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

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
⊢ ∏ i ∈ s, x ^ w i = x

Tactic:
rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]
      have : (∑ i ∈ s, w i) ≠ 0 := by
        rw [hw']
        exact one_ne_zero
      obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this
      rw [← hx i his hi]
      exact hz i his

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : ∑ i ∈ s, f i ^ p = 0
⊢ ∀ i ∈ s, f i = 0 ∨ g i = 0

Tactic:
intro i his

Goal state after:
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : ∑ i ∈ s, f i ^ p = 0
i : ι
his : i ∈ s
⊢ f i = 0 ∨ g i = 0
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : ∑ i ∈ s, f i ^ p = 0
i : ι
his : i ∈ s
⊢ f i = 0

Tactic:
sum_eq_zero_iff

Goal state after:
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : ∀ i ∈ s, f i ^ p = 0
i : ι
his : i ∈ s
⊢ f i = 0
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ ∫⁻ (a : α), ↑(f a * g a) ∂μ ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q)

Tactic:
ENNReal.coe_mul

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ ∫⁻ (a : α), ↑(f a) * ↑(g a) ∂μ ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: case refine_1
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ eLpNorm' (f n) p μ

pp: case refine_2
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ eLpNorm' (f n) p μ

pp: case refine_3
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x ↦
    (StrictMono.orderIsoOfSurjective (fun x ↦ x ^ p) h_rpow_mono h_rpow_surj) (eLpNorm' (f x) p μ)

pp: case refine_4
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x ↦
    (StrictMono.orderIsoOfSurjective (fun x ↦ x ^ p) h_rpow_mono h_rpow_surj) (eLpNorm' (f x) p μ)

Tactic:
all_goals isBoundedDefault

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
⊢ 𝔼 i ∈ s, w i * f i ≤ (𝔼 i ∈ s, w i) ^ (1 - p⁻¹) * (𝔼 i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
simp_rw [expect_eq_sum_div_card]

Goal state after:
ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
⊢ (∑ i ∈ s, w i * f i) / ↑(#s) ≤ ((∑ i ∈ s, w i) / ↑(#s)) ^ (1 - p⁻¹) * ((∑ i ∈ s, w i * f i ^ p) / ↑(#s)) ^ p⁻¹
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 g : ↥(Lp E p μ)
h : ↑↑f =ᵐ[μ] ↑↑g
⊢ f = g

Tactic:
ext
  exact h

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

Goal state before:
pp: case refine_1
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 : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖ₑ

pp: case refine_2
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 : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖ₑ

pp: case refine_3
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 : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x_1 ↦
    (StrictMono.orderIsoOfSurjective (fun x ↦ x ^ p) h_rpow_mono h_rpow_surj)
      (∑ i ∈ Finset.range (x_1 + 1), ‖f (i + 1) x - f i x‖ₑ)

pp: case refine_4
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 : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x ↦ x ^ p
h_rpow_surj : Function.Surjective fun y ↦ y ^ p
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x_1 ↦
    (StrictMono.orderIsoOfSurjective (fun x ↦ x ^ p) h_rpow_mono h_rpow_surj)
      (∑ i ∈ Finset.range (x_1 + 1), ‖f (i + 1) x - f i x‖ₑ)

Tactic:
all_goals isBoundedDefault

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
g : E → F
c✝¹ : ℝ≥0
inst✝ : NormedSpace ℝ E
ν : Measure α
c✝ : ℝ≥0∞
hc : c✝ ≠ ∞
h : μ ≤ c✝ • ν
c : ℝ
f : ↥(Lp E p ν)
this : μ ≪ ν
⊢ ↑↑(c • f) =ᵐ[ν] (RingHom.id ℝ) c • ↑↑f

Tactic:
Lp.coeFn_smul

Goal state after:
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
g : E → F
c✝¹ : ℝ≥0
inst✝ : NormedSpace ℝ E
ν : Measure α
c✝ : ℝ≥0∞
hc : c✝ ≠ ∞
h : μ ≤ c✝ • ν
c : ℝ
f : ↥(Lp E p ν)
this : μ ≪ ν
⊢ c • ↑↑f =ᵐ[ν] (RingHom.id ℝ) c • ↑↑f
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
f : α ⊕ β ↪ γ
a : α
b : β
h : (inl.trans f) a = (inr.trans f) b
⊢ False

Tactic:
have : Sum.inl a = Sum.inr b := f.injective h

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α ⊕ β ↪ γ
a : α
b : β
h : (inl.trans f) a = (inr.trans f) b
this : Sum.inl a = Sum.inr b
⊢ False
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

pp: case inl.inr
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 : α
b₂ : β
f_eq : f a₁ = g b₂
⊢ Sum.inl a₁ = Sum.inr b₂

pp: case inr.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)
b₁ : β
a₂ : α
f_eq : g b₁ = f a₂
⊢ Sum.inr b₁ = Sum.inl a₂

pp: case inr.inr
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)
b₁ b₂ : β
f_eq : g b₁ = g b₂
⊢ Sum.inr b₁ = Sum.inr b₂

Tactic:
· rw [f.injective f_eq]

Goal state after:
case inl.inr
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 : α
b₂ : β
f_eq : f a₁ = g b₂
⊢ Sum.inl a₁ = Sum.inr b₂

case inr.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)
b₁ : β
a₂ : α
f_eq : g b₁ = f a₂
⊢ Sum.inr b₁ = Sum.inl a₂

case inr.inr
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)
b₁ b₂ : β
f_eq : g b₁ = g b₂
⊢ Sum.inr b₁ = Sum.inr b₂
Module: Mathlib.Topology.Algebra.Module.Simple

Goal state before:
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:
LinearMap.ker_zero

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 ↑⊤ ∨ Dense ↑⊤
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
aux : ContinuousMultilinearMap 𝕜 E₁ F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F := ⋯
⊢ Filter.Tendsto (⇑aux) (𝓝 0) (𝓝 0)

Tactic:
hasBasis_nhds_zero.tendsto_iff hasBasis_nhds_zero

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
aux : ContinuousMultilinearMap 𝕜 E₁ F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F :=
  { toFun := fun g ↦ g.compContinuousLinearMap f, map_add' := ⋯, map_smul' := ⋯ }
⊢ ∀ (ib : Set ((i : ι) → E i) × Set F),
    IsVonNBounded 𝕜 ib.1 ∧ ib.2 ∈ 𝓝 0 →
      ∃ ia, (IsVonNBounded 𝕜 ia.1 ∧ ia.2 ∈ 𝓝 0) ∧ ∀ x ∈ {f | MapsTo (⇑f) ia.1 ia.2}, aux x ∈ {f | MapsTo (⇑f) ib.1 ib.2}
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
a✝ : α
ha : a✝ ∈ {x}
⊢ f a✝ = f x

Tactic:
simp only [mem_singleton_iff.1 ha]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
⊢ IntegrableOn f t μ

Tactic:
have A : IntegrableOn f v μ := by
    rw [IntegrableOn, hu.restrict_toMeasurable]
    · exact hu
    · intro x hx; simpa using hx.2

Goal state after:
α : Type u_1
mα : MeasurableSpace α
s t : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
hf : IntegrableOn f s μ
ht : NullMeasurableSet t μ
h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0
u : Set α := {x | x ∈ s ∧ f x ≠ 0}
hu : IntegrableOn f u μ
v : Set α := toMeasurable μ u
A : IntegrableOn f v μ
⊢ IntegrableOn f t μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
l : Filter α
inst✝ : ContinuousAdd ε'
f g : α → ε'
hg : IntegrableAtFilter g l μ
s : Set α
sl : s ∈ l
hs : IntegrableOn f s μ
⊢ IntegrableAtFilter (f + g) l μ

Tactic:
rcases hg with ⟨t, tl, ht⟩

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
l : Filter α
inst✝ : ContinuousAdd ε'
f g : α → ε'
s : Set α
sl : s ∈ l
hs : IntegrableOn f s μ
t : Set α
tl : t ∈ l
ht : IntegrableOn g t μ
⊢ IntegrableAtFilter (f + g) l μ
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
⊢ IntegrableAtFilter f ⊤ μ ↔ Integrable f μ

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ h.integrableAtFilter ⊤⟩
  obtain ⟨s, hsf, hs⟩ := h
  exact (integrableOn_iff_integrable_of_support_subset fun _ _ ↦ hsf _).mp hs

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
E : Type u_4
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
f : E →ₗ[𝕜] F
s : Submodule 𝕜 E
inst✝ : s.CoFG
h : IsClosed[inst✝⁶] ↑(Submodule.map f s)
⊢ IsClosed[inst✝⁶] ↑f.range

Tactic:
obtain ⟨t, s_compl_t⟩ := Submodule.exists_isCompl s

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
E : Type u_4
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
f : E →ₗ[𝕜] F
s : Submodule 𝕜 E
inst✝ : s.CoFG
h : IsClosed[inst✝⁶] ↑(Submodule.map f s)
t : Submodule 𝕜 E
s_compl_t : IsCompl s t
⊢ IsClosed[inst✝⁶] ↑f.range
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✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : UpgradedIsCompletelyPseudoMetrizableSpace E := upgradeIsCompletelyPseudoMetrizable E
s : Set E := closure[inst✝³] (⋃ i, range (f i))
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}

Tactic:
have : SecondCountableTopology s := @UniformSpace.secondCountable_of_separable s _ _
    (IsSeparable.iUnion (fun i ↦ (hf i).isSeparable_range)).closure.separableSpace

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✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : UpgradedIsCompletelyPseudoMetrizableSpace E := upgradeIsCompletelyPseudoMetrizable E
s : Set E := closure[inst✝³] (⋃ i, range (f i))
this : SecondCountableTopology ↑s
⊢ 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‖₊ * ‖z‖₊ ≤ ‖(f.smulRight z) m‖₊

Tactic:
rw [smulRight_apply, nnnorm_smul]

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

Goal state before:
pp: case pos
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ∞
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : eLpNorm (fun x ↦ f x - y₀) p μ < ∞
hp_zero : p = 0
⊢ Tendsto (fun n ↦ eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)

pp: case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ∞
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : eLpNorm (fun x ↦ f x - y₀) p μ < ∞
hp_zero : ¬p = 0
⊢ Tendsto (fun n ↦ eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)

Tactic:
· simpa only [hp_zero, eLpNorm_exponent_zero] using tendsto_const_nhds

Goal state after:
case neg
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hp_ne_top : p ≠ ∞
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : eLpNorm (fun x ↦ f x - y₀) p μ < ∞
hp_zero : ¬p = 0
⊢ Tendsto (fun n ↦ eLpNorm (⇑(approxOn f hf s y₀ h₀ n) - f) p μ) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : LocallyIntegrableOn f s μ
hg : ContinuousOn g s
hs : IsLocallyClosed s
⊢ LocallyIntegrableOn (fun x ↦ f x * g x) s μ

Tactic:
by
  rw [MeasureTheory.locallyIntegrableOn_iff hs] at hf ⊢
  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c

Goal state after:
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)
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range Subtype.val) ∧ Tendsto x atTop (𝓝 f)

Tactic:
have hfi' : MemLp f p μ := 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
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range Subtype.val) ∧ Tendsto x atTop (𝓝 f)
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: 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'
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * μ.real s
f : α →ₛ F
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ C * ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) * ‖x‖

Tactic:
by simp_rw [mul_sum, ← mul_assoc]; rfl

Goal state after:
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'
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
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ C * ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) * ‖x‖

Tactic:
by simp_rw [mul_sum, ← mul_assoc]; rfl

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
huv : Tendsto (u / v) l (𝓝 1)
this : ∀ᶠ (x : α) in l, v x = 0 → u x = 0
⊢ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v

Tactic:
exact ⟨u / v, huv, this.mono fun x hz' ↦ (div_mul_cancel_of_imp hz').symm⟩

Goal state after: