mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ ∀ (n : ℕ), 0 ≤ x n

Tactic:
intro n

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
n : ℕ
⊢ 0 ≤ x n
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 ≤ (x n).toLp ⋯

Tactic:
intro n

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
⊢ 0 ≤ (x n).toLp ⋯
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] ↑↑↑((x n).toLp ⋯)

Tactic:
Lp.simpleFunc.toLp_eq_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
⊢ ↑↑↑0 ≤ᵐ[μ] ↑↑(MemLp.toLp ⇑(x n) ⋯)
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
a : α
⊢ ↑↑↑0 a = 0 a → ↑↑(MemLp.toLp ⇑(x n) ⋯) a = (x n) a → ↑↑↑0 a ≤ ↑↑(MemLp.toLp ⇑(x n) ⋯) a

Tactic:
ha0

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
⊢ ↑↑(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: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
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
⊢ C * ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) * ‖x‖ = C * integral μ (map norm f)

Tactic:
rw [map_integral f norm hf norm_zero]; simp_rw [smul_eq_mul]

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

Goal state before:
pp: 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
⊢ integral μ f ≤ integral ν f

Tactic:
simp only [integral_eq]

Goal state after:
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 ∈ f.range, μ.real (⇑f ⁻¹' {x}) • x ≤ ∑ x ∈ f.range, ν.real (⇑f ⁻¹' {x}) • 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 := ⋯
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 ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
n : ℕ
⊢ ↑↑↑((x n).toLp ⋯) =ᵐ[μ] ⇑(x n)

Tactic:
Lp.simpleFunc.toLp_eq_toLp

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
⊢ ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᵐ[μ] ⇑(x n)
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 g : α → G
hf : Integrable f μ
hg : Integrable g μ
⊢ dist (∫ (a : α), f a ∂μ) (∫ (a : α), g a ∂μ) ≤ (∫⁻ (a : α), edist (f a) (g a) ∂μ).toReal

Tactic:
dist_eq_norm,

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
f g : α → G
hf : Integrable f μ
hg : Integrable g μ
⊢ ‖∫ (a : α), f a ∂μ - ∫ (a : α), g a ∂μ‖ ≤ (∫⁻ (a : α), edist (f a) (g a) ∂μ).toReal
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 g : α → G
hf : Integrable f μ
hg : Integrable g μ
⊢ ENNReal.ofReal (dist (∫ (a : α), f a ∂μ) (∫ (a : α), g a ∂μ)) ≤ ∫⁻ (a : α), edist (f a) (g a) ∂μ

Tactic:
exact ENNReal.ofReal_le_of_le_toReal (dist_integral_le_lintegral_edist hf hg)

Goal state after:
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
⊢ x ∉ {0}

Tactic:
rw [Finset.mem_singleton]; exact hx0

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
s : Set E
inst✝ : SeparableSpace ↑s
hfi : Integrable f μ
hfm : Measurable f
hs : ∀ᵐ (x : α) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
y₀ : E
h₀ : y₀ ∈ s
h₀i : Integrable (fun x ↦ y₀) μ
⊢ Tendsto (fun n ↦ SimpleFunc.integral μ (approxOn f hfm s y₀ h₀ n)) atTop (𝓝 (∫ (x : α), f x ∂μ))

Tactic:
have hfi' := SimpleFunc.integrable_approxOn hfm hfi h₀ h₀i
  simp only [SimpleFunc.integral_eq_integral _ (hfi' _), integral, L1.integral]
  exact tendsto_setToFun_approxOn_of_measurable (dominatedFinMeasAdditive_weightedSMul μ)
    hfi hfm hs h₀ h₀i

Goal state after:
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v₁ v₂ : V
x : P
⊢ ‖(v₁ +ᵥ x) -ᵥ (v₂ +ᵥ x)‖ = dist v₁ v₂

Tactic:
dist_eq_norm,

Goal state after:
V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v₁ v₂ : V
x : P
⊢ ‖(v₁ +ᵥ x) -ᵥ (v₂ +ᵥ x)‖ = ‖v₁ - v₂‖
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
x y z : P
⊢ dist (x -ᵥ y) (x -ᵥ z) = dist y z

Tactic:
dist_eq_norm,

Goal state after:
V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
x y z : P
⊢ ‖x -ᵥ y - (x -ᵥ z)‖ = dist y z
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄

Tactic:
dist_eq_norm,

Goal state after:
V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ ‖p₁ -ᵥ p₂ - (p₃ -ᵥ p₄)‖ ≤ dist p₁ p₃ + dist p₂ p₄
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
⊢ ∀ ⦃a_1 b_1 : ℝ⦄,
    0 ≤ a_1 →
      0 ≤ b_1 →
        a_1 + b_1 = 1 →
          (fun b ↦ ∫ (x : α), f x b ∂μ) (a_1 • a + b_1 • b) ≤
            a_1 • (fun b ↦ ∫ (x : α), f x b ∂μ) a + b_1 • (fun b ↦ ∫ (x : α), f x b ∂μ) b

Tactic:
p

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p : ℝ
⊢ ∀ ⦃b_1 : ℝ⦄,
    0 ≤ p →
      0 ≤ b_1 →
        p + b_1 = 1 →
          (fun b ↦ ∫ (x : α), f x b ∂μ) (p • a + b_1 • b) ≤
            p • (fun b ↦ ∫ (x : α), f x b ∂μ) a + b_1 • (fun b ↦ ∫ (x : α), f x b ∂μ) b
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: case inr
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x : F
s : Set F
hx : x ∈ s
h₀ : infDist x sᶜ ≠ 0
⊢ closedBall x (infDist x sᶜ) ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s

Tactic:
· rw [← closure_ball x h₀]
    exact closure_mono ball_infDist_compl_subset

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → G
C : ℝ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ ∫ (x : α), C ∂μ = C * μ.real univ

Tactic:
integral_const,

Goal state after:
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → G
C : ℝ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ μ.real univ • C = C * μ.real univ
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
φ₀ : E →L[𝕜] F
hφ₀ : φ₀ ∈ {L | Injective ⇑L}
K : ℝ≥0
K_pos : K > 0
H : AntilipschitzWith K ⇑↑φ₀
⊢ ∀ᶠ (y : E →L[𝕜] F) in 𝓝 φ₀, y ∈ {L | Injective ⇑L}

Tactic:
have : ∀ᶠ φ in 𝓝 φ₀, ‖φ - φ₀‖₊ < K⁻¹ := eventually_nnnorm_sub_lt _ <| inv_pos_of_pos K_pos

Goal state after:
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
φ₀ : E →L[𝕜] F
hφ₀ : φ₀ ∈ {L | Injective ⇑L}
K : ℝ≥0
K_pos : K > 0
H : AntilipschitzWith K ⇑↑φ₀
this : ∀ᶠ (φ : E →L[𝕜] F) in 𝓝 φ₀, ‖φ - φ₀‖₊ < K⁻¹
⊢ ∀ᶠ (y : E →L[𝕜] F) in 𝓝 φ₀, y ∈ {L | Injective ⇑L}
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
⊢ dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C

Tactic:
fconstructor
  · intro w x
    exact lt_of_le_of_lt (dist_coe_le_dist x) w
  · by_cases h : Nonempty α
    · exact dist_lt_of_nonempty_compact
    · rintro -
      convert! C0
      apply le_antisymm _ dist_nonneg'
      rw [dist_eq]
      exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempty.intro x))⟩

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ).toReal ≤
    (∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ).toReal ^ (1 / p) *
      (∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ).toReal ^ (1 / q)

Tactic:
ENNReal.toReal_rpow,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ).toReal ≤
    ((∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ) ^ (1 / p)).toReal *
      (∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ).toReal ^ (1 / q)
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ).toReal ≤
    ((∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ) ^ (1 / p)).toReal *
      (∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ).toReal ^ (1 / q)

Tactic:
ENNReal.toReal_rpow,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
⊢ (∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ).toReal ≤
    ((∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ) ^ (1 / p)).toReal *
      ((∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ) ^ (1 / q)).toReal
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ

Tactic:
refine lintegral_congr fun x => ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
x : α
⊢ ENNReal.ofReal (‖f x‖ ^ p) = ‖f x‖ₑ ^ p
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ

Tactic:
refine lintegral_congr fun x => ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ
x : α
⊢ ENNReal.ofReal (‖g x‖ ^ q) = ‖g x‖ₑ ^ q
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
f g : C(α, β)
C : ℝ
inst✝ : Nonempty α
⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C

Tactic:
by
  simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
    mkOfCompact_apply]

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε
⊢ ∀ i ≥ N, thickenedIndicatorAux (δseq i) E x = 0

Tactic:
intro n

Goal state after:
case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε
n : ℕ
⊢ n ≥ N → thickenedIndicatorAux (δseq n) E x = 0
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case inr
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
this :
  ∀ (x y : α),
    infEDist x E ≤ infEDist y E →
      edist ((thickenedIndicator δ_pos E) x) ((thickenedIndicator δ_pos E) y) ≤ ↑δ.toNNReal⁻¹ * edist x y
h : ¬infEDist x E ≤ infEDist y E
⊢ edist ((thickenedIndicator δ_pos E) x) ((thickenedIndicator δ_pos E) y) ≤ ↑δ.toNNReal⁻¹ * edist x y

pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ edist ((thickenedIndicator δ_pos E) x) ((thickenedIndicator δ_pos E) y) ≤ ↑δ.toNNReal⁻¹ * edist x y

Tactic:
· specialize this y x (le_of_not_ge h)
    rwa [edist_comm, edist_comm x]

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ edist ((thickenedIndicator δ_pos E) x) ((thickenedIndicator δ_pos E) y) ≤ ↑δ.toNNReal⁻¹ * edist x y
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case e'_5
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
C : ℝ
hT : DominatedFinMeasAdditive μ T C
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) atTop (𝓝 (f a))
hF : CompleteSpace F
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n) μ
f_int : Integrable f μ
this :
  Tendsto (fun n ↦ (L1.setToL1 hT) (Integrable.toL1 (fs n) ⋯)) atTop (𝓝 ((L1.setToL1 hT) (Integrable.toL1 f f_int)))
⊢ setToFun μ T hT f = (L1.setToL1 hT) (Integrable.toL1 f f_int)

Tactic:
· exact setToFun_eq hT f_int

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

Goal state before:
pp: case hν
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s \ t))

Tactic:
· exact Integrable.mono_measure hfs (Measure.restrict_mono sdiff_subset le_rfl)

Goal state after: