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

Goal state before:
pp: case inr.inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : ∑ i ∈ s, g i ^ q = 0
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

pp: case inr.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

Tactic:
· calc
      ∑ i ∈ s, f i * g i = ∑ i ∈ s, g i * f i := by
        congr with i
        rw [mul_comm]
      _ ≤ (∑ i ∈ s, g i ^ q) ^ (1 / q) * (∑ i ∈ s, f i ^ p) ^ (1 / p) :=
        (inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hg)
      _ = (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) := mul_comm _ _

Goal state after:
case inr.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : 0 < ε
δ : ℝ≥0
hδ : 0 < ↑δ
t : Set α
left✝ : MeasurableSet t
ht : μ t ≤ ↑δ
hunif : ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ tᶜ, edist (g x) (f n x) < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ∀ x ∈ tᶜ, edist (g x) (f b x) < ε
n : ℕ
hn : n ≥ N
⊢ tᶜ ⊆ {x | ε ≤ edist (f n x) (g x)}ᶜ

Tactic:
intro x hx

Goal state after:
case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : 0 < ε
δ : ℝ≥0
hδ : 0 < ↑δ
t : Set α
left✝ : MeasurableSet t
ht : μ t ≤ ↑δ
hunif : ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, ∀ x ∈ tᶜ, edist (g x) (f n x) < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ∀ x ∈ tᶜ, edist (g x) (f b x) < ε
n : ℕ
hn : n ≥ N
x : α
hx : x ∈ tᶜ
⊢ x ∈ {x | ε ≤ edist (f n x) (g x)}ᶜ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞
this :
  ∑ i ∈ s, ↑(f i).toNNReal * ↑(g i).toNNReal ≤
    (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ * (∑ x ∈ s, ↑(g x).toNNReal ^ q) ^ q⁻¹
⊢ ∑ i ∈ s, f i * g i = ∑ i ∈ s, ↑(f i).toNNReal * ↑(g i).toNNReal

Tactic:
· refine Finset.sum_congr rfl fun i hi => ?_
      simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞
this :
  ∑ i ∈ s, ↑(f i).toNNReal * ↑(g i).toNNReal ≤
    (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ * (∑ x ∈ s, ↑(g x).toNNReal ^ q) ^ q⁻¹
⊢ ∑ i ∈ s, f i ^ p = ∑ x ∈ s, ↑(f x).toNNReal ^ p

Tactic:
· refine Finset.sum_congr rfl fun i hi => ?_
      simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]

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
β : Type u_7
inst✝ : MeasurableSpace β
μb : Measure β
g : β →ₘ[μb] E
hg : g ∈ Lp E p μb
f : α → β
hf : MeasurePreserving f μ μb
⊢ g.compMeasurePreserving f hf ∈ Lp E p μ

Tactic:
by
  rw [Lp.mem_Lp_iff_eLpNorm_lt_top] at hg ⊢
  rwa [eLpNorm_compMeasurePreserving]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞
this :
  ∑ i ∈ s, ↑(f i).toNNReal * ↑(g i).toNNReal ≤
    (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ * (∑ x ∈ s, ↑(g x).toNNReal ^ q) ^ q⁻¹
⊢ ∑ i ∈ s, g i ^ q = ∑ x ∈ s, ↑(g x).toNNReal ^ q

Tactic:
· refine Finset.sum_congr rfl fun i hi => ?_
      simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Unique α
x✝ : α ↪ β
x : α
⊢ ((fun x ↦ { toFun := fun x_1 ↦ x, inj' := ⋯ }) ((fun f ↦ f default) x✝)) x = x✝ x

Tactic:
simp_rw [Function.Embedding.coeFn_mk]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : Unique α
x✝ : α ↪ β
x : α
⊢ x✝ default = x✝ x
Module: Mathlib.Analysis.Normed.Module.Span

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
x : E
hx : ‖x‖ = 1
x✝ : 𝕜
⊢ ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) x✝‖ = ‖x✝‖

Tactic:
rw [LinearEquiv.toSpanNonzeroSingleton_homothety, hx, one_mul]

Goal state after:
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✝ : ContinuousSub M
h : IsTopCompl p q
x : M
⊢ (q.projectionL p ⋯) x = x - (p.projectionL q h) x

Tactic:
rw [eq_sub_iff_add_eq, projectionL_add_projectionL_eq_self]

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

Goal state before:
pp: case insert
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
μ : Measure α
f : β → Set α
a✝ : β
s✝ : Finset β
hx : a✝ ∉ s✝
IH : μ.real (⋃ b ∈ s✝, f b) ≤ ∑ p ∈ s✝, μ.real (f p)
⊢ μ.real (⋃ b ∈ insert a✝ s✝, f b) ≤ ∑ p ∈ insert a✝ s✝, μ.real (f p)

Tactic:
simp only [hx, Finset.mem_insert, iUnion_iUnion_eq_or_left, not_false_eq_true,
      Finset.sum_insert]

Goal state after:
case insert
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
μ : Measure α
f : β → Set α
a✝ : β
s✝ : Finset β
hx : a✝ ∉ s✝
IH : μ.real (⋃ b ∈ s✝, f b) ≤ ∑ p ∈ s✝, μ.real (f p)
⊢ μ.real (f a✝ ∪ ⋃ x ∈ s✝, f x) ≤ μ.real (f a✝) + ∑ p ∈ s✝, μ.real (f p)
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
hp₁ : Fact (1 ≤ p)
β : Type u_3
l : Filter β
t : β → Set α
ht : ∀ (b : β), MeasurableSet (t b)
hμt : ∀ (b : β), μ (t b) ≠ ∞
hp : p ≠ ∞
h : Tendsto (fun b ↦ μ (t b ∆ s)) l (𝓝 0)
hp₀ : p ≠ 0
⊢ Tendsto (fun b ↦ dist (indicatorConstLp p ⋯ ⋯ c) (indicatorConstLp p hs hμs c)) l (𝓝 0)

Tactic:
simp only [dist_indicatorConstLp_eq_norm, norm_indicatorConstLp hp₀ hp]

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
hp₁ : Fact (1 ≤ p)
β : Type u_3
l : Filter β
t : β → Set α
ht : ∀ (b : β), MeasurableSet (t b)
hμt : ∀ (b : β), μ (t b) ≠ ∞
hp : p ≠ ∞
h : Tendsto (fun b ↦ μ (t b ∆ s)) l (𝓝 0)
hp₀ : p ≠ 0
⊢ Tendsto (fun b ↦ ‖c‖ * μ.real (t b ∆ s) ^ (1 / p.toReal)) l (𝓝 0)
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
a : α
f : α → ε
hf : StronglyMeasurable f
hfa : ‖f a‖ₑ < ∞
⊢ HasFiniteIntegral f (Measure.dirac a)

Tactic:
by simpa [HasFiniteIntegral, lintegral_dirac' _ hf.enorm]

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✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
h : Integrable (f + g) μ
⊢ g = f + g + -f

Tactic:
simp only [add_neg_cancel_comm]

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hf : IntegrableOn f s μ
h's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0
u : ℕ → ℝ≥0∞
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set α := ⋯
A : ∀ (n : ℕ), μ (s ∩ v n) ≠ ∞
⊢ s ⊆ ⋃ n, v n

Tactic:
intro x hx

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hf : IntegrableOn f s μ
h's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0
u : ℕ → ℝ≥0∞
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set α := fun n ↦ toMeasurable (μ.restrict s) {x | u n ≤ ‖f x‖ₑ}
A : ∀ (n : ℕ), μ (s ∩ v n) ≠ ∞
x : α
hx : x ∈ s
⊢ x ∈ ⋃ n, v n
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
⊢ ∀ x ∈ u, ‖f x‖ₑ ≠ 0

Tactic:
intro x hx

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
x : α
hx : x ∈ u
⊢ ‖f x‖ₑ ≠ 0
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
⊢ ∀ (a : α), f a < ∞ → ENNReal.ofReal (f a).toReal = f a

Tactic:
intro x hx

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
x : α
hx : f x < ∞
⊢ ENNReal.ofReal (f x).toReal = f x
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
⊢ ∀ (a : α), f a < ∞ → ↑(f a).toNNReal = f a

Tactic:
intro x hx

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
x : α
hx : f x < ∞
⊢ ↑(f x).toNNReal = f x
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → E
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable g (μ.withDensity fun x ↦ ↑(AEMeasurable.mk f hf x))

Tactic:
suffices (fun x => (f x : ℝ≥0∞)) =ᵐ[μ] (fun x => (hf.mk f x : ℝ≥0)) by
        rw [withDensity_congr_ae this]
      filter_upwards [hf.ae_eq_mk] with x hx
      simp [hx]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Integrable g (μ.withDensity f) ↔ Integrable (fun x ↦ (f x).toReal • g x) μ

Tactic:
rw [← withDensity_congr_ae (coe_toNNReal_ae_eq hflt),
    integrable_withDensity_iff_integrable_coe_smul₀]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Integrable (fun x ↦ ↑(f x).toNNReal • g x) μ ↔ Integrable (fun x ↦ (f x).toReal • g x) μ

case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ AEMeasurable (fun x ↦ (f x).toNNReal) μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
u v : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hu : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
hv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x = f x • ↑↑v x
huv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x = f x • ↑↑(u + v) x
h' :
  ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x =
    (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
h'' : ↑(f x) ≠ 0 → ↑↑(u + v) x = (↑↑u + ↑↑v) x
⊢ f x • ↑↑(u + v) x = ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x

Tactic:
h',

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
u v : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hu : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
hv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x = f x • ↑↑v x
huv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x = f x • ↑↑(u + v) x
h' :
  ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x =
    (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
h'' : ↑(f x) ≠ 0 → ↑↑(u + v) x = (↑↑u + ↑↑v) x
⊢ f x • ↑↑(u + v) x = (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
r : ℝ
u : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
h : ↑(f x) ≠ 0 → ↑↑(r • u) x = (r • ↑↑u) x
h' : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(r • u) x) ⋯) x = f x • ↑↑(r • u) x
h'' : ↑↑(r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = (r • ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯)) x
h''' : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
⊢ ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(r • u) x) ⋯) x = ↑↑(r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x

Tactic:
h',

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
r : ℝ
u : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
h : ↑(f x) ≠ 0 → ↑↑(r • u) x = (r • ↑↑u) x
h' : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(r • u) x) ⋯) x = f x • ↑↑(r • u) x
h'' : ↑↑(r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = (r • ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯)) x
h''' : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
⊢ f x • ↑↑(r • u) x = ↑↑(r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x
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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
⊢ 𝓤 (ContinuousMultilinearMap 𝕜 E G) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖f.1 - f.2‖ < r}

Tactic:
refine (ContinuousMultilinearMap.seminorm 𝕜 E G).uniformity_eq_of_hasBasis
    (ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis Metric.nhds_basis_closedBall)
    ?_ fun (s, r) ⟨hs, hr⟩ ↦ ?_

Goal state after:
case refine_1
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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
⊢ ∃ r, (ContinuousMultilinearMap.seminorm 𝕜 E G).closedBall 0 r ∈ 𝓝 0

case refine_2
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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
x✝¹ : Set ((i : ι) → E i) × ℝ
s : Set ((i : ι) → E i)
r : ℝ
x✝ : Bornology.IsVonNBounded 𝕜 (s, r).1 ∧ 0 < (s, r).2
hs : Bornology.IsVonNBounded 𝕜 (s, r).1
hr : 0 < (s, r).2
⊢ ∃ r_1 > 0, (ContinuousMultilinearMap.seminorm 𝕜 E G).ball 0 r_1 ⊆ {f | Set.MapsTo ⇑f (s, r).1 (closedBall 0 (s, r).2)}
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case succ
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝² : Fintype ι
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this : FiniteDimensional 𝕜 E
H₁ : ∀ (s : Submodule 𝕜 E), finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s
H₂ : ∀ (f : E →ₗ[𝕜] 𝕜), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
i : ι
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun a ↦ ξ.equivFun a i

Tactic:
change Continuous (ξ.coord i)

Goal state after:
case succ
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝² : Fintype ι
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this : FiniteDimensional 𝕜 E
H₁ : ∀ (s : Submodule 𝕜 E), finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s
H₂ : ∀ (f : E →ₗ[𝕜] 𝕜), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
i : ι
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑(ξ.coord i)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
ε' : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : X → ε
μ : Measure X
s : Set X
hf : LocallyIntegrableOn f s μ
g : X → ε'
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : X) ∂μ, ‖g x‖ₑ ≤ ‖f x‖ₑ
⊢ LocallyIntegrableOn g s μ

Tactic:
intro x hx

Goal state after:
X : Type u_1
ε : Type u_3
ε' : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : X → ε
μ : Measure X
s : Set X
hf : LocallyIntegrableOn f s μ
g : X → ε'
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : X) ∂μ, ‖g x‖ₑ ≤ ‖f x‖ₑ
x : X
hx : x ∈ s
⊢ IntegrableAtFilter g (𝓝[s] x) μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
F : Type u_7
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : Measure X
s : Set X
f : X → E
hf : LocallyIntegrableOn f s μ
g : X → F
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : X) ∂μ, ‖g x‖ ≤ ‖f x‖
⊢ LocallyIntegrableOn g s μ

Tactic:
intro x hx

Goal state after:
X : Type u_1
E : Type u_6
F : Type u_7
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : Measure X
s : Set X
f : X → E
hf : LocallyIntegrableOn f s μ
g : X → F
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : X) ∂μ, ‖g x‖ ≤ ‖f x‖
x : X
hx : x ∈ s
⊢ IntegrableAtFilter g (𝓝[s] x) μ
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 ν : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrableOn f s μ
h : ν.restrict s ≤ μ.restrict s
⊢ LocallyIntegrableOn f s ν

Tactic:
intro x hx

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 ν : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrableOn f s μ
h : ν.restrict s ≤ μ.restrict s
x : X
hx : x ∈ s
⊢ IntegrableAtFilter f (𝓝[s] x) ν
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 g : X → ε
μ : Measure X
s : Set X
h : f =ᵐ[μ.restrict s] g
hf : LocallyIntegrableOn f s μ
⊢ LocallyIntegrableOn g s μ

Tactic:
intro x hx

Goal state after:
X : Type u_1
ε : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f g : X → ε
μ : Measure X
s : Set X
h : f =ᵐ[μ.restrict s] g
hf : LocallyIntegrableOn f s μ
x : X
hx : x ∈ s
⊢ IntegrableAtFilter g (𝓝[s] x) μ