mathlib-initiative/mathlib-tactics huggingface dataset:
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 : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
⊢ ‖f m₁ - f m₂‖ ≤ C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖

Tactic:
have A :
    ∀ i : ι,
      ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤
        ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by
    intro i
    calc
      ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤
          ∏ j : ι, Function.update (fun _ => max ‖m₁‖ ‖m₂‖) i ‖m₁ - m₂‖ j := by
        gcongr with j
        rcases eq_or_ne j i with rfl | h
        · simp only [ite_true, Function.update_self]
          exact norm_le_pi_norm (m₁ - m₂) _
        · simp [h, -le_sup_iff, -sup_le_iff, sup_le_sup, norm_le_pi_norm]
      _ = ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) := by
        rw [prod_update_of_mem (Finset.mem_univ _)]
        simp [card_univ_sdiff]

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
A :
  ∀ (i : ι),
    (∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)
⊢ ‖f m₁ - f m₂‖ ≤ C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : CommMonoid E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : ContinuousMul E
L : SummationFilter ι
inst✝³ : L.NeBot
inst✝² : L.filter.IsCountablyGenerated
inst✝¹ : IsCompletelyPseudoMetrizableSpace E
inst✝ : Countable ι
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEStronglyMeasurable (f i) μ
⊢ AEStronglyMeasurable (fun x ↦ ∏'[L] (i : ι), f i x) μ

Tactic:
choose g hg_meas hg_eq_f using h
  use (fun x => ∏'[L] i, g i x), StronglyMeasurable.tprod hg_meas
  filter_upwards [ae_all_iff.mpr hg_eq_f] with x h_eq using tprod_congr h_eq

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α →ₘ[μ] β
⊢ MeasureTheory.Integrable (↑f) μ ↔ f ∈ Lp β 1 μ

Tactic:
← memLp_one_iff_integrable,

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α →ₘ[μ] β
⊢ MemLp (↑f) 1 μ ↔ f ∈ Lp β 1 μ
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

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

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

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

Goal state before:
pp: case h₁
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
this : ∀ (x : ι), x ∈ s.toFinsetᶜ ↔ (fun x ↦ x ∉ s) x
⊢ ∏ i ∈ s.toFinsetᶜ, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖ = ∏ i, ‖x i‖

Tactic:
prod_subtype _ this

Goal state after:
case h₁
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
this : ∀ (x : ι), x ∈ s.toFinsetᶜ ↔ (fun x ↦ x ∉ s) x
⊢ ∏ a, ‖if h : ↑a ∈ s then m (e.symm ⟨↑a, ⋯⟩) ↑a else x ⟨↑a, ⋯⟩‖ = ∏ i, ‖x i‖

𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
this : ∀ (x : ι), x ∈ s.toFinsetᶜ ↔ (fun x ↦ x ∉ s) x
⊢ Fintype { x // x ∉ s }
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: 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'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ setToSimpleFunc T f = 0

Tactic:
simp_rw [setToSimpleFunc]
  refine sum_eq_zero fun x _ => ?_
  by_cases hx0 : x = 0
  · simp [hx0]
  rw [h_zero (f ⁻¹' ({x} : Set E)) (measurableSet_fiber _ _)
      (measure_preimage_lt_top_of_integrable f hf hx0), zero_apply]

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

Goal state before:
pp: 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}) ≠ ∞
⊢ setToSimpleFunc T (map g f) = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) (g x)

Tactic:
simp only [setToSimpleFunc, range_map]

Goal state after:
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}) ≠ ∞
⊢ ∑ x ∈ Finset.image g f.range, (T (⇑(map g f) ⁻¹' {x})) x = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) (g x)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝ : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f : ↥(Lp E p μ)
hfi' : MemLp (↑↑f) p μ
this : SeparableSpace ↑(Set.range ↑↑f ∪ {0})
⊢ f = MemLp.toLp (↑↑f) hfi'

Tactic:
toLp_coeFn f (Lp.memLp f)

Goal state after:
α : Type u_1
E : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝ : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f : ↥(Lp E p μ)
hfi' : MemLp (↑↑f) p μ
this : SeparableSpace ↑(Set.range ↑↑f ∪ {0})
⊢ f = f
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}
⊢ SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})

Tactic:
apply IsSeparable.separableSpace

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 := ⋯
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
⊢ IsSeparable ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
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:
← Lp.simpleFunc.coeFn_le,

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] ↑↑↑((x n).toLp ⋯)
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' T'' : Set α → E →L[ℝ] F
h_add : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T'' s = T s + T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
this : ∀ x ∈ {x ∈ f.range | x ≠ 0}, T'' (⇑f ⁻¹' {x}) = T (⇑f ⁻¹' {x}) + T' (⇑f ⁻¹' {x})
x : E
hx : x ∈ {x ∈ f.range | x ≠ 0}
⊢ (T'' (⇑f ⁻¹' {x})) x = (T (⇑f ⁻¹' {x})) x + (T' (⇑f ⁻¹' {x})) x

Tactic:
this x hx

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' T'' : Set α → E →L[ℝ] F
h_add : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T'' s = T s + T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
this : ∀ x ∈ {x ∈ f.range | x ≠ 0}, T'' (⇑f ⁻¹' {x}) = T (⇑f ⁻¹' {x}) + T' (⇑f ⁻¹' {x})
x : E
hx : x ∈ {x ∈ f.range | x ≠ 0}
⊢ (T (⇑f ⁻¹' {x}) + T' (⇑f ⁻¹' {x})) x = (T (⇑f ⁻¹' {x})) x + (T' (⇑f ⁻¹' {x})) 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 : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : ∀ (y : E), y ≠ 0 → μ (⇑g ⁻¹' {y}) < ∞
x : E
hx_ne : x ≠ 0
⊢ μ (⇑(-g) ⁻¹' {x}) < ∞

Tactic:
rw [SimpleFunc.coe_neg, Pi.neg_def, ← Function.comp_def, preimage_comp, neg_preimage,
    Set.neg_singleton]

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

Tactic:
· exact setToSimpleFunc_const' T x

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
c : β
hu : u ~[l] const α c
⊢ Tendsto u l (𝓝 c)

Tactic:
by
  rcases em <| c = 0 with rfl | h
  · exact (tendsto_congr' <| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds
  · exact (isEquivalent_const_iff_tendsto h).mp hu

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
u v : α → β
l : Filter α
inst✝ : OrderTopology β
huv : u ~[l] v
hu : Tendsto u l atBot
⊢ Tendsto v l atBot

Tactic:
convert! tendsto_neg_atTop_atBot.comp (huv.neg.tendsto_atTop <| tendsto_neg_atBot_atTop.comp hu)
  ext
  simp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_4
inst✝² : RCLike 𝕜
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
r : ℝ
hr : r < 1
hx₀ : 0 ∉ F
h : ∀ y ∈ F, r * ‖0‖ ≤ ‖0 - y‖
⊢ False

Tactic:
simp at hx₀

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
⊢ ↑1 * ‖setToL1SCLM α E μ hT‖ = ‖setToL1SCLM α E μ hT‖

Tactic:
by rw [NNReal.coe_one, one_mul]

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

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
hs : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
ν : Measure X
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ↑(nndist (∫ (x : X), f x ∂∑ i ∈ s, μ i) (∫ (x : X), f x ∂(∑ i ∈ s, μ i + ν))) < ↑ε

Tactic:
← hν,

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
hs : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
ν : Measure X
hf : Integrable f (∑ i ∈ s, μ i + ν)
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ↑(nndist (∫ (x : X), f x ∂∑ i ∈ s, μ i) (∫ (x : X), f x ∂(∑ i ∈ s, μ i + ν))) < ↑ε
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
hs : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
ν : Measure X
hf : Integrable f (∑ i ∈ s, μ i) ∧ Integrable f ν
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ∫⁻ (x : X), ‖f x‖ₑ ∂ν < ↑ε

Tactic:
← hν,

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
ν : Measure X
hs : ∫⁻ (x : X), ‖f x‖ₑ ∂(∑ i ∈ s, μ i + ν) < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
hf : Integrable f (∑ i ∈ s, μ i) ∧ Integrable f ν
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ∫⁻ (x : X), ‖f x‖ₑ ∂ν < ↑ε
Module: Mathlib.MeasureTheory.Integral.SetToL1

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
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T hT f = 0

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
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = 0

Tactic:
· rw [setToFun_eq hT hf]; exact L1.setToL1_zero_left' hT h_zero _

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
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ (L1.setToL1 hT) (indicatorConstLp 1 hs hμs x) = (T s) x

Tactic:
exact L1.setToL1_indicatorConstLp hT hs hμs x

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

Goal state before:
pp: G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
α : Type u_6
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → G
⊢ ∫ (x : ↑s), f ↑x ∂Measure.comap Subtype.val μ = ∫ (x : α) in s, f x ∂μ

Tactic:
by
  rw [← map_comap_subtype_coe hs]
  exact ((MeasurableEmbedding.subtype_coe hs).integral_map _).symm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

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
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : StronglyMeasurable f
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
i : ℕ
⊢ ∑ x ∈ (SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).range,
      (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {x})) x =
    ∑ x ∈ ((SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).comp φ hφ).range,
      (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {x})) x

Tactic:
refine (Finset.sum_subset (SimpleFunc.range_comp_subset_range _ hφ) fun y _ hy => ?_).symm

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝¹ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : StronglyMeasurable f
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
i : ℕ
y : E
x✝ : y ∈ (SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).range
hy : y ∉ ((SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).comp φ hφ).range
⊢ (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {y})) y = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case refine_2
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
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ ≤
    (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ) ^ (1 / q)

Tactic:
· exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf.1.nnnorm.aemeasurable.coe_nnreal_ennreal
      hg.1.nnnorm.aemeasurable.coe_nnreal_ennreal

Goal state after: