mathlib-initiative/mathlib-tactics huggingface dataset:
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
h_add_m : AEMeasurable (fun a ↦ (f + g) a ^ (p - 1)) μ
h_add_apply : ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), (f a + g a) * (f + g) a ^ (p - 1) ∂μ
⊢ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ =
    ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ

Tactic:
simp_rw [h_add_apply, add_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
h_add_m : AEMeasurable (fun a ↦ (f + g) a ^ (p - 1)) μ
h_add_apply : ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), (f a + g a) * (f + g) a ^ (p - 1) ∂μ
⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) + g a * (f + g) a ^ (p - 1) ∂μ =
    ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case empty
ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s : Finset ι
hf : ∀ i ∈ ∅, MemLp (f i) (p i) μ
⊢ MemLp (∏ i ∈ ∅, f i) (∑ i ∈ ∅, (p i)⁻¹)⁻¹ μ

Tactic:
by_cases hμ : μ = 0

Goal state after:
case pos
ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s : Finset ι
hf : ∀ i ∈ ∅, MemLp (f i) (p i) μ
hμ : μ = 0
⊢ MemLp (∏ i ∈ ∅, f i) (∑ i ∈ ∅, (p i)⁻¹)⁻¹ μ

case neg
ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s : Finset ι
hf : ∀ i ∈ ∅, MemLp (f i) (p i) μ
hμ : ¬μ = 0
⊢ MemLp (∏ i ∈ ∅, f i) (∑ i ∈ ∅, (p i)⁻¹)⁻¹ μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ
p : ℝ
hp : 1 ≤ p
this : (∑ i ∈ s, ↑⟨|f i|, ⋯⟩) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ x ∈ s, ↑⟨|f x|, ⋯⟩ ^ p
⊢ (∑ i ∈ s, |f i|) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, |f i| ^ p

Tactic:
exact this

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → E
g : α → E
C : ℝ≥0∞
bound : ∀ᶠ (n : ι) in u, eLpNorm (f n) p μ ≤ C
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 ↦ f x_1 x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
this : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f (v n) x) atTop (𝓝 (g x))
⊢ liminf (fun n ↦ eLpNorm (f (v n)) p μ) atTop ≤ C

Tactic:
refine liminf_le_of_le (by isBoundedDefault) (fun b hb => ?_)

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → E
g : α → E
C : ℝ≥0∞
bound : ∀ᶠ (n : ι) in u, eLpNorm (f n) p μ ≤ C
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 ↦ f x_1 x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
this : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f (v n) x) atTop (𝓝 (g x))
b : ℝ≥0∞
hb : ∀ᶠ (n : ℕ) in atTop, b ≤ eLpNorm (f (v n)) p μ
⊢ b ≤ 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 : ↥(Lp E p μ)
hp : 0 < p
⊢ ‖f‖₊ = 0 ↔ f = 0

Tactic:
by
  refine ⟨fun hf => ?_, fun hf => by simp [hf]⟩
  simp_rw [nnnorm_def, ENNReal.toNNReal_eq_zero_iff, eLpNorm_ne_top, or_false] at hf
  simp_rw [eq_zero_iff_ae_eq_zero, ← eLpNorm_eq_zero_iff (Lp.aestronglyMeasurable f) hp.ne.symm, hf]

Goal state after:
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✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
f : ↥(Lp E p μ)
C : ℝ≥0
hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C
⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C

Tactic:
by_cases hμ : μ = 0

Goal state after:
case pos
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
f : ↥(Lp E p μ)
C : ℝ≥0
hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C
hμ : μ = 0
⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C

case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
f : ↥(Lp E p μ)
C : ℝ≥0
hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C
hμ : ¬μ = 0
⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * 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✝¹ : NormedAddCommGroup E
β : Type u_7
inst✝ : MeasurableSpace β
μb : Measure β
g : β →ₘ[μb] E
hg : eLpNorm (↑g) p μb < ∞
f : α → β
hf : MeasurePreserving f μ μb
⊢ eLpNorm (↑(g.compMeasurePreserving f hf)) p μ < ∞

Tactic:
eLpNorm_compMeasurePreserving

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
β : Type u_7
inst✝ : MeasurableSpace β
μb : Measure β
g : β →ₘ[μb] E
hg : eLpNorm (↑g) p μb < ∞
f : α → β
hf : MeasurePreserving f μ μb
⊢ eLpNorm (↑g) p μb < ∞
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✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
ν : Measure α
c : ℝ≥0∞
hc : c ≠ ∞
h : μ ≤ c • ν
f : ↥(Lp E p ν)
⊢ ↑↑((LpToLpOfMeasureLeSMulₗ hc h) f) =ᵐ[μ] ↑↑f

Tactic:
simp [LpToLpOfMeasureLeSMulₗ, MemLp.coeFn_toLp]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝³ : NormedAddCommGroup E
inst✝² : Lattice E
inst✝¹ : HasSolidNorm E
inst✝ : IsOrderedAddMonoid E
f g : α →ₘ[μ] E
hf : f ∈ Lp E p μ
hg : g ∈ Lp E p μ
⊢ f ⊓ g ∈ Lp E p μ

Tactic:
rw [mem_Lp_iff_memLp] at *
    exact (memLp_congr_ae (AEEqFun.coeFn_inf _ _)).mpr (hf.inf hg)

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
a : α ↪ γ
f : β ↪ γ
⊢ Disjoint (Set.range ⇑a) (Set.range ⇑f) ↔ ∀ (a_1 : β), f a_1 ∈ (Set.range ⇑a)ᶜ

Tactic:
← Set.range_subset_iff,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
a : α ↪ γ
f : β ↪ γ
⊢ Disjoint (Set.range ⇑a) (Set.range ⇑f) ↔ Set.range ⇑f ⊆ (Set.range ⇑a)ᶜ
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 := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
⊢ ∀ᶠ (c : 𝕜) in 𝓝 0, MapsTo (fun x ↦ c • x) (⇑f '' s) V

Tactic:
NormedAddGroup.nhds_zero_basis_norm_lt.eventually_iff

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 := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
⊢ ∃ i, 0 < i ∧ ∀ ⦃x : 𝕜⦄, x ∈ {y | ‖y‖ < i} → MapsTo (fun x_1 ↦ x • x_1) (⇑f '' s) V
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 := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
hc₀' : ∏ i ∈ I, c i ≠ 0
a : 𝕜
ha : a ∈ {y | ‖y‖ < ‖∏ i ∈ I, c i‖}
x : (i : ι) → E i
hx : x ∈ s
i₀ : ι
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
⊢ (a / ∏ i ∈ I, c i) • f y = a • f x

Tactic:
f.map_piecewise_smul,

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 := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
hc₀' : ∏ i ∈ I, c i ≠ 0
a : 𝕜
ha : a ∈ {y | ‖y‖ < ‖∏ i ∈ I, c i‖}
x : (i : ι) → E i
hx : x ∈ s
i₀ : ι
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
⊢ (a / ∏ i ∈ I, c i) • (∏ i ∈ I, c i) • f x = a • f x
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun y ↦ f (y.1 - x.1, y.2) + f (x.1, y.2 - x.2)) (𝓝 x) (𝓝 (0 + 0))

Tactic:
apply Tendsto.add

Goal state after:
case hf
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun x_1 ↦ f (x_1.1 - x.1, x_1.2)) (𝓝 x) (𝓝 0)

case hg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun x_1 ↦ f (x.1, x_1.2 - x.2)) (𝓝 x) (𝓝 0)
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

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

Goal state after:
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✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι✝) → TopologicalSpace (E i)
inst✝⁵ : (i : ι✝) → AddCommGroup (E i)
inst✝⁴ : (i : ι✝) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGroup F
ι : Type u_5
p : ι → Prop
b : ι → Set F
h : (𝓝 0).HasBasis p b
⊢ (𝓝 0).HasBasis (fun Si ↦ IsVonNBounded 𝕜 Si.1 ∧ p Si.2) fun Si ↦ {f | MapsTo (⇑f) Si.1 (b Si.2)}

Tactic:
let : UniformSpace F := IsTopologicalAddGroup.rightUniformSpace F

Goal state after:
𝕜 : Type u_1
ι✝ : Type u_2
E : ι✝ → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι✝) → TopologicalSpace (E i)
inst✝⁵ : (i : ι✝) → AddCommGroup (E i)
inst✝⁴ : (i : ι✝) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGroup F
ι : Type u_5
p : ι → Prop
b : ι → Set F
h : (𝓝 0).HasBasis p b
this : UniformSpace F := IsTopologicalAddGroup.rightUniformSpace F
⊢ (𝓝 0).HasBasis (fun Si ↦ IsVonNBounded 𝕜 Si.1 ∧ p Si.2) fun Si ↦ {f | MapsTo (⇑f) Si.1 (b Si.2)}
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
g_emb : IsEmbedding g
x✝¹ x✝ : X
⊢ (ker (g ∘ f)) x✝¹ x✝ ↔ (ker f) x✝¹ x✝

Tactic:
by simp [g_emb.injective.eq_iff]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ε' : Type u_8
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
μ : Measure α
⊢ Integrable 0 μ

Tactic:
simp [Integrable, aestronglyMeasurable_zero]

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f : α → E
⊢ IntegrableAtFilter (-f) l μ ↔ IntegrableAtFilter f l μ

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ h.neg⟩

Goal state after:
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f : α → E
h : IntegrableAtFilter (-f) l μ
⊢ IntegrableAtFilter f l μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
H : Type u_8
inst✝ : NormedAddCommGroup H
m0 : MeasurableSpace α
μ' : Measure α
f : α → H
hm : m ≤ m0
hf_int : Integrable f (μ'.trim hm)
⊢ Integrable f μ'

Tactic:
obtain ⟨hf_meas_ae, hf⟩ := hf_int

Goal state after:
α : Type u_1
m : MeasurableSpace α
H : Type u_8
inst✝ : NormedAddCommGroup H
m0 : MeasurableSpace α
μ' : Measure α
f : α → H
hm : m ≤ m0
hf_meas_ae : AEStronglyMeasurable f (μ'.trim hm)
hf : HasFiniteIntegral f (μ'.trim hm)
⊢ Integrable f μ'
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝¹ : Fintype ι
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
n : ℕ
hn : Fintype.card ι = n
⊢ Continuous[inst✝⁵, Pi.topologicalSpace] ⇑ξ.equivFun

Tactic:
induction n generalizing ι E with

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

Goal state before:
pp: 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
s t : Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 (E ⧸ s)
s_closed : IsClosed[inst✝⁴] ↑s
s_le_t : s ≤ t
⊢ IsClosed[inst✝⁴] ↑t

Tactic:
rw [show t = comap s.mkQ (map s.mkQ t) by simpa]
  exact (map s.mkQ t).closed_of_finiteDimensional.preimage continuous_quot_mk

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

Goal state before:
pp: case neg
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
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed[inst✝³] s
hf : LocallyIntegrableOn f s μ
x : X
h : x ∉ s
⊢ IntegrableAtFilter f (𝓝 x) (μ.restrict s)

Tactic:
← isOpen_compl_iff

Goal state after:
case neg
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
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen[inst✝³] sᶜ
hf : LocallyIntegrableOn f s μ
x : X
h : x ∉ s
⊢ IntegrableAtFilter f (𝓝 x) (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case refine_4
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
hf : LocallyIntegrable f μ
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
u : Set X
ux : u ∈ 𝓝 x
hu : IntegrableOn f u μ
v : Set X
vu : v ⊆ u
v_open : IsOpen[inst✝³] v
xv : x ∈ v
⊢ ∃ t ∈ 𝓝[k] x, ∃ u, IsOpen[inst✝³] u ∧ t ⊆ u ∧ IntegrableOn f u μ

Tactic:
exact ⟨v, nhdsWithin_le_nhds (v_open.mem_nhds xv), v, v_open, Subset.rfl, hu.mono_set vu⟩

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

Goal state before:
pp: X : Type u_1
ε'' : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace ε''
inst✝³ : ESeminormedAddMonoid ε''
μ : Measure X
inst✝² : PseudoMetrizableSpace ε''
f : X → ε''
inst✝¹ : LinearOrder X
inst✝ : CompactIccSpace X
⊢ Integrable f μ ↔ (IntegrableAtFilter f atBot μ ∧ IntegrableAtFilter f atTop μ) ∧ LocallyIntegrable f μ

Tactic:
by
  constructor
  · exact fun hf ↦ ⟨⟨hf.integrableAtFilter _, hf.integrableAtFilter _⟩, hf.locallyIntegrable⟩
  · refine fun h ↦ integrable_iff_integrableAtFilter_cocompact.mpr ⟨?_, h.2⟩
    exact (IntegrableAtFilter.sup_iff.mpr h.1).filter_mono cocompact_le_atBot_atTop

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
y : E
x✝ : y ∈ f.range
hy0 : y = 0
⊢ ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y}) < ∞

Tactic:
simp [hy0, ENNReal.toReal_pos hp0 hp_top]

Goal state after:
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✝² : MeasurableSpace G := borel G
this✝¹ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this✝ : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n ↦ approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
h_toLp : ∀ (n : ℕ), ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᵐ[μ] ⇑(x n)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
this : Tendsto (fun b ↦ dist ↑((x b).toLp ⋯) ↑g) atTop (𝓝 0)
⊢ Tendsto (fun b ↦ dist ↑(coeSimpleFuncNonnegToLpNonneg p μ G ⟨(x b).toLp ⋯, ⋯⟩) ↑g) atTop (𝓝 0)

Tactic:
exact this

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
f : α → G
hf : f =ᵐ[μ] 0
⊢ ∫ (a : α), f a ∂μ = 0

Tactic:
by
  simp [integral_congr_ae hf, integral_zero]

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

Goal state before:
pp: case inr.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
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
⊢ ∑ x_1 ∈ (piecewise s hs (const α x) (const α 0)).range,
      (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =
    (T s) x

Tactic:
range_indicator hs hs_empty hs_univ

Goal state after:
case inr.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
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
⊢ ∑ x_1 ∈ {x, 0}, (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = (T s) x
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h✝ : c ≠ 0
h : Tendsto (u - fun x ↦ c) l (𝓝 0)
this : Tendsto (fun x ↦ u x) l (𝓝 c)
⊢ Tendsto u l (𝓝 c)

Tactic:
exact this

Goal state after:
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
⊢ ∫ (x : α), |f x| ∂μ = 2 * ∫ (x : α), (f x)⁻ ∂μ + ∫ (x : α), f x ∂μ

Tactic:
by
  simp only [NegPart.negPart]
  have h_eq : ∀ x, |f x| = 2 * max (-f x) 0 + f x := by grind
  rw [integral_congr_ae (Eventually.of_forall h_eq), integral_add (by fun_prop) hf,
    integral_const_mul]

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
hp_pos : p ≠ 0
H : ∀ (f' : α →ₛ E) (δ : ℝ≥0∞), δ ≠ 0 → MemLp (⇑f') p μ → ∃ g, eLpNorm (⇑f' - g) p μ ≤ δ ∧ P g
⊢ ∃ g, eLpNorm (f - g) p μ ≤ ε ∧ P g

Tactic:
obtain ⟨η, ηpos, hη⟩ := exists_Lp_half E μ p hε
    rcases hf.exists_simpleFunc_eLpNorm_sub_lt hp_ne_top ηpos.ne' with ⟨f', hf', f'_mem⟩
    rcases H f' η ηpos.ne' f'_mem with ⟨g, hg, Pg⟩
    refine ⟨g, ?_, Pg⟩
    convert!
      (hη _ _ (hf.aestronglyMeasurable.sub f'.aestronglyMeasurable)
          (f'.aestronglyMeasurable.sub (h2P g Pg)) hf'.le hg).le using 2
    simp only [sub_add_sub_cancel]

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

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

Tactic:
simp [dist_eq_norm_vsub V _ x]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
this : Tendsto (fun x ↦ ∫ (a : α), f x a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))

Tactic:
exact this

Goal state after: