eLpNorm' (f n) p μ) ^ p
      = atTop.liminf fun n ↦ eLpNorm' (f n) p μ ^ p := by
    have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hp_pos
    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_pow_liminf]
  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.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
⊢ TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ), 0 < ε → Tendsto (fun i ↦ μ.real {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)

Tactic:
tendstoInMeasure_iff_dist

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), 0 < ε → Tendsto (fun i ↦ μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)) ↔
    ∀ (ε : ℝ), 0 < ε → Tendsto (fun i ↦ μ.real {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_3.a
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞
this :
  ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ i ∈ s, ↑(w i).toNNReal) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
i : ι
hi : i ∈ s
⊢ ↑(w i).toNNReal = w i

Tactic:
exact coe_toNNReal (H'.1 _ hi)

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
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 0)
⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)

Tactic:
ENNReal.tendsto_nhds_zero

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 : ∀ ε_1 > 0, ∀ᶠ (x : ι) in l, 1 / ε ^ p.toReal * eLpNorm (f x - g) p μ ^ p.toReal ≤ ε_1
⊢ ∀ ε_1 > 0, ∀ᶠ (x : ι) in l, μ {x_1 | ε ≤ edist (f x x_1) (g x_1)} ≤ ε_1
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
m : MeasurableSpace α
μ : Measure α
E : Type u_5
inst✝ : SeminormedAddCommGroup E
f : ι → α → E
g : α → E
l : Filter ι
δ : ℝ≥0∞
hδ : 0 < δ
hδ_top : δ ≠ ∞
hfg : Tendsto (fun n ↦ essSup (fun x ↦ ‖(f n - g) x‖ₑ) μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ μ {x | δ ≤ edist (f i x) (g x)}) l (𝓝 0)

Tactic:
ENNReal.tendsto_nhds_zero

Goal state after:
α : Type u_1
ι : Type u_2
m : MeasurableSpace α
μ : Measure α
E : Type u_5
inst✝ : SeminormedAddCommGroup E
f : ι → α → E
g : α → E
l : Filter ι
δ : ℝ≥0∞
hδ : 0 < δ
hδ_top : δ ≠ ∞
hfg : ∀ ε > 0, ∀ᶠ (x : ι) in l, essSup (fun x_1 ↦ ‖(f x - g) x_1‖ₑ) μ ≤ ε
⊢ ∀ ε > 0, ∀ᶠ (x : ι) in l, μ {x_1 | δ ≤ edist (f x x_1) (g x_1)} ≤ ε
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
h_is_o : (fun t ↦ (↑x + t)⁻¹ʳ - ↑x⁻¹) =o[𝓝 0] fun x ↦ 1
⊢ ContinuousAt inverse ↑x

Tactic:
have h_lim : Tendsto (fun y : R => y - x) (𝓝 x) (𝓝 0) := by
    refine tendsto_zero_iff_norm_tendsto_zero.mpr ?_
    exact tendsto_iff_norm_sub_tendsto_zero.mp tendsto_id

Goal state after:
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
h_is_o : (fun t ↦ (↑x + t)⁻¹ʳ - ↑x⁻¹) =o[𝓝 0] fun x ↦ 1
h_lim : Tendsto (fun y ↦ y - ↑x) (𝓝 ↑x) (𝓝 0)
⊢ ContinuousAt inverse ↑x
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E
C : ℝ
hC_pos : C > 0
hC : ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
y : F
hx : max ‖x‖ 1 = ‖x‖ ∧ 1 ≤ ‖x‖
⊢ C * ‖x‖ * ‖y‖ ≤ C * max ‖x‖ 1 * ‖y‖

Tactic:
hx.1

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E
C : ℝ
hC_pos : C > 0
hC : ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
y : F
hx : max ‖x‖ 1 = ‖x‖ ∧ 1 ≤ ‖x‖
⊢ C * ‖x‖ * ‖y‖ ≤ C * ‖x‖ * ‖y‖
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := ⋯
h_O : Continuous O
f' : E →L[𝕜] F
⊢ f' ∈ range toContinuousLinearMap → f' ∈ O ⁻¹' {x | IsUnit x}

pp: case mpr
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := ⋯
h_O : Continuous O
f' : E →L[𝕜] F
⊢ f' ∈ O ⁻¹' {x | IsUnit x} → f' ∈ range toContinuousLinearMap

Tactic:
· rintro ⟨e', rfl⟩
    exact ⟨(e'.trans e.symm).toUnit, rfl⟩

Goal state after:
case mpr
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := ⋯
h_O : Continuous O
f' : E →L[𝕜] F
⊢ f' ∈ O ⁻¹' {x | IsUnit x} → f' ∈ range toContinuousLinearMap
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: case inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
η : ℝ≥0
hη_pos : 0 < η
hη_le : ‖c‖ₑ * ↑η ^ (1 / p.toReal) ≤ ε
s : Set α
hs : μ s ≤ ↑η
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) ≤ ε

Tactic:
← hη_le,

Goal state after:
case inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
η : ℝ≥0
hη_pos : 0 < η
hη_le : ‖c‖ₑ * ↑η ^ (1 / p.toReal) ≤ ε
s : Set α
hs : μ s ≤ ↑η
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) ≤ ‖c‖ₑ * ↑η ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

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

Tactic:
mul_one,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
hμs_ne_zero : μ s ≠ 0
⊢ ‖c‖ₑ.toReal = ‖c‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : Finite ι
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : TopologicalSpace F
inst✝² : AddCommGroup F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : Module 𝕜 F
⊢ Continuous[instTopologicalSpaceProd, inst✝³] fun fx ↦ fx.1 fx.2

Tactic:
cases nonempty_fintype ι
    let _ := IsTopologicalAddGroup.rightUniformSpace F
    have := isUniformAddGroup_of_addCommGroup (G := F)
    refine (UniformOnFun.continuousOn_eval₂ fun m ↦ ?_).comp_continuous
      (isEmbedding_toUniformOnFun.continuous.prodMap continuous_id) fun (f, x) ↦ f.cont.continuousAt
    exact ⟨ball m 1, NormedSpace.isVonNBounded_of_isBounded _ isBounded_ball,
      ball_mem_nhds _ one_pos⟩

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSingletonClass α
a : α
f : α → ε
hfa : ‖f a‖ₑ < ∞
⊢ HasFiniteIntegral f (Measure.dirac a)

Tactic:
by simpa [HasFiniteIntegral]

Goal state after:
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
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
f : β → ε
μ : Measure α
s : Set β
⊢ IntegrableOn f s (Measure.map (⇑e) μ) ↔ IntegrableOn (f ∘ ⇑e) (⇑e ⁻¹' s) μ

Tactic:
simp only [IntegrableOn, e.restrict_map, integrable_map_equiv e]

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝² : NormedAddCommGroup E
μ : Measure α
l : Filter α
𝕜 : Type u_7
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → E
c : 𝕜
hc : c = 0
⊢ IntegrableAtFilter (c • f) l μ ↔ c = 0 ∨ IntegrableAtFilter f l μ

pp: case neg
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝² : NormedAddCommGroup E
μ : Measure α
l : Filter α
𝕜 : Type u_7
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → E
c : 𝕜
hc : ¬c = 0
⊢ IntegrableAtFilter (c • f) l μ ↔ c = 0 ∨ IntegrableAtFilter f l μ

Tactic:
· simp [hc]

Goal state after:
case neg
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝² : NormedAddCommGroup E
μ : Measure α
l : Filter α
𝕜 : Type u_7
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → E
c : 𝕜
hc : ¬c = 0
⊢ IntegrableAtFilter (c • f) l μ ↔ c = 0 ∨ IntegrableAtFilter f l μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case mpr
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
l l' : Filter α
⊢ IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ → IntegrableAtFilter f (l ⊔ l') μ

Tactic:
· exact fun ⟨⟨s, hsl, hs⟩, ⟨t, htl, ht⟩⟩ ↦ ⟨s ∪ t, union_mem_sup hsl htl, hs.union ht⟩

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

Goal state before:
pp: case a
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : IsEmpty ι
x✝ : G
x : (i : ι) → E i
⊢ ‖(constOfIsEmpty 𝕜 E x✝) x‖ ≤ ‖x✝‖ * 1

Tactic:
mul_one,

Goal state after:
case a
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : IsEmpty ι
x✝ : G
x : (i : ι) → E i
⊢ ‖(constOfIsEmpty 𝕜 E x✝) x‖ ≤ ‖x✝‖
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f g : X → ε
μ : Measure X
s : Set X
h : f =ᵐ[μ.restrict s] g
hf : LocallyIntegrableOn f s μ
x : X
hx : x ∈ s
⊢ IntegrableAtFilter g (𝓝[s] x) μ

Tactic:
obtain ⟨t, hxt, hft⟩ := hf x hx

Goal state after:
X : Type u_1
ε : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f g : X → ε
μ : Measure X
s : Set X
h : f =ᵐ[μ.restrict s] g
hf : LocallyIntegrableOn f s μ
x : X
hx : x ∈ s
t : Set X
hxt : t ∈ 𝓝[s] x
hft : IntegrableOn f t μ
⊢ IntegrableAtFilter g (𝓝[s] x) μ
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 𝕜
z : G
⊢ ‖f.smulRight z‖₊ = ‖f‖₊ * ‖z‖₊

Tactic:
refine le_antisymm ?_ ?_
  · refine opNNNorm_le_iff.2 fun m => (nnnorm_smul_le _ _).trans ?_
    rw [mul_right_comm]
    gcongr
    exact le_opNNNorm _ _
  · obtain hz | hz := eq_zero_or_pos ‖z‖₊
    · simp [hz]
    rw [← le_div_iff₀ hz, opNNNorm_le_iff]
    intro m
    rw [div_mul_eq_mul_div, le_div_iff₀ hz]
    refine le_trans ?_ ((f.smulRight z).le_opNNNorm m)
    rw [smulRight_apply, nnnorm_smul]

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
⊢ ∃ u, (∀ (n : ℕ), IsOpen[inst✝³] (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ

Tactic:
rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
⊢ ∃ u, (∀ (n : ℕ), IsOpen[inst✝³] (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
μ : Measure X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A μ
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
⊢ IntegrableOn (fun x ↦ g x * g' x) A μ

Tactic:
rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩

Goal state after:
X : Type u_1
R : Type u_8
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
μ : Measure X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A μ
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
⊢ IntegrableOn (fun x ↦ g x * g' x) A μ
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case pos
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
⊢ (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = ∑ j ∈ f.range with g j = g (f a), (T (⇑f ⁻¹' {j})) (g j)

Tactic:
simp_rw [h0]

Goal state after:
case pos
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
⊢ (T (⇑(map g f) ⁻¹' {0})) 0 = ∑ x ∈ f.range with g x = 0, (T (⇑f ⁻¹' {x})) (g x)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝² : MeasurableSpace G := borel G
this✝¹ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this✝ : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n ↦ approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
h_toLp : ∀ (n : ℕ), ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᵐ[μ] ⇑(x n)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
this : Tendsto (fun n ↦ ↑((x n).toLp ⋯)) atTop (𝓝 ↑g)
⊢ Tendsto (fun n ↦ coeSimpleFuncNonnegToLpNonneg p μ G ⟨(x n).toLp ⋯, ⋯⟩) atTop (𝓝 g)

Tactic:
rw [tendsto_iff_dist_tendsto_zero] at this ⊢
    simp_rw [Subtype.dist_eq]
    exact this

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
R : Type u_6
inst✝³ : NormedRing R
inst✝² : Module R G
inst✝¹ : IsBoundedSMul R G
inst✝ : SMulCommClass ℝ R G
c : R
f : α → G
hf : Integrable f μ
⊢ ∫ (a : α), c • f a ∂μ = c • ∫ (a : α), f a ∂μ

Tactic:
by
  by_cases hG : CompleteSpace G
  · simpa only [integral, hG, hf, hf.fun_smul c] using! L1.integral_smul c (toL1 f hf)
  · simp [integral, hG]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
⊢ u ~[l] u

Tactic:
rw [IsEquivalent, sub_self]
  exact isLittleO_zero _ _

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_mono : ∀ᵐ (x : α) ∂μ, MonotoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ MonotoneOn (fun b ↦ ∫ (x : α), f x b ∂μ) s

Tactic:
by
  intro a ha b hb hab
  refine integral_mono_ae (hf_int a ha) (hf_int b hb) ?_
  filter_upwards [hf_mono] with x hx
  exact hx ha hb hab

Goal state after: