mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
p q : ℝ
hpq : p.HolderConjugate q
f g : ι → ℝ≥0
A : ℝ≥0
hf_sum : HasSum (fun i ↦ ↑(f i) ^ p) (↑A ^ p)
B : ℝ≥0
hg_sum : HasSum (fun i ↦ ↑(g i) ^ q) (↑B ^ q)
⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A * ↑B ∧ HasSum (fun i ↦ ↑(f i) * ↑(g i)) C

Tactic:
norm_cast at hf_sum hg_sum

Goal state after:
ι : Type u
p q : ℝ
hpq : p.HolderConjugate q
f g : ι → ℝ≥0
A B : ℝ≥0
hf_sum : HasSum (fun a ↦ f a ^ p) (A ^ p)
hg_sum : HasSum (fun a ↦ g a ^ q) (B ^ q)
⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A * ↑B ∧ HasSum (fun i ↦ ↑(f i) * ↑(g i)) C
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✝ : NormedAddCommGroup E
f : α → E
hf : MemLp f p μ
⊢ edist (MemLp.toLp f hf) 0 = eLpNorm f p μ

Tactic:
simpa using edist_toLp_toLp f 0 hf .zero

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < B N
n : ℕ
⊢ eLpNorm' (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i

Tactic:
let f_norm_diff i x := ‖f (i + 1) x - f i x‖

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < B N
n : ℕ
f_norm_diff : ℕ → α → ℝ := fun i x ↦ ‖f (i + 1) x - f i x‖
⊢ eLpNorm' (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : Fact (1 ≤ p)
⊢ MetricSpace.toEMetricSpace.toEDist = instEDist

Tactic:
by
  with_reducible_and_instances rfl

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : Fact (1 ≤ p)
⊢ SeminormedAddGroup.toNNNorm = instNNNorm

Tactic:
by
  with_reducible_and_instances rfl

Goal state after:
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: case inl.inl
α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
a₁ a₂ : α
f_eq : f a₁ = f a₂
⊢ Sum.inl a₁ = Sum.inl a₂

Tactic:
f.injective f_eq

Goal state after:
case inl.inl
α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
a₁ a₂ : α
f_eq : f a₁ = f a₂
⊢ Sum.inl a₂ = Sum.inl a₂
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: case pos.refine_2
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → essSup (fun x ↦ ‖(f n - f m_1) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
⊢ Tendsto (fun n ↦ (B n).toReal) atTop (𝓝 0)

Tactic:
rw [← ENNReal.toReal_zero]
      exact
        Tendsto.comp (g := ENNReal.toReal) (ENNReal.tendsto_toReal ENNReal.zero_ne_top)
          (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)

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

Goal state before:
pp: case inl
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
⊢ IsClosed ↑l.ker ∨ Dense ↑l.ker

pp: case inr
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
⊢ IsClosed ↑(ker 0) ∨ Dense ↑(ker 0)

Tactic:
· exact (LinearMap.ker l).isClosed_or_dense_of_isCoatom (LinearMap.isCoatom_ker_of_surjective hl)

Goal state after:
case inr
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
⊢ IsClosed ↑(ker 0) ∨ Dense ↑(ker 0)
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ IsTopCompl p q ↔
    Continuous[QuotientModule.Quotient.topologicalSpace p, instTopologicalSpaceSubtype] ⇑(p.quotientEquivOfIsCompl q h)

Tactic:
rw [p.isQuotientMap_mkQL.continuous_iff, isTopCompl_comm]

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsCompl p q
⊢ IsTopCompl q p ↔ Continuous[inst✝³, instTopologicalSpaceSubtype] (⇑(p.quotientEquivOfIsCompl q h) ∘ ⇑p.mkQL)
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:

Tactic:
μ.real s₁ ≤ μ.real (s₂ ∪ s₁) := measureReal_le_measureReal_union_right h
    _ = μ.real (s₂ ∪ s₁ \ s₂) := congr_arg μ.real union_sdiff_self.symm
    _ ≤ μ.real s₂ + μ.real (s₁ \ s₂) := measureReal_union_le _ _

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
ι : Type u_3
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ a ∈ s, μ (t a) ≤ μ univ

Tactic:
exact sum_measure_le_measure_univ (fun i mi ↦ (h i mi).nullMeasurableSet) H.aedisjoint

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ eLpNorm ((s.indicator fun x ↦ c) - t.indicator fun x ↦ c) p μ = ‖MemLp.toLp ((s ∆ t).indicator fun x ↦ c) ⋯‖ₑ

Tactic:
eLpNorm_indicator_sub_indicator,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ eLpNorm ((s ∆ t).indicator fun x ↦ c) p μ = ‖MemLp.toLp ((s ∆ t).indicator fun x ↦ c) ⋯‖ₑ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
p : ℝ≥0∞
hf : MemLp f p μ
⊢ Integrable (fun x ↦ ‖f x‖ ^ p.toReal) μ

Tactic:
by_cases h_zero : p = 0
  · simp [h_zero]
  by_cases h_top : p = ∞
  · simp [h_top]
  exact hf.integrable_norm_rpow h_zero h_top

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
ν : Measure δ
g : δ → ε
f : α → δ
hf : MeasurePreserving f μ ν
hg : AEStronglyMeasurable g ν
⊢ Integrable (g ∘ f) μ ↔ Integrable g ν

Tactic:
← hf.map_eq

Goal state after:
α : Type u_1
δ : Type u_4
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
ν : Measure δ
g : δ → ε
f : α → δ
hf : MeasurePreserving f μ ν
hg : AEStronglyMeasurable g (Measure.map f μ)
⊢ Integrable (g ∘ f) μ ↔ Integrable g (Measure.map f μ)
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h_meas : AEStronglyMeasurable f μ
hf : f ≤ᵐ[μ] 0
hg : g ≤ᵐ[μ] 0
⊢ Integrable (-(f + g)) μ ↔ Integrable (-f) μ ∧ Integrable g μ

Tactic:
← integrable_neg_iff (f := g),

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h_meas : AEStronglyMeasurable f μ
hf : f ≤ᵐ[μ] 0
hg : g ≤ᵐ[μ] 0
⊢ Integrable (-(f + g)) μ ↔ Integrable (-f) μ ∧ Integrable (-g) μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
⊢ Integrable (↑↑(indicatorConstLp p hs hμs c)) μ

Tactic:
rw [integrable_congr indicatorConstLp_coeFn, integrable_indicator_iff hs, IntegrableOn,
    integrable_const_iff, isFiniteMeasure_restrict]
  exact .inr hμs

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
R : Type u_8
inst✝² : NormedRing R
inst✝¹ : Module R β
inst✝ : IsBoundedSMul R β
f : α → β
hf : MemLp f 1 μ
g : α → R
g_aestronglyMeasurable : AEStronglyMeasurable g μ
ess_sup_g : essSup (fun x ↦ ‖g x‖ₑ) μ ≠ ∞
⊢ eLpNorm (fun x ↦ g x • f x) 1 μ < ∞

Tactic:
have hg' : eLpNorm g ∞ μ ≠ ∞ := by rwa [eLpNorm_exponent_top]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
R : Type u_8
inst✝² : NormedRing R
inst✝¹ : Module R β
inst✝ : IsBoundedSMul R β
f : α → β
hf : MemLp f 1 μ
g : α → R
g_aestronglyMeasurable : AEStronglyMeasurable g μ
ess_sup_g : essSup (fun x ↦ ‖g x‖ₑ) μ ≠ ∞
hg' : eLpNorm g ∞ μ ≠ ∞
⊢ eLpNorm (fun x ↦ g x • f x) 1 μ < ∞
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:

Tactic:
eLpNorm (fun x : α => g x • f x) 1 μ ≤ _ := by
      simpa using! MeasureTheory.eLpNorm_smul_le_mul_eLpNorm hf.1 g_aestronglyMeasurable
        (p := ∞) (q := 1)
    _ < ∞ := ENNReal.mul_lt_top hg'.lt_top hf.2

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
𝕜 : Type u_8
inst✝² : NormedRing 𝕜
inst✝¹ : MulActionWithZero 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
f : α → 𝕜
hf : MemLp f 1 μ
g : α → β
g_aestronglyMeasurable : AEStronglyMeasurable g μ
ess_sup_g : essSup (fun x ↦ ‖g x‖ₑ) μ ≠ ∞
⊢ eLpNorm (fun x ↦ f x • g x) 1 μ < ∞

Tactic:
have hg' : eLpNorm g ∞ μ ≠ ∞ := by rwa [eLpNorm_exponent_top]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
𝕜 : Type u_8
inst✝² : NormedRing 𝕜
inst✝¹ : MulActionWithZero 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
f : α → 𝕜
hf : MemLp f 1 μ
g : α → β
g_aestronglyMeasurable : AEStronglyMeasurable g μ
ess_sup_g : essSup (fun x ↦ ‖g x‖ₑ) μ ≠ ∞
hg' : eLpNorm g ∞ μ ≠ ∞
⊢ eLpNorm (fun x ↦ f x • g x) 1 μ < ∞
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 : ContinuousMultilinearMap 𝕜 E G
⊢ IsClosed {c | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i, ‖m i‖}

Tactic:
simp only [Set.ofPred_and, Set.ofPred_forall]

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 : ContinuousMultilinearMap 𝕜 E G
⊢ IsClosed ({a | 0 ≤ a} ∩ ⋂ i, {x | ‖f i‖ ≤ x * ∏ i_1, ‖i i_1‖})
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 g : ContinuousMultilinearMap 𝕜 E G
x : (i : ι) → E i
⊢ ‖(f + g) x‖ ≤ (‖f‖ + ‖g‖) * ∏ i, ‖x i‖

Tactic:
rw [add_mul]
    exact norm_add_le_of_le (le_opNorm _ _) (le_opNorm _ _)

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

Goal state before:
pp: case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : a ≤ b
⊢ IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ

Tactic:
← Ioc_union_left hab,

Goal state after:
case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : a ≤ b
⊢ IntegrableOn f (Ioc a b ∪ {a}) μ ↔ IntegrableOn f (Ioc a b) μ
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
c : E
hc : c = 0
⊢ IsClosedMap fun x ↦ x • c

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
c : E
hc : ¬c = 0
⊢ IsClosedMap fun x ↦ x • c

Tactic:
· simp_rw [hc, smul_zero]
    exact isClosedMap_const

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
c : E
hc : ¬c = 0
⊢ IsClosedMap fun x ↦ x • c
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
a : 𝕜
f₀ : E i →L[𝕜] E₁ i
x : (i : ι) → E i
⊢ (g.compContinuousLinearMap (update f i (a • f₀))) x = (a • g.compContinuousLinearMap (update f i f₀)) x

Tactic:
simp only [compContinuousLinearMap_apply, smul_apply]

Goal state after:
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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
a : 𝕜
f₀ : E i →L[𝕜] E₁ i
x : (i : ι) → E i
⊢ (g fun i_1 ↦ (update f i (a • f₀) i_1) (x i_1)) = a • g fun i_1 ↦ (update f i f₀ i_1) (x i_1)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝ : PseudoMetrizableSpace ε
x✝ : IntegrableAtFilter f (cocompact X) μ ∧ LocallyIntegrable f μ
s : Set X
hsc : s ∈ cocompact X
hs : IntegrableOn f s μ
hloc : LocallyIntegrable f μ
t : Set X
htc : IsCompact t
ht : sᶜ ⊆ t
⊢ Integrable f μ

Tactic:
← integrableOn_univ,

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝ : PseudoMetrizableSpace ε
x✝ : IntegrableAtFilter f (cocompact X) μ ∧ LocallyIntegrable f μ
s : Set X
hsc : s ∈ cocompact X
hs : IntegrableOn f s μ
hloc : LocallyIntegrable f μ
t : Set X
htc : IsCompact t
ht : sᶜ ⊆ t
⊢ IntegrableOn f univ μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

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

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

Tactic:
· intro c s hs hf
    by_cases hc : c = 0
    · convert! indicatorConst 0 MeasurableSet.empty (by simp) using 1
      ext1
      simp [hc]
    exact indicatorConst c hs
      (SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs hf)

Goal state after:
case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
⊢ ∀ ⦃f g : α →ₛ E⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (∀ (hf : MemLp (⇑f) p μ), P (f.toLp hf)) →
        (∀ (hf : MemLp (⇑g) p μ), P (g.toLp hf)) → ∀ (hf : MemLp (⇑(f + g)) p μ), P ((f + g).toLp hf)
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:
rw [h (f ⁻¹' {x}) (SimpleFunc.measurableSet_fiber _ _)
        (SimpleFunc.measure_preimage_lt_top_of_integrable _ hf hx0)]

Goal state after:
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:
· rw [h (f ⁻¹' {x}) (SimpleFunc.measurableSet_fiber _ _)
        (SimpleFunc.measure_preimage_lt_top_of_integrable _ hf hx0)]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
f : ↥(α →₁ₛ[μ] E)
⊢ ‖{ toFun := integral, map_add' := ⋯, map_smul' := ⋯ } f‖ ≤ 1 * ‖f‖

Tactic:
simpa [one_mul] using norm_integral_le_norm f

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : ↥(Lp E 1 μ)
⊢ integralCLM (f + g) = integralCLM f + integralCLM g

Tactic:
exact map_add integralCLM f g

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

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ¬(AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ)
⊢ 0 = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal

Tactic:
hfm,

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ¬(True ∧ HasFiniteIntegral f μ)
⊢ 0 = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace E
f : α →ₛ E
hfi : Integrable (⇑f) μ
⊢ integral μ f = ∫ (x : α), f x ∂μ

Tactic:
by
  rw [MeasureTheory.integral_eq f hfi, ← L1.SimpleFunc.toLp_one_eq_toL1,
    L1.SimpleFunc.integral_L1_eq_integral, L1.SimpleFunc.integral_eq_integral]
  exact SimpleFunc.integral_congr hfi (Lp.simpleFunc.toSimpleFunc_toLp _ _).symm

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ g, eLpNorm (f - g) p μ ≤ ε ∧ P g

Tactic:
rcases eq_or_ne p 0 with (rfl | hp_pos)

Goal state after:
case inl
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
P : (α → E) → Prop
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
ε : ℝ≥0∞
hε : ε ≠ 0
hp_ne_top : 0 ≠ ∞
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) 0 μ ≤ ε ∧ P g
hf : MemLp f 0 μ
⊢ ∃ g, eLpNorm (f - g) 0 μ ≤ ε ∧ P g

case inr
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hp_pos : p ≠ 0
⊢ ∃ g, eLpNorm (f - g) p μ ≤ ε ∧ P g