mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.MeasurableSpace.Pi

Goal state before:
pp: case pos
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
val✝ : Encodable ι
i : ι
s : Set (α i)
hs : s ∈ C i
t : (i : ι) → ℕ → Set (α i)
h1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i
h2t : ∀ (i : ι), ⋃ n, t i n = univ
x✝ : (i : ι) → α i
i' : ι
h : i' = i
⊢ x✝ i' ∈ update (fun i' ↦ iUnion (t i')) i (⋃ x, s) i' ↔ x✝ i' ∈ ⋃ j, update (fun i' ↦ t i' j) i s i'

Tactic:
subst h

Goal state after:
case pos
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
val✝ : Encodable ι
t : (i : ι) → ℕ → Set (α i)
h1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i
h2t : ∀ (i : ι), ⋃ n, t i n = univ
x✝ : (i : ι) → α i
i' : ι
s : Set (α i')
hs : s ∈ C i'
⊢ x✝ i' ∈ update (fun i' ↦ iUnion (t i')) i' (⋃ x, s) i' ↔ x✝ i' ∈ ⋃ j, update (fun i' ↦ t i' j) i' s i'
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
r : β → γ → Prop
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ LiftRel r f g ↔ ∀ᵐ (a : α) ∂μ, r (↑f a) (↑g a)

Tactic:
rw [← liftRel_mk_mk (hf := f.aestronglyMeasurable) (hg := g.aestronglyMeasurable),
    mk_coeFn, mk_coeFn]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑g a✝

Tactic:
exact inf_le_right

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

Goal state before:
pp: case refine_1
⊢ Measurable fun p ↦ (↑p.1, p.2).1 ^ (↑p.1, p.2).2

Tactic:
simp_rw [ENNReal.coe_rpow_def]
    exact Measurable.ite (by measurability) measurable_const (by fun_prop)

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

Goal state before:
pp: α : Type u_1
F : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F
𝕜 : Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
c : 𝕜
f : α → F
⊢ eLpNormEssSup (c • f) μ = ‖c‖ₑ * eLpNormEssSup f μ

Tactic:
simp_rw [eLpNormEssSup_eq_essSup_enorm, Pi.smul_apply, enorm_smul,
    ENNReal.essSup_const_mul]

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

Goal state before:
pp: case pos
α : Type u_1
F : Type u_3
G : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
g : α → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ≥0∞
h0 : ¬p = 0
h_top : p = ∞
⊢ eLpNorm f p μ ≤ c • eLpNorm g p μ

pp: case neg
α : Type u_1
F : Type u_3
G : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
g : α → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ≥0∞
h0 : ¬p = 0
h_top : ¬p = ∞
⊢ eLpNorm f p μ ≤ c • eLpNorm g p μ

Tactic:
· rw [h_top]
    exact eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul h

Goal state after:
case neg
α : Type u_1
F : Type u_3
G : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
g : α → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ≥0∞
h0 : ¬p = 0
h_top : ¬p = ∞
⊢ eLpNorm f p μ ≤ c • eLpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
C : ℝ
hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ eLpNorm f p μ ≤ ENNReal.ofReal C * μ Set.univ ^ p.toReal⁻¹

Tactic:
exact eLpNorm_le_of_ae_nnnorm_bound (hfC.mono fun x hx => hx.trans C.le_coe_toNNReal)

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case inr.inr.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) / p < log (1 + s)

Tactic:
← div_lt_div_right_of_neg hs'

Goal state after:
case inr.inr.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) / s < log (1 + p * s) / p / s
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case inr.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) < log (1 + p * s) / p

Tactic:
← div_lt_div_right_of_neg hs'

Goal state after:
case inr.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) / p / s < log (1 + s) / s
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
μ ν : Measure α
p : ℝ≥0∞
⊢ eLpNorm f p ν ≤ eLpNorm f p (μ + ν)

Tactic:
grw [← Measure.le_add_left le_rfl]

Goal state after:
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁵ : Semifield α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Semifield β
inst✝¹ : LinearOrder β
inst✝ : IsStrictOrderedRing β
s : Set ι
f : ι → α
g : ι → β
hf : ∀ (i : ι), i ∈ s → 0 < f i
hg : ∀ (i : ι), i ∈ s → 0 < g i
⊢ MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s

Tactic:
rw [monovaryOn_inv_left₀ hf, antivaryOn_inv_right₀ hg]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4
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
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ w i * z i = w i * rexp (log (z i))

Tactic:
· rcases eq_or_lt_of_le (hz i hi) with hz | hz
      · simp [A i hi hz.symm]
      · rw [exp_log hz]

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

Goal state before:
pp: case pos
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
hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0
hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0
hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ∞
hg_top : ∫⁻ (a : α), g a ^ q ∂μ = ∞
⊢ ∫⁻ (a : α), (g * f) a ∂μ ≤ (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p)

Tactic:
exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.symm.pos hpq.nonneg hg_top hf_zero

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

Goal state before:
pp: case hpq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ p i₀ + (1 - p i₀) = 1

Tactic:
apply add_sub_cancel

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

Goal state before:
pp: α : Type u_2
inst✝ : MeasurableSpace α
p q r : ℝ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp0_ne : p ≠ 0
hp0 : 0 ≤ p
hq0_lt : 0 < q
hq0_ne : q ≠ 0
h_one_div_r : 1 / r = 1 / p - 1 / q
p2 : ℝ := q / p
q2 : ℝ := p2.conjExponent
hp2q2 : p2.HolderConjugate q2
⊢ 0 ≤ 1 / p

Tactic:
by simp [hp0]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
hpq : p.HolderConjugate p.conjExponent
this :
  IsGreatest ((fun a ↦ ∑ x ∈ s, f x * a x + ∑ x ∈ s, g x * a x) '' {g | ∑ i ∈ s, g i ^ p.conjExponent ≤ 1})
    ((∑ x ∈ s, (f x + g x) ^ p) ^ (1 / p))
φ : ι → ℝ≥0
hφ : φ ∈ {g | ∑ i ∈ s, g i ^ p.conjExponent ≤ 1}
H : (fun a ↦ ∑ x ∈ s, f x * a x + ∑ x ∈ s, g x * a x) φ = (∑ x ∈ s, (f x + g x) ^ p) ^ (1 / p)
⊢ (fun a ↦ ∑ x ∈ s, f x * a x + ∑ x ∈ s, g x * a x) φ ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)

Tactic:
exact add_le_add ((isGreatest_Lp s f hpq).2 ⟨φ, hφ, rfl⟩) ((isGreatest_Lp s g hpq).2 ⟨φ, hφ, rfl⟩)

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

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

Tactic:
refine cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) ?_ ?_

Goal state after:
case pos.refine_1
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → essSup (fun x ↦ ‖(f n - f m_1) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n x) (f m x) ≤ (B N).toReal

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

Goal state before:
pp: case pos.refine_1
α : Type u_1
m✝ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x ↦ ‖(f n - f m) x‖ₑ) μ < B N
x : α
n m N : ℕ
hx : ‖(f n - f m) x‖ₑ < B N
hnN : N ≤ n
hmN : N ≤ m
⊢ ‖f n x - f m x‖ₑ ≤ B N

Tactic:
exact hx.le

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Basic

Goal state before:
pp: R : Type u
ι : Type v
M₁ : ι → Type w₁
M₂ : Type w₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
inst✝¹ : (i : ι) → TopologicalSpace (M₁ i)
inst✝ : TopologicalSpace M₂
f : MultilinearMap R M₁ M₂
hf : Continuous f.toFun
g : MultilinearMap R M₁ M₂
hg : Continuous g.toFun
h : { toMultilinearMap := f, cont := hf }.toMultilinearMap = { toMultilinearMap := g, cont := hg }.toMultilinearMap
⊢ { toMultilinearMap := f, cont := hf } = { toMultilinearMap := g, cont := hg }

Tactic:
subst h

Goal state after:
R : Type u
ι : Type v
M₁ : ι → Type w₁
M₂ : Type w₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
inst✝¹ : (i : ι) → TopologicalSpace (M₁ i)
inst✝ : TopologicalSpace M₂
f : MultilinearMap R M₁ M₂
hf : Continuous f.toFun
hg : Continuous { toMultilinearMap := f, cont := hf }.toFun
⊢ { toMultilinearMap := f, cont := hf } =
    { toMultilinearMap := { toMultilinearMap := f, cont := hf }.toMultilinearMap, cont := hg }
Module: Mathlib.Analysis.Normed.Operator.Mul

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : NormedAlgebra 𝕜 R
inst✝³ : Module R E
inst✝² : NormSMulClass R E
inst✝¹ : IsScalarTower 𝕜 R E
inst✝ : Nontrivial E
N : ℝ
x✝ : N ≥ 0
h : ∀ (x : R), ‖(lsmul 𝕜 R) x‖ ≤ N * ‖x‖
⊢ 1 ≤ N

Tactic:
obtain ⟨y, hy⟩ := exists_ne (0 : E)

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : NormedAlgebra 𝕜 R
inst✝³ : Module R E
inst✝² : NormSMulClass R E
inst✝¹ : IsScalarTower 𝕜 R E
inst✝ : Nontrivial E
N : ℝ
x✝ : N ≥ 0
h : ∀ (x : R), ‖(lsmul 𝕜 R) x‖ ≤ N * ‖x‖
y : E
hy : y ≠ 0
⊢ 1 ≤ N
Module: Mathlib.Analysis.Normed.Operator.Mul

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : NormedAlgebra 𝕜 R
inst✝³ : Module R E
inst✝² : NormSMulClass R E
inst✝¹ : IsScalarTower 𝕜 R E
inst✝ : Nontrivial E
a : R
N : ℝ
x✝ : N ≥ 0
h : ∀ (x : E), ‖((lsmul 𝕜 R) a) x‖ ≤ N * ‖x‖
⊢ ‖a‖ ≤ N

Tactic:
obtain ⟨y, hy⟩ := exists_ne (0 : E)

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : NormedAlgebra 𝕜 R
inst✝³ : Module R E
inst✝² : NormSMulClass R E
inst✝¹ : IsScalarTower 𝕜 R E
inst✝ : Nontrivial E
a : R
N : ℝ
x✝ : N ≥ 0
h : ∀ (x : E), ‖((lsmul 𝕜 R) a) x‖ ≤ N * ‖x‖
y : E
hy : y ≠ 0
⊢ ‖a‖ ≤ N
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 : Measurable f
g : α → E
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable (fun x ↦ ↑(f x) • g x) μ

Tactic:
by
  by_cases H : AEStronglyMeasurable (fun x : α => (f x : ℝ) • g x) μ
  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, hasFiniteIntegral_iff_enorm, H,
      true_and]
    rw [lintegral_withDensity_eq_lintegral_mul₀' hf.coe_nnreal_ennreal.aemeasurable]
    · simp [enorm_smul]
    · simpa [aemeasurable_withDensity_ennreal_iff hf, enorm_smul] using H.enorm
  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G
⊢ IsLeast {c | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i, ‖m i‖} ‖f‖

Tactic:
refine IsClosed.isLeast_csInf ?_ bounds_nonempty bounds_bddBelow
  simp only [Set.ofPred_and, Set.ofPred_forall]
  exact isClosed_Ici.inter (isClosed_iInter fun m ↦ isClosed_le continuous_const (by fun_prop))

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 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
ε : ℝ
hε : ∀ x ∈ (s, r).1, ‖x‖ ≤ ε
δ : ℝ
hδ₀ : 0 < δ
hδ : ε ^ Fintype.card ι * δ < (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)}

Tactic:
refine ⟨δ, hδ₀, fun f hf x hx ↦ ?_⟩

Goal state after:
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
ε : ℝ
hε : ∀ x ∈ (s, r).1, ‖x‖ ≤ ε
δ : ℝ
hδ₀ : 0 < δ
hδ : ε ^ Fintype.card ι * δ < (s, r).2
f : ContinuousMultilinearMap 𝕜 E G
hf : f ∈ (ContinuousMultilinearMap.seminorm 𝕜 E G).ball 0 δ
x : (i : ι) → E i
hx : x ∈ (s, r).1
⊢ f x ∈ closedBall 0 (s, r).2
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
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen[inst✝³] (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) μ
this : s ⊆ ⋃ x, u x
T : Set ↑s
hT_count : T.Countable
hT_un : ⋃ i ∈ T, u i = ⋃ i, u i
⊢ s ⊆ ⋃ i ∈ T, u i

Tactic:
by rwa [hT_un]

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

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
h_ind : ∀ (n : ℕ), U ⊆ ↑M + c ^ n • U
h_small : Tendsto (fun n ↦ c ^ n • U) atTop (𝓝 0).smallSets
x : Eᵤ
hx : x ∈ U
m : ℕ → Eᵤ
hm : ∀ (n : ℕ), m n ∈ ↑M
u : ℕ → Eᵤ
hu : ∀ (n : ℕ), u n ∈ c ^ n • U
h_eq : ∀ (n : ℕ), (fun x1 x2 ↦ x1 + x2) (m n) (u n) = x
hu_tendsto : Tendsto u atTop (𝓝 0)
hm_tendsto : Tendsto m atTop (𝓝 x)
⊢ x ∈ ↑M

Tactic:
exact M.closed_of_finiteDimensional.mem_of_tendsto hm_tendsto (Eventually.of_forall hm)

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ ∫⁻ (a : α), ‖↑↑(f - g) a‖ₑ ∂μ = ∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ

Tactic:
lintegral_congr_ae

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ ∫⁻ (a : α), ?m.52 a ∂μ = ∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ (fun a ↦ ‖↑↑(f - g) a‖ₑ) =ᵐ[μ] ?m.52

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ α → ℝ≥0∞
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')
x : G
m : (i : ι) → E i
⊢ ‖({ toFun := fun x ↦ { toFun := fun m ↦ (f m) x, map_update_add' := ⋯, map_update_smul' := ⋯ }, map_add' := ⋯ } x)
        m‖ ≤
    ‖f‖ * ‖x‖ * ∏ i, ‖m i‖

Tactic:
AddHom.coe_mk,

Goal state after:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')
x : G
m : (i : ι) → E i
⊢ ‖{ toFun := fun m ↦ (f m) x, map_update_add' := ⋯, map_update_smul' := ⋯ } m‖ ≤ ‖f‖ * ‖x‖ * ∏ i, ‖m i‖
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_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
f : ↥(simpleFunc E p μ)
⊢ P f

Tactic:
by
  suffices ∀ f : α →ₛ E, ∀ hf : MemLp f p μ, P (toLp f hf) by
    rw [← toLp_toSimpleFunc f]
    apply this
  clear f
  apply SimpleFunc.induction
  · intro c s hs hf
    by_cases hc : c = 0
    · convert! indicatorConst 0 MeasurableSet.empty (by simp) using 1
      ext1
      simp [hc]
    exact indicatorConst c hs
      (SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs hf)
  · intro f g hfg hf hg hfg'
    obtain ⟨hf', hg'⟩ : MemLp f p μ ∧ MemLp g p μ :=
      (memLp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable).mp hfg'
    exact add hf' hg' hfg (hf hf') (hg hg')

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

Goal state before:
pp: case refine_2
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
y : F
hy : y ∈ {x ∈ f.range | x ≠ 0}
⊢ μ.real (⇑(piecewise s hs f 0) ⁻¹' {y}) • y = (μ.restrict s).real (⇑f ⁻¹' {y}) • y

Tactic:
dsimp
    rw [Set.piecewise_eq_indicator, indicator_preimage_of_notMem,
      measureReal_restrict_apply (f.measurableSet_preimage _)]
    exact fun h₀ => (mem_filter.1 hy).2 (Eq.symm h₀)

Goal state after: