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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ 0 ≤ 1 / p.toReal

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ ‖T s + T' s‖ ≤ max C C' * (μ + ν).real s

Tactic:
measureReal_add_apply hμs.ne hνs.ne,

Goal state after:
α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ ‖T s + T' s‖ ≤ max C C' * (μ.real s + ν.real s)
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case refine_1.inr
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup Eₗ
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 Eₗ
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : CompleteSpace F
e : E →L[𝕜] Eₗ
N : ℝ≥0
inst✝ : RingHomIsometric σ₁₂
h_dense : DenseRange ⇑e
h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖
hN : N ≤ 0
this : Unique E
⊢ 0 ≤ ↑N * ‖0‖

Tactic:
simp

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

Goal state before:
pp: case inl
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'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
hx : 0 ∉ f.range ∨ 0 = 0
⊢ (T (⇑f ⁻¹' {0})) 0 = 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(simpleFunc E p μ)
⊢ ⇑(toSimpleFunc (k • f)) =ᵐ[μ] k • ⇑(toSimpleFunc f)

Tactic:
by
  filter_upwards [toSimpleFunc_eq_toFun (k • f), toSimpleFunc_eq_toFun f,
    Lp.coeFn_smul k (f : Lp E p μ)] with _
  simp only [Pi.smul_apply, coe_smul]
  repeat intro h; rw [h]

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
Fₗ : Type u_6
inst✝¹³ : NormedField 𝕜
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜₂ F
inst✝⁷ : NormedAddCommGroup Eₗ
inst✝⁶ : NormedSpace 𝕜 Eₗ
inst✝⁵ : CompleteSpace Eₗ
inst✝⁴ : NormedAddCommGroup Fₗ
inst✝³ : NormedSpace 𝕜₂ Fₗ
inst✝² : CompleteSpace Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : E ≃ₛₗ[σ₁₂] F
e₁ : E →ₗ[𝕜] Eₗ
e₂ : F →ₗ[𝕜₂] Fₗ
h_dense₁ : DenseRange ⇑e₁
h_dense₂ : DenseRange ⇑e₂
h_norm : ∀ (x : E), ‖e₂ (f x)‖ = ‖e₁ x‖
h_norm₂ : ∀ (x : F), ‖e₁ (f.symm x)‖ = ‖e₂ x‖
y : E
⊢ ∃ C, ∀ (x : E), ‖(e₂ ∘ₛₗ ↑f) x‖ ≤ C * ‖e₁ x‖

Tactic:
use 1; simp [h_norm]

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

Goal state before:
pp: case const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
c : E
s : Set α
hs : MeasurableSet s
⊢ ∀ (hf : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ),
    P ((SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)).toLp hf)

Tactic:
hf

Goal state after:
case const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
c : E
s : Set α
hs : MeasurableSet s
hf : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
⊢ P ((SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)).toLp hf)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
c : E
s : Set α
hs : MeasurableSet s
hf : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : c = 0
⊢ μ ∅ < ∞

Tactic:
simp

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

Goal state before:
pp: case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
f g : α →ₛ E
hfg : Disjoint (support ⇑f) (support ⇑g)
⊢ (∀ (hf : MemLp (⇑f) p μ), P (f.toLp hf)) →
    (∀ (hf : MemLp (⇑g) p μ), P (g.toLp hf)) → ∀ (hf : MemLp (⇑(f + g)) p μ), P ((f + g).toLp hf)

Tactic:
hf

Goal state after:
case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
f g : α →ₛ E
hfg : Disjoint (support ⇑f) (support ⇑g)
hf : ∀ (hf : MemLp (⇑f) p μ), P (f.toLp hf)
⊢ (∀ (hf : MemLp (⇑g) p μ), P (g.toLp hf)) → ∀ (hf : MemLp (⇑(f + g)) p μ), P ((f + g).toLp hf)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
x : F
⊢ μ.real ∅ • x = 0 x

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
s : Set α
h_zero : μ s = 0
x : F
⊢ ENNReal.toReal 0 • x = 0 x

Tactic:
simp

Goal state after:
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✝¹ : MeasurableSpace G := borel G
this✝ : 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})
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)

Tactic:
have g_meas : Measurable (g : α → G) := (Lp.stronglyMeasurable (g : Lp G p μ)).measurable

Goal state after:
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✝¹ : MeasurableSpace G := borel G
this✝ : 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 : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case inl
α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
hx : 0 ∉ f.range ∨ 0 = 0
⊢ μ.real (⇑f ⁻¹' {0}) • 0 = 0

Tactic:
simp

Goal state after:
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✝¹ : MeasurableSpace G := borel G
this✝ : 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
⊢ eLpNorm (fun x ↦ 0) p μ < ∞

Tactic:
simp

Goal state after:
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✝¹ : MeasurableSpace G := borel G
this✝ : 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)
n : ℕ
⊢ ↑↑↑0 ≤ᵐ[μ] ↑↑(MemLp.toLp ⇑(x n) ⋯)

Tactic:
filter_upwards [Lp.simpleFunc.coeFn_zero p μ G, h_toLp n] with a ha0 ha_toLp

Goal state after:
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✝¹ : MeasurableSpace G := borel G
this✝ : 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)
n : ℕ
a : α
ha0 : ↑↑↑0 a = 0 a
ha_toLp : ↑↑(MemLp.toLp ⇑(x n) ⋯) a = (x n) a
⊢ ↑↑↑0 a ≤ ↑↑(MemLp.toLp ⇑(x n) ⋯) a
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case neg
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
f : α →ₛ F
hf : 0 ≤ᵐ[μ] ⇑f
y : α
hy : ¬0 ≤ f y
⊢ 0 ≤ μ.real (⇑f ⁻¹' {f y}) • f y

Tactic:
· suffices μ (f ⁻¹' {f y}) = 0 by simp [this, measureReal_def]
    rw [← nonpos_iff_eq_zero]
    refine le_of_le_of_eq (measure_mono fun x hx ↦ ?_) (ae_iff.mp hf)
    simp only [Set.mem_preimage, mem_singleton_iff, mem_ofPred_eq] at hx ⊢
    exact hx ▸ hy

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
a : α
h₁ : (toSimpleFunc (posPart (-f))) a = (toSimpleFunc (-f)).posPart a
h₂ : (toSimpleFunc (-f)) a = (-⇑(toSimpleFunc f)) a
⊢ max ((-⇑(toSimpleFunc f)) a) 0 = max ((-toSimpleFunc f) a) 0

Tactic:
simp

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 : Integrable (⇑g) μ
hp_pair : Integrable (⇑(f.pair g)) μ
⊢ 0.1 + 0.2 = 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
G' : Type u_7
G'' : Type u_8
inst✝⁶ : NormedAddCommGroup G''
inst✝⁵ : PartialOrder G''
inst✝⁴ : IsOrderedAddMonoid G''
inst✝³ : NormedSpace ℝ G''
inst✝² : NormedAddCommGroup G'
inst✝¹ : PartialOrder G'
inst✝ : NormedSpace ℝ G'
m : MeasurableSpace α
T : Set α → G' →L[ℝ] G''
hT_nonneg : ∀ (s : Set α) (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α →ₛ G'
hf : 0 ≤ f
i : G'
hi : i ∈ Set.range ⇑f
y : α
hy : f y = i
⊢ 0 ≤ 0 y

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
⊢ μ ∅ < ∞

Tactic:
simp

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
f : α → G
⊢ 0 ≤ 1

Tactic:
simp

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
f : α → G
⊢ ↑(NNReal.mk 1 ⋯) * (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal

Tactic:
simp

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
f : α → G
⊢ ↑(NNReal.mk 1 ⋯) * ∫⁻ (x : α), ‖f x‖ₑ ∂μ ≤ ∫⁻ (a : α), ‖f a‖ₑ ∂μ

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
x✝ : α
⊢ (SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = ∅.indicator (fun x ↦ 0) x✝

Tactic:
simp

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

Goal state before:
pp: case inl
α : Type u_1
E : Type u_2
F' : Type u_4
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'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * μ.real s
f : α →ₛ E
hf : Integrable (⇑f) μ
hb : 0 ∈ f.range
⊢ ‖T (⇑f ⁻¹' {0})‖ * ‖0‖ ≤ C * μ.real (⇑f ⁻¹' {0}) * ‖0‖

Tactic:
simp

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

Goal state before:
pp: case pos
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
hx0 : x = 0
⊢ ∑ x ∈ {0, 0}, (T (⇑(piecewise s hs (const α 0) (const α 0)) ⁻¹' {x})) x = (T s) 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖

Tactic:
L1.norm_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = (∫⁻ (a : α), ‖↑↑(Lp.posPart f₁) a‖ₑ ∂μ).toReal
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
⊢ (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal = ‖Lp.negPart f₁‖

Tactic:
L1.norm_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
⊢ (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal = (∫⁻ (a : α), ‖↑↑(Lp.negPart f₁) a‖ₑ ∂μ).toReal
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case neg
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
hx0 : ¬x = 0
⊢ s.ite Set.univ ∅ = s

Tactic:
simp

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

Goal state before:
pp: case e'_8
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u v : α → β
l : Filter α
h : u ~[l] v
x✝ : α
⊢ u x✝ = (u - v) x✝ + v x✝

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ (fun a ↦ ENNReal.ofReal (-f a)) =ᵐ[μ] 0

pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (-f a)) μ

Tactic:
· refine hf.mono ?_
        simp only [Pi.zero_apply]
        intro a h
        simp only [h, neg_nonpos, ofReal_eq_zero]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (-f a)) μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
f g : α →ₛ E
hfg : Disjoint (support ⇑f) (support ⇑g)
⊢ (MemLp (⇑f) p μ → motive ⇑f) → (MemLp (⇑g) p μ → motive ⇑g) → MemLp (⇑(f + g)) p μ → motive ⇑(f + g)

Tactic:
hf

Goal state after:
case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
f g : α →ₛ E
hfg : Disjoint (support ⇑f) (support ⇑g)
hf : MemLp (⇑f) p μ → motive ⇑f
⊢ (MemLp (⇑g) p μ → motive ⇑g) → MemLp (⇑(f + g)) p μ → motive ⇑(f + g)
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
P : Type u_7
inst✝ : NormedAddCommGroup P
f : α → P
hf : Integrable f μ
⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂μ ≠ ∞

Tactic:
exact lt_top_iff_ne_top.mp (hasFiniteIntegral_iff_enorm.mpr hf.2)

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

Goal state before:
pp: case e'_3
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
hequiv : u ~[l] v
this : Tendsto (fun x ↦ u x / v x - v x / v x) l (𝓝 0)
key : Tendsto (fun x ↦ v x / v x) l (𝓝 1)
x✝ : α
⊢ (u / v) x✝ = u x✝ / v x✝ - v x✝ / v x✝ + v x✝ / v x✝

Tactic:
simp

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

Goal state before:
pp: case e'_5
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
hequiv : u ~[l] v
this : Tendsto (fun x ↦ u x / v x - v x / v x) l (𝓝 0)
key : Tendsto (fun x ↦ v x / v x) l (𝓝 1)
⊢ 1 = 0 + 1

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedDivisionRing 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : CompleteSpace F
G : Type u_5
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedSpace ℝ E
f✝ : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : FirstCountableTopology X
inst✝³ : CompleteSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
s : Set E
inst✝ : SeparableSpace ↑s
hs : range f ∪ {0} ⊆ s
n : ℕ
⊢ 0 ∈ range f ∪ {0}

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹³ : NormedAddCommGroup E✝
inst✝¹² : NormedDivisionRing 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : CompleteSpace F
G : Type u_5
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : NormedSpace ℝ E✝
f✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : FirstCountableTopology X
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
inst✝ : SeparableSpace ↑(range f ∪ {0})
n : ℕ
x : α
⊢ 0 ∈ range f ∪ {0}

Tactic:
simp

Goal state after: