mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
ε C : ℝ
ε_pos : 0 < ε
hC : 0 ≤ C
c : 𝕜
hc : 1 < ‖c⁻¹‖
hf : ∀ (x : E), ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
h0 : ¬c = 0
⊢ ∀ (x : E), ε / ‖c⁻¹‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖

Tactic:
norm_inv,

Goal state after:
case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
ε C : ℝ
ε_pos : 0 < ε
hC : 0 ≤ C
c : 𝕜
hc : 1 < ‖c⁻¹‖
hf : ∀ (x : E), ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
h0 : ¬c = 0
⊢ ∀ (x : E), ε / ‖c‖⁻¹ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
Module: Mathlib.Analysis.Normed.Operator.NNNorm

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
x : E
⊢ ‖f x‖ₑ ≤ ‖f‖ₑ * ‖x‖ₑ

Tactic:
dsimp [enorm]; exact mod_cast le_opNNNorm ..

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u_1
pi : ι → Prop
s : ι → Set (β × β)
h : (𝓤 β).HasBasis pi s
⊢ (𝓤 C(α, β)).HasBasis (fun p ↦ IsCompact p.1 ∧ pi p.2) fun p ↦ {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2}

Tactic:
by
  rw [← isUniformEmbedding_toUniformOnFunIsCompact.comap_uniformity]
  exact .comap _ <| UniformOnFun.hasBasis_uniformity_of_basis _ _ {K | IsCompact K}
    ⟨∅, isCompact_empty⟩ (directedOn_of_sup_mem fun _ _ ↦ IsCompact.union) h

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : ConditionallyCompleteLattice β
f : α → β
inst✝ : Nonempty α
hμ : ∀ (a : α), μ {a} ≠ 0
hf : BddBelow (range f)
⊢ essInf f μ = ⨅ a, f a

Tactic:
by rw [essInf, ae_eq_top.2 hμ, liminf_top_eq_ciInf hf]

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : NeZero μ
b : β
x : α
h : ∃ b_1, const α b = mk (Function.const α b_1) ⋯
b' : β := Classical.choose h
this : const α b = mk (Function.const α b') ⋯
⊢ Function.const α b' x = b

Tactic:
const,

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : NeZero μ
b : β
x : α
h : ∃ b_1, const α b = mk (Function.const α b_1) ⋯
b' : β := Classical.choose h
this : mk (fun x ↦ b) ⋯ = mk (Function.const α b') ⋯
⊢ Function.const α b' x = b
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
i : ι
s : Set (α i)
hs : (μ i) s = 0
⊢ (Measure.pi μ) (eval i ⁻¹' s) = 0

Tactic:
rcases exists_measurable_superset_of_null hs with ⟨t, hst, _, hμt⟩

Goal state after:
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
i : ι
s : Set (α i)
hs : (μ i) s = 0
t : Set (α i)
hst : s ⊆ t
left✝ : MeasurableSet t
hμt : (μ i) t = 0
⊢ (Measure.pi μ) (eval i ⁻¹' s) = 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s : Set α
c : ε
hμs : μ s ≠ 0
⊢ eLpNormEssSup (s.indicator fun x ↦ c) μ = ‖c‖ₑ

Tactic:
refine le_antisymm (eLpNormEssSup_indicator_const_le s c) ?_
  by_contra! h
  have h' := ae_iff.mp (ae_lt_of_essSup_lt h)
  push Not at h'
  refine hμs (measure_mono_null (fun x hx_mem => ?_) h')
  rw [Set.mem_ofPred_eq, Set.indicator_of_mem hx_mem]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_5
ε' : Type u_6
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : α → ε
g : α → ε'
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ
p : ℝ
hp : 0 < p
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ ≤ ↑(c ^ p) * ∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ

Tactic:
← lintegral_const_mul' _ _ ENNReal.coe_ne_top

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_5
ε' : Type u_6
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : α → ε
g : α → ε'
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ
p : ℝ
hp : 0 < p
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ ≤ ∫⁻ (a : α), ↑(c ^ p) * ‖g a‖ₑ ^ p ∂μ
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0
h : p.HolderConjugate q
⊢ p * q = p + q

Tactic:
by
  simpa [mul_add, add_mul, h.ne_zero, h.symm.ne_zero, add_comm q] using congr(p * $(h.inv_eq) * q)

Goal state after:
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case inl.inr
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : a < y
⊢ (f x - f a) / (x - a) < (f y - f a) / (y - a)

Tactic:
· convert! hf.slope_strict_mono_adjacent hx hy hxa hya using 1
      rw [← neg_div_neg_eq]; simp

Goal state after:
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y : 𝕜
hx : x ∈ s
hxy : x < y
hxy' : f x < f y
u : 𝕜
hu : u ∈ s ∩ Set.Ici y
v : 𝕜
hv : v ∈ s ∩ Set.Ici y
huv : u < v
z : 𝕜
hz : z ∈ s ∩ Set.Ioi y
⊢ y ∈ Set.Ioo x z

Tactic:
exact ⟨hxy, hz.2⟩

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case mp
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
⊢ ∃ j ∈ s, ∃ k ∈ s, z j ≠ z k

Tactic:
by_contra! h_contra

Goal state after:
case mp
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
h_contra : ∀ j ∈ s, ∀ k ∈ s, z j = z k
⊢ False
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case pos
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : p = q
⊢ eLpNorm' f q μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / q - 1 / q)

Tactic:
sub_self,

Goal state after:
case pos
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : p = q
⊢ eLpNorm' f q μ ≤ eLpNorm' f q μ * μ Set.univ ^ 0
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h : a ≠ ∞ ∧ b ≠ ∞
⊢ p.toNNReal ≠ 0

Tactic:
simp [hpq.pos]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h0 : a ≠ 0 ∧ b ≠ 0
h : a ≠ ∞ ∧ b ≠ ∞
⊢ p.toNNReal ≠ 0

Tactic:
simp [hpq.pos]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr : p.HolderTriple q r
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ

Tactic:
by
  have hpqr := hpqr.one_div_eq
  obtain (rfl | rfl | hp) := ENNReal.trichotomy p
  · simp_all
  · have : r = q := by simpa using hpqr
    exact this ▸ eLpNorm_le_eLpNorm_top_mul_eLpNorm r f hg b c h
  obtain (rfl | rfl | hq) := ENNReal.trichotomy q
  · simp_all
  · have : r = p := by simpa using hpqr
    exact this ▸ eLpNorm_le_eLpNorm_mul_eLpNorm_top p hf g b c h
  obtain ⟨hp₁, hp₂⟩ := ENNReal.toReal_pos_iff.mp hp
  obtain ⟨hq₁, hq₂⟩ := ENNReal.toReal_pos_iff.mp hq
  have hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal := by
    have := congr(ENNReal.toReal $(hpqr))
    rw [ENNReal.toReal_add (by simpa using hp₁.ne') (by simpa using hq₁.ne')] at this
    simpa
  have hr : 0 < r.toReal := one_div_pos.mp <| by rw [hpqr']; positivity
  obtain ⟨hr₁, hr₂⟩ := ENNReal.toReal_pos_iff.mp hr
  have hrp : r.toReal < p.toReal := lt_of_one_div_lt_one_div hp <|
    hpqr' ▸ lt_add_of_pos_right _ (by positivity)
  rw [eLpNorm_eq_eLpNorm', eLpNorm_eq_eLpNorm', eLpNorm_eq_eLpNorm']
  · exact eLpNorm'_le_eLpNorm'_mul_eLpNorm' hf hg b c h hr hrp hpqr'
  all_goals first | positivity | finiteness

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ p) (B ^ p)
hp' : p ≠ 0
H₁ : Summable fun i ↦ (f i + g i) ^ p
H₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
⊢ ∃ C ≤ A + B, HasSum (fun i ↦ (f i + g i) ^ p) (C ^ p)

Tactic:
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']

Goal state after:
ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ p) (B ^ p)
hp' : p ≠ 0
H₁ : Summable fun i ↦ (f i + g i) ^ p
H₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
hB : B = (∑' (i : ι), g i ^ p) ^ (1 / p)
⊢ ∃ C ≤ A + B, HasSum (fun i ↦ (f i + g i) ^ p) (C ^ p)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → α → E
f_ℒp : ∀ (n : ι), MemLp (f n) p μ
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
n : ι
a✝ : α
⊢ ↑↑(MemLp.toLp (f n - f_lim) ⋯) a✝ = (f n - f_lim) a✝ →
    ↑↑(MemLp.toLp (f n) ⋯ - MemLp.toLp f_lim f_lim_ℒp) a✝ =
        (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) a✝ →
      (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) a✝ = (f n - f_lim) a✝

Tactic:
hx₁

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → α → E
f_ℒp : ∀ (n : ι), MemLp (f n) p μ
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
n : ι
a✝ : α
hx₁ : ↑↑(MemLp.toLp (f n - f_lim) ⋯) a✝ = (f n - f_lim) a✝
⊢ ↑↑(MemLp.toLp (f n) ⋯ - MemLp.toLp f_lim f_lim_ℒp) a✝ = (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) a✝ →
    (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) a✝ = (f n - f_lim) a✝
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case hx
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
⊢ 0 ≤ ∑ i ∈ s, w i * f i ^ p

Tactic:
exact sum_nonneg fun i _ ↦ by have := hw i; have := hf i; positivity

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MemLp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ∞ →
          (∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N) →
            ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
hf : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → dist (f n) (f m_1) < B N
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n ↦ ENNReal.ofReal (B n)
⊢ ∑' (n : ℕ), ENNReal.ofReal (B n) = ENNReal.ofReal (∑' (b : ℕ), B b)

Tactic:
exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).symm

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

Goal state before:
pp: case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝ : {ξ₀}ᶜ ∈ 𝓝 0
this : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
⊢ ‖ξ₀‖ * ‖ξ⁻¹‖ ≤ 1

Tactic:
norm_inv,

Goal state after:
case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝ : {ξ₀}ᶜ ∈ 𝓝 0
this : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
⊢ ‖ξ₀‖ * ‖ξ‖⁻¹ ≤ 1
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 := fun n ↦ toMeasurable (μ.restrict s) {x | u n ≤ ‖f x‖ₑ}
A : ∀ (n : ℕ), μ (s ∩ v n) ≠ ∞
x : α
hx : x ∈ s
n : ℕ
hn : u n < ‖f x‖ₑ
⊢ x ∈ ⋃ n, v n

Tactic:
exact mem_iUnion.2 ⟨n, subset_toMeasurable _ _ hn.le⟩

Goal state after:
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'
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

Tactic:
refine ⟨s ∩ t, inter_mem sl tl, ?_⟩

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
⊢ IntegrableOn (f + g) (s ∩ t) 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)))
⊢ ∀ (a : α),
    ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) a = f a • ↑↑u a →
      ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) a = f a • ↑↑v a →
        ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) a = f a • ↑↑(u + v) a →
          ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) a =
              (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) a →
            (↑(f a) ≠ 0 → ↑↑(u + v) a = (↑↑u + ↑↑v) a) →
              ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) a =
                ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) a

Tactic:
intro x hu hv huv h' 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
⊢ ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x =
    ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v 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
f : MultilinearMap 𝕜 E G
C : ℝ
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
D : ℝ := max C 1
D_pos : 0 ≤ D
⊢ Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f

Tactic:
replace H (m) : ‖f m‖ ≤ D * ∏ i, ‖m i‖ :=
    (H m).trans (mul_le_mul_of_nonneg_right (le_max_left _ _) <| by positivity)

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 : MultilinearMap 𝕜 E G
C : ℝ
D : ℝ := max C 1
D_pos : 0 ≤ D
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ D * ∏ i, ‖m i‖
⊢ Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
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
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
n : ℕ
⊢ u n ∈ range u

Tactic:
exact mem_range_self n

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

Goal state before:
pp: case refine_1.inl
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✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen[inst✝³] (u n)

pp: case refine_1.inr
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✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n ∈ T
⊢ IsOpen[inst✝³] (u n)

Tactic:
· rw [h]
      exact isOpen_empty

Goal state after:
case refine_1.inr
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✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n ∈ T
⊢ IsOpen[inst✝³] (u n)
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
G' : Type wG'
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : G →ₗ[𝕜] MultilinearMap 𝕜 E G'
C : ℝ
H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖
⊢ ‖mkContinuousLinear f C H‖ ≤ max C 0

Tactic:
dsimp only [mkContinuousLinear]

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
G' : Type wG'
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : G →ₗ[𝕜] MultilinearMap 𝕜 E G'
C : ℝ
H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖
⊢ ‖{ toFun := fun x ↦ (f x).mkContinuous (C * ‖x‖) ⋯, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (max C 0) ⋯‖ ≤
    max C 0
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case refine_2
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✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
⊢ s ⊆ ⋃ n, u n

Tactic:
intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
    have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv
    obtain ⟨n, rfl⟩ : ∃ n, u n = v := by simpa only [mem_range] using this
    exact mem_iUnion_of_mem _ h'v

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

Goal state before:
pp: case refine_3
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
⊢ ∀ ⦃s t : Set X⦄,
    (∃ u, IsOpen[inst✝³] u ∧ s ⊆ u ∧ IntegrableOn f u μ) →
      (∃ u, IsOpen[inst✝³] u ∧ t ⊆ u ∧ IntegrableOn f u μ) → ∃ u, IsOpen[inst✝³] u ∧ s ∪ t ⊆ u ∧ IntegrableOn f u μ

Tactic:
rintro s t ⟨u, u_open, su, hu⟩ ⟨v, v_open, tv, hv⟩
    exact ⟨u ∪ v, u_open.union v_open, union_subset_union su tv, hu.union hv⟩

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

Goal state before:
pp: case refine_2
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 : BorelSpace E
f : β → E
hp_ne_top : p ≠ ∞
μ : Measure β
fmeas : Measurable f
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
hf : eLpNorm f p μ < ∞
⊢ eLpNorm (fun x ↦ f x - 0) p μ < ∞

Tactic:
· simpa using hf

Goal state after: