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

Goal state before:
pp: α : Type u_1
m✝ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MemLp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ∞ →
          (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
            ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n ↦ ENNReal.ofReal (B n)
hB1_has : HasSum B1 (ENNReal.ofReal M)
hB1 : ∑' (i : ℕ), B1 i < ∞
f1 : ℕ → α → E := fun n ↦ ↑↑(f n)
N n m : ℕ
hf : (eLpNorm (↑↑(f n) - ↑↑(f m)) p μ).toReal < B N
hn : N ≤ n
hm : N ≤ m
⊢ eLpNorm (f1 n - f1 m) p μ < B1 N

Tactic:
dsimp only [f1]

Goal state after:
α : Type u_1
m✝ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MemLp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ∞ →
          (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
            ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n ↦ ENNReal.ofReal (B n)
hB1_has : HasSum B1 (ENNReal.ofReal M)
hB1 : ∑' (i : ℕ), B1 i < ∞
f1 : ℕ → α → E := fun n ↦ ↑↑(f n)
N n m : ℕ
hf : (eLpNorm (↑↑(f n) - ↑↑(f m)) p μ).toReal < B N
hn : N ≤ n
hm : N ≤ m
⊢ eLpNorm (↑↑(f n) - ↑↑(f m)) p μ < B1 N
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L L' : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ ↑↑((L + L').compLp f) =ᵐ[μ] ↑↑(L.compLp f) + ↑↑(L'.compLp f)

Tactic:
coeFn_compLp',

Goal state after:
α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L L' : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ (fun a ↦ (L + L') (↑↑f a)) =ᵐ[μ] ↑↑(L.compLp f) + ↑↑(L'.compLp f)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L L' : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ (fun a ↦ (L + L') (↑↑f a)) =ᵐ[μ] ↑↑(L.compLp f) + ↑↑(L'.compLp f)

Tactic:
coeFn_compLp',

Goal state after:
α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L L' : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ (fun a ↦ (L + L') (↑↑f a)) =ᵐ[μ] (fun a ↦ L (↑↑f a)) + ↑↑(L'.compLp f)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜'
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝⁴ : RingHomIsometric σ
𝕜'' : Type u_8
inst✝³ : NormedRing 𝕜''
inst✝² : Module 𝕜'' F
inst✝¹ : IsBoundedSMul 𝕜'' F
inst✝ : SMulCommClass 𝕜' 𝕜'' F
c : 𝕜''
L : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ ↑↑((c • L).compLp f) =ᵐ[μ] c • ↑↑(L.compLp f)

Tactic:
coeFn_compLp',

Goal state after:
α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜'
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝⁴ : RingHomIsometric σ
𝕜'' : Type u_8
inst✝³ : NormedRing 𝕜''
inst✝² : Module 𝕜'' F
inst✝¹ : IsBoundedSMul 𝕜'' F
inst✝ : SMulCommClass 𝕜' 𝕜'' F
c : 𝕜''
L : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ (fun a ↦ (c • L) (↑↑f a)) =ᵐ[μ] c • ↑↑(L.compLp f)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
ν : Measure α
c : ℝ≥0∞
hc : c ≠ ∞
h : μ ≤ c • ν
inst✝ : Fact (1 ≤ p)
f : ↥(Lp E p ν)
⊢ ‖(LpToLpOfMeasureLeSMulₗ hc h) f‖ₑ.toReal ≤ (c ^ (1 / p).toReal).toReal * ‖f‖ₑ.toReal

Tactic:
← ENNReal.toReal_mul

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
ν : Measure α
c : ℝ≥0∞
hc : c ≠ ∞
h : μ ≤ c • ν
inst✝ : Fact (1 ≤ p)
f : ↥(Lp E p ν)
⊢ ‖(LpToLpOfMeasureLeSMulₗ hc h) f‖ₑ.toReal ≤ (c ^ (1 / p).toReal * ‖f‖ₑ).toReal
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
f : ↥(Lp ℝ p μ)
⊢ ∀ᵐ (a : α) ∂μ, ↑↑(negPart f) a = max (-↑↑f a) 0

Tactic:
rw [negPart]

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
f : ↥(Lp ℝ p μ)
⊢ ∀ᵐ (a : α) ∂μ, ↑↑(posPart (-f)) a = max (-↑↑f a) 0
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
ε : ℝ≥0∞
hε : ε > 0
⊢ ∃ N, ∀ (n : ℕ), N ≤ n → B n ≤ ε

Tactic:
exact (ENNReal.tendsto_atTop_zero.mp (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)) ε hε

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
hfV : ∃ I, I.Finite ∧ ∃ t, (∀ (i : ι), t i ∈ 𝓝 (0 i)) ∧ I.pi t ⊆ ⇑f ⁻¹' V
⊢ ∃ I t, (∀ (i : ι), t i ∈ 𝓝 0) ∧ (↑I).pi t ⊆ ⇑f ⁻¹' V

Tactic:
exists_finite_iff_finset

Goal state after:
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
hfV : ∃ s t, (∀ (i : ι), t i ∈ 𝓝 (0 i)) ∧ (↑s).pi t ⊆ ⇑f ⁻¹' V
⊢ ∃ I t, (∀ (i : ι), t i ∈ 𝓝 0) ∧ (↑I).pi t ⊆ ⇑f ⁻¹' V
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →SL[σ₁₂] F
inst✝ : RingHomIsometric σ₁₂
hn : ‖f‖ = 0
x : E
⊢ ‖f‖ * ‖x‖ = 0

Tactic:
rw [hn, zero_mul]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁸ : Ring R
E : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module R E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : TopologicalSpace F
inst✝² : AddCommGroup F
inst✝¹ : Module R F
inst✝ : ContinuousAdd F
p q : Submodule R E
h : IsTopCompl p q
⊢ ofIsTopCompl h 0 0 = 0

Tactic:
by
  ext; simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁸ : Ring R
E : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module R E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : TopologicalSpace F
inst✝² : AddCommGroup F
inst✝¹ : Module R F
inst✝ : ContinuousAdd F
p q : Submodule R E
h : IsTopCompl p q
φ₁ φ₂ : ↥p →L[R] F
ψ₁ ψ₂ : ↥q →L[R] F
⊢ ofIsTopCompl h (φ₁ + φ₂) (ψ₁ + ψ₂) = ofIsTopCompl h φ₁ ψ₁ + ofIsTopCompl h φ₂ ψ₂

Tactic:
by
  ext; simp

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Fintype ι
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖1‖

Tactic:
apply le_antisymm
  · apply opNorm_le_bound <;> simp
  · convert! ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => 1
    simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: case inr
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
inst✝ : IsCompletelyPseudoMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
this✝³ : MeasurableSpace E := borel E
this✝² : BorelSpace E
this✝¹ : UpgradedIsCompletelyPseudoMetrizableSpace E := upgradeIsCompletelyPseudoMetrizable E
s : Set E := closure[inst✝³] (⋃ i, range (f i))
this✝ : SecondCountableTopology ↑s
this : IsCompletelyPseudoMetrizableSpace ↑s
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}

Tactic:
let g i x : s := ⟨f i x, subset_closure <| mem_iUnion.2 ⟨i, ⟨x, rfl⟩⟩⟩

Goal state after:
case inr
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
inst✝ : IsCompletelyPseudoMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
this✝³ : MeasurableSpace E := borel E
this✝² : BorelSpace E
this✝¹ : UpgradedIsCompletelyPseudoMetrizableSpace E := upgradeIsCompletelyPseudoMetrizable E
s : Set E := closure[inst✝³] (⋃ i, range (f i))
this✝ : SecondCountableTopology ↑s
this : IsCompletelyPseudoMetrizableSpace ↑s
g : ι → X → ↑s := fun i x ↦ ⟨f i x, ⋯⟩
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case refine_2.inr
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
hz : 0 < ‖z‖₊
⊢ ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ ‖f.smulRight z‖₊ / ‖z‖₊ * ∏ i, ‖m i‖₊

Tactic:
intro m

Goal state after:
case refine_2.inr
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
hz : 0 < ‖z‖₊
m : (i : ι) → E i
⊢ ‖f m‖₊ ≤ ‖f.smulRight z‖₊ / ‖z‖₊ * ∏ i, ‖m i‖₊
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ s = ⋃ n, u n ∩ s

Tactic:
rw [← iUnion_inter]

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
⊢ s = (⋃ i, u i) ∩ s
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : LocallyCompactSpace X
hs : IsLocallyClosed s
⊢ LocallyIntegrableOn f s μ ↔ ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ

Tactic:
refine ⟨fun hf k hk ↦ hf.integrableOn_compact_subset hk, fun hf x hx ↦ ?_⟩

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : LocallyCompactSpace X
hs : IsLocallyClosed s
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
x : X
hx : x ∈ s
⊢ IntegrableAtFilter f (𝓝[s] x) μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : X → ε
μ : Measure X
⊢ LocallyIntegrableOn f univ μ ↔ LocallyIntegrable f μ

Tactic:
by
  simp only [LocallyIntegrableOn, nhdsWithin_univ, mem_univ, true_imp_iff]; rfl

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
⊢ ∏ i ∈ s.toFinsetᶜ, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖ ≤ ∏ i, ‖x i‖

pp: case h₂
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
⊢ ∏ i ∈ s.toFinset, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖ ≤ ∏ i, ‖m i‖

Tactic:
· apply le_of_eq
      have : ∀ x, x ∈ s.toFinsetᶜ ↔ (fun x ↦ x ∉ s) x := by simp
      rw [prod_subtype _ this]
      congr with i
      simp [i.2]

Goal state after:
case h₂
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
⊢ ∏ i ∈ s.toFinset, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖ ≤ ∏ i, ‖m i‖
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
g : β →ₛ E
hg : eLpNorm (f' - ⇑g) p μ < ε
g_mem : MemLp (⇑g) p μ
⊢ eLpNorm (f - ⇑g) p μ < ε

Tactic:
suffices eLpNorm (f - ⇑g) p μ = eLpNorm (f' - ⇑g) p μ by rwa [this]

Goal state after:
β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
g : β →ₛ E
hg : eLpNorm (f' - ⇑g) p μ < ε
g_mem : MemLp (⇑g) p μ
⊢ eLpNorm (f - ⇑g) p μ = eLpNorm (f' - ⇑g) p μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
⊢ ∀ {a b : ↥(Lp E p μ)},
    a ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f} → b ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f} → a + b ∈ {f | ∃ s, AEEqFun.mk ⇑s ⋯ = ↑f}

Tactic:
rintro f g ⟨s, hs⟩ ⟨t, ht⟩
    use s + t
    simp only [← hs, ← ht, AEEqFun.mk_add_mk, AddSubgroup.coe_add,
      SimpleFunc.coe_add]

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

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(Lp E p μ)
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hs : AEEqFun.mk ⇑s ⋯ = ↑f
⊢ AEEqFun.mk ⇑(k • s) ⋯ = ↑(k • ↑⟨f, ⋯⟩)

Tactic:
apply Eq.trans (AEEqFun.smul_mk k s s.aestronglyMeasurable).symm _

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(Lp E p μ)
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hs : AEEqFun.mk ⇑s ⋯ = ↑f
⊢ k • AEEqFun.mk ⇑s ⋯ = ↑(k • ↑⟨f, ⋯⟩)
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case refine_1.inl
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
f : E →SL[σ₁₂] 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 : 0 ≤ N
⊢ 0 ≤ ↑N * ‖f‖

Tactic:
| inl hN =>

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α →ₛ ℝ
⊢ map norm f.negPart = f.negPart

Tactic:
rw [negPart]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α →ₛ ℝ
⊢ map norm (-f).posPart = (-f).posPart
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
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)

Tactic:
have hx_nonneg : ∀ n, 0 ≤ x n := by
    intro n a
    change x n a ∈ { y : G | 0 ≤ y }
    have A : (range (g : α → G) ∪ {0} : Set G) ∩ { y | 0 ≤ y } ⊆ { y | 0 ≤ y } :=
      inter_subset_right
    apply A
    exact SimpleFunc.approxOn_mem g_meas _ n a

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
⊢ ∃ 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: 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
⊢ 0 ≤ integral μ f

Tactic:
rw [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
f : α →ₛ F
hf : 0 ≤ᵐ[μ] ⇑f
⊢ 0 ≤ ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) • 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
⊢ ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal - (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal

Tactic:
by
  let f₁ := hf.toL1 f
  -- Go to the `L¹` space
  have eq₁ : ENNReal.toReal (∫⁻ a, ENNReal.ofReal (f a) ∂μ) = ‖Lp.posPart f₁‖ := by
    rw [L1.norm_def]
    congr 1
    apply lintegral_congr_ae
    filter_upwards [Lp.coeFn_posPart f₁, hf.coeFn_toL1] with _ h₁ h₂
    rw [h₁, h₂, ENNReal.ofReal]
    congr 1
    apply NNReal.eq
    rw [Real.nnnorm_of_nonneg (le_max_right _ _)]
    rw [Real.coe_toNNReal', NNReal.coe_mk]
  -- Go to the `L¹` space
  have eq₂ : ENNReal.toReal (∫⁻ a, ENNReal.ofReal (-f a) ∂μ) = ‖Lp.negPart f₁‖ := by
    rw [L1.norm_def]
    congr 1
    apply lintegral_congr_ae
    filter_upwards [Lp.coeFn_negPart f₁, hf.coeFn_toL1] with _ h₁ h₂
    rw [h₁, h₂, ENNReal.ofReal]
    congr 1
    apply NNReal.eq
    simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg]
    rw [Real.norm_of_nonpos (min_le_right _ _), ← max_neg_neg, neg_zero]
  rw [eq₁, eq₂, integral, dif_pos, dif_pos]
  exact L1.integral_eq_norm_posPart_sub _

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

Goal state before:
pp: case cons
α : Type u_1
ι : Type u_2
β : Type u_3
inst✝ : NormedField β
l : Filter α
f g : ι → α → β
i : ι
L : List ι
ihL :
  (∀ i ∈ L, f i ~[l] g i) →
    (fun x ↦ (List.map (fun x_1 ↦ f x_1 x) L).prod) ~[l] fun x ↦ (List.map (fun x_1 ↦ g x_1 x) L).prod
h : f i ~[l] g i ∧ ∀ x ∈ L, f x ~[l] g x
⊢ (fun x ↦ f i x * (List.map (fun x_1 ↦ f x_1 x) L).prod) ~[l] fun x ↦ g i x * (List.map (fun x_1 ↦ g x_1 x) L).prod

Tactic:
exact h.1.mul (ihL h.2)

Goal state after: