mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
⊢ (ENNReal.toReal ∘ g) 0 = 0

Tactic:
·

Goal state after:
α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
⊢ (ENNReal.toReal ∘ g) 0 = 0
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case hi
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 := ⋯
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 := ⋯
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 ⋯
⊢ eLpNorm (fun x ↦ ↑↑↑g x - 0) p μ < ∞

Tactic:
·

Goal state after:
case hi
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 := ⋯
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 := ⋯
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 ⋯
⊢ eLpNorm (fun x ↦ ↑↑↑g x - 0) p μ < ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case pos
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

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:
·

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

Goal state before:
pp: case pos
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
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
eq : f a = g a
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2

pp: case neg
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
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
eq : ¬f a = g a
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2

Tactic:
·

Goal state after:
case pos
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
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
eq : f a = g a
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2
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:
·

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

Goal state before:
pp: case neg
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
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
eq : ¬f a = g a
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2

Tactic:
·

Goal state after:
case neg
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
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
eq : ¬f a = g a
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case pos
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : 0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : ¬0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

Tactic:
·

Goal state after:
case pos
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : 0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case refine_1
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)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
n : ℕ
⊢ ⇑(x n) =ᵐ[μ] ↑↑↑((x n).toLp ⋯)

pp: case refine_2
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)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
n : ℕ
⊢ ↑↑↑g =ᵐ[μ] ↑↑↑g

Tactic:
·

Goal state after:
case refine_1
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)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
n : ℕ
⊢ ⇑(x n) =ᵐ[μ] ↑↑↑((x n).toLp ⋯)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case pos.inl
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx✝ : 0 ≤ f x
hx : 0 = f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

pp: case pos.inr
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx✝ : 0 ≤ f x
hx : 0 < f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

Tactic:
·

Goal state after:
case pos.inl
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx✝ : 0 ≤ f x
hx : 0 = f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x
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 T' : Set α → E →L[ℝ] F
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ setToSimpleFunc T f = setToSimpleFunc T' f

Tactic:
simp_rw [setToSimpleFunc]

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 T' : Set α → E →L[ℝ] F
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x = ∑ x ∈ f.range, (T' (⇑f ⁻¹' {x})) x
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : ¬0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

Tactic:
·

Goal state after:
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : ¬0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case pos
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 T' : Set α → E →L[ℝ] F
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : x = 0
⊢ (T (⇑f ⁻¹' {x})) x = (T' (⇑f ⁻¹' {x})) x

pp: case neg
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 T' : Set α → E →L[ℝ] F
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ (T (⇑f ⁻¹' {x})) x = (T' (⇑f ⁻¹' {x})) x

Tactic:
·

Goal state after:
case pos
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 T' : Set α → E →L[ℝ] F
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : x = 0
⊢ (T (⇑f ⁻¹' {x})) x = (T' (⇑f ⁻¹' {x})) x
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case refine_2
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)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
n : ℕ
⊢ ↑↑↑g =ᵐ[μ] ↑↑↑g

Tactic:
·

Goal state after:
case refine_2
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)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
n : ℕ
⊢ ↑↑↑g =ᵐ[μ] ↑↑↑g
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case neg
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 T' : Set α → E →L[ℝ] F
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ (T (⇑f ⁻¹' {x})) x = (T' (⇑f ⁻¹' {x})) x

Tactic:
·

Goal state after:
case neg
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 T' : Set α → E →L[ℝ] F
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ (T (⇑f ⁻¹' {x})) x = (T' (⇑f ⁻¹' {x})) x
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
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : ¬0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

Tactic:
· suffices ν (f ⁻¹' {f x}) = 0 by
      have A : μ (f ⁻¹' {f x}) = 0 := by simpa using (hμν _ |>.trans_eq this)
      simp [measureReal_def, A, this]
    rw [← nonpos_iff_eq_zero, ← ae_iff.mp hf]
    refine measure_mono fun y hy ↦ ?_
    simp_all

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

Goal state before:
pp: case refine_1
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 : ↥(Lp E p μ) → Prop
indicatorConst :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α → E⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    Disjoint (support f) (support g) →
      motive (MemLp.toLp f hf) → motive (MemLp.toLp g hg) → motive (MemLp.toLp f hf + MemLp.toLp g hg)
isClosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive f}
f : ↥(Lp E p μ)
⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)

pp: case refine_2
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 : ↥(Lp E p μ) → Prop
indicatorConst :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α → E⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    Disjoint (support f) (support g) →
      motive (MemLp.toLp f hf) → motive (MemLp.toLp g hg) → motive (MemLp.toLp f hf + MemLp.toLp g hg)
isClosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive f}
f : ↥(Lp E p μ)
⊢ ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → motive ↑(f.toLp hf) → motive ↑(g.toLp hg) → motive ↑(f.toLp hf + g.toLp hg)

Tactic:
·

Goal state after:
case refine_1
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 : ↥(Lp E p μ) → Prop
indicatorConst :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α → E⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    Disjoint (support f) (support g) →
      motive (MemLp.toLp f hf) → motive (MemLp.toLp g hg) → motive (MemLp.toLp f hf + MemLp.toLp g hg)
isClosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive f}
f : ↥(Lp E p μ)
⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case refine_2
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 : ↥(Lp E p μ) → Prop
indicatorConst :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α → E⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    Disjoint (support f) (support g) →
      motive (MemLp.toLp f hf) → motive (MemLp.toLp g hg) → motive (MemLp.toLp f hf + MemLp.toLp g hg)
isClosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive f}
f : ↥(Lp E p μ)
⊢ ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → motive ↑(f.toLp hf) → motive ↑(g.toLp hg) → motive ↑(f.toLp hf + g.toLp hg)

Tactic:
·

Goal state after:
case refine_2
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 : ↥(Lp E p μ) → Prop
indicatorConst :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α → E⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    Disjoint (support f) (support g) →
      motive (MemLp.toLp f hf) → motive (MemLp.toLp g hg) → motive (MemLp.toLp f hf + MemLp.toLp g hg)
isClosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive f}
f : ↥(Lp E p μ)
⊢ ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → motive ↑(f.toLp hf) → motive ↑(g.toLp hg) → motive ↑(f.toLp hf + g.toLp hg)
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)
ae_eq₁ : ⇑(toSimpleFunc f).posPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))
ae_eq₂ : ⇑(toSimpleFunc f).negPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (negPart f)))
⊢ MeasureTheory.SimpleFunc.integral μ (toSimpleFunc f) =
    MeasureTheory.SimpleFunc.integral μ
      (SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f)))

pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
ae_eq₁ : ⇑(toSimpleFunc f).posPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))
ae_eq₂ : ⇑(toSimpleFunc f).negPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (negPart f)))
⊢ Integrable (⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))) μ

pp: case hg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
ae_eq₁ : ⇑(toSimpleFunc f).posPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))
ae_eq₂ : ⇑(toSimpleFunc f).negPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (negPart f)))
⊢ Integrable (⇑(SimpleFunc.map norm (toSimpleFunc (negPart f)))) μ

Tactic:
·

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
ae_eq₁ : ⇑(toSimpleFunc f).posPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))
ae_eq₂ : ⇑(toSimpleFunc f).negPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (negPart f)))
⊢ MeasureTheory.SimpleFunc.integral μ (toSimpleFunc f) =
    MeasureTheory.SimpleFunc.integral μ
      (SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f)))