mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε'' : Type u_8
inst✝¹ : TopologicalSpace ε''
inst✝ : ESeminormedAddMonoid ε''
c : ε''
hc' : ‖c‖ₑ ≠ ∞
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hp : 0 < p.toReal
hμ : μ = 0
⊢ eLpNorm (fun x ↦ c) p μ < ∞ ↔ ‖c‖ₑ = 0 ∨ μ Set.univ < ∞

Tactic:
simp only [hμ, Measure.coe_zero, Pi.zero_apply, or_true, ENNReal.zero_lt_top,
      eLpNorm_measure_zero]

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

Goal state before:
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:
eLpNorm_eq_eLpNorm' h0 h_top

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.toReal μ ≤ c • eLpNorm' g p.toReal μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: 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
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c * ‖g x‖
hc : c < 0
p : ℝ≥0∞
⊢ eLpNorm f p μ = 0 ∧ eLpNorm g p μ = 0

Tactic:
simp_rw [le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg (norm_nonneg _) hc (norm_nonneg _),
    norm_eq_zero, eventually_and] at h

Goal state after:
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
hc : c < 0
p : ℝ≥0∞
h : (∀ᵐ (x : α) ∂μ, f x = 0) ∧ ∀ᵐ (x : α) ∂μ, g x = 0
⊢ eLpNorm f p μ = 0 ∧ eLpNorm g p μ = 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
c : ε'
hc : ‖c‖ₑ ≠ ∞
⊢ eLpNorm (fun x ↦ c) ∞ μ < ∞

Tactic:
by_cases h : μ = 0

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
c : ε'
hc : ‖c‖ₑ ≠ ∞
h : μ = 0
⊢ eLpNorm (fun x ↦ c) ∞ μ < ∞

case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
c : ε'
hc : ‖c‖ₑ ≠ ∞
h : ¬μ = 0
⊢ eLpNorm (fun x ↦ c) ∞ μ < ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
𝕜 : Type u_5
inst✝ : RCLike 𝕜
f : α → 𝕜
hf : MemLp f p μ
⊢ MemLp (fun x ↦ RCLike.re (f x)) p μ

Tactic:
have : ∀ x, ‖RCLike.re (f x)‖ ≤ 1 * ‖f x‖ := by
    intro x
    rw [one_mul]
    exact RCLike.norm_re_le_norm (f x)

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
𝕜 : Type u_5
inst✝ : RCLike 𝕜
f : α → 𝕜
hf : MemLp f p μ
this : ∀ (x : α), ‖RCLike.re (f x)‖ ≤ 1 * ‖f x‖
⊢ MemLp (fun x ↦ RCLike.re (f x)) p μ
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 : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
⊢ (f z - f x) / (z - x) < (f z - f y) / (z - y)

Tactic:
have hyz' : 0 < z - y := by linarith
  have hxz' : 0 < z - x := by linarith
  field_simp
  linarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz]

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

Goal state before:
pp: p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < z - y
⊢ p * y ^ (p - 1) * (z - y) / y ^ p < (z / y) ^ p - 1

Tactic:
lt_sub_iff_add_lt',

Goal state after:
p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < z - y
⊢ 1 + p * y ^ (p - 1) * (z - y) / y ^ p < (z / y) ^ p
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : α → ε
hf : AEStronglyMeasurable f μ
h0 : p ≠ 0
h_top : ¬p = ∞
⊢ eLpNorm f p μ = 0 ↔ f =ᵐ[μ] 0

Tactic:
eLpNorm_eq_eLpNorm' h0 h_top

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : α → ε
hf : AEStronglyMeasurable f μ
h0 : p ≠ 0
h_top : ¬p = ∞
⊢ eLpNorm' f p.toReal μ = 0 ↔ f =ᵐ[μ] 0
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
f : ι → α
g : ι → β
hf : StrongLT 0 f
hg : StrongLT 0 g
⊢ Antivary f⁻¹ g⁻¹ ↔ Antivary f g

Tactic:
antivary_inv_left₀ hf,

Goal state after:
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
f : ι → α
g : ι → β
hf : StrongLT 0 f
hg : StrongLT 0 g
⊢ Monovary f g⁻¹ ↔ Antivary f g
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p⁻¹⁻¹

Tactic:
inv_inv

Goal state after:
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p⁻¹⁻¹

Tactic:
inv_inv

Goal state after:
p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: z₁ z₂ p : ℝ≥0∞
p_zero : ¬p = 0
⊢ p = p⁻¹⁻¹

Tactic:
inv_inv

Goal state after:
z₁ z₂ p : ℝ≥0∞
p_zero : ¬p = 0
⊢ p = p
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ ∫⁻ (a : α), liminf (fun x ↦ ‖f x a‖ₑ ^ p) atTop ∂μ ≤ liminf (fun n ↦ eLpNorm' (f n) p μ) atTop ^ p⁻¹⁻¹

Tactic:
inv_inv

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ ∫⁻ (a : α), liminf (fun x ↦ ‖f x a‖ₑ ^ p) atTop ∂μ ≤ liminf (fun n ↦ eLpNorm' (f n) p μ) atTop ^ p
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

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

Tactic:
simp_rw [Pi.mul_apply, ENNReal.coe_mul]
  exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf.coe_nnreal_ennreal hg.coe_nnreal_ennreal

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_pow : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
⊢ ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p⁻¹⁻¹

Tactic:
inv_inv

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_pow : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
⊢ ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: p : ℝ≥0∞
α : Type u_6
E : Type u_7
F : Type u_8
K : ℝ≥0
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : α → E
g : E → F
hg : LipschitzWith K g
g0 : g 0 = 0
hL : MemLp f p μ
x : α
⊢ ‖g (f x)‖ ≤ ↑K * ‖f x‖

Tactic:
by
    -- TODO: add `LipschitzWith.nnnorm_sub_le` and `LipschitzWith.nnnorm_le`
    simpa [g0] using hg.norm_sub_le (f x) 0

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p⁻¹⁻¹
hp_pos : 0 < p
h_tsum_lt_top : (∑' (i : ℕ), B i) ^ p < ∞
⊢ ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p

Tactic:
inv_inv

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_lt_top : (∑' (i : ℕ), B i) ^ p < ∞
⊢ ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
⊢ ∀ᵐ (x : α) ∂μ, ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞

Tactic:
have hp_pos : 0 < p := zero_lt_one.trans_le hp1
  have h_integral : ∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞ := by
    have h_tsum_lt_top : (∑' i, B i) ^ p < ∞ := ENNReal.rpow_lt_top_of_nonneg hp_pos.le hB
    refine lt_of_le_of_lt ?_ h_tsum_lt_top
    rwa [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv] at h
  have rpow_ae_lt_top : ∀ᵐ x ∂μ, (∑' i, ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞ := by
    refine ae_lt_top' (AEMeasurable.pow_const ?_ _) h_integral.ne
    exact AEMeasurable.tsum fun n => ((hf (n + 1)).sub (hf n)).enorm
  refine rpow_ae_lt_top.mono fun x hx => ?_
  rwa [← ENNReal.lt_rpow_inv_iff hp_pos,
    ENNReal.top_rpow_of_pos (by simp [hp_pos] : 0 < p⁻¹)] at hx

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

Goal state before:
pp: α : Type u_1
𝕜✝ : Type u_2
𝕜'✝ : Type u_3
E : Type u_4
F✝ : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F✝
g✝ : E → F✝
c✝ : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜'
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' F✝
σ : 𝕜 →+* 𝕜'
inst✝⁴ : RingHomIsometric σ
F : Type u_8
G : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] E →L[𝕜] F
c : 𝕜
g : G
⊢ compLpₗ p μ (B (c • g)) = (RingHom.id 𝕜) c • compLpₗ p μ (B g)

Tactic:
by
    ext f
    filter_upwards [(c • B g).coeFn_compLp f, (B g).coeFn_compLp f,
      Lp.coeFn_smul c ((B g).compLp f)] with x hx hg hsmul
    simp [hx, hsmul, hg]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
ι : Type u_5
inst✝² : SeminormedAddGroup E
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → E
g : α → E
C p : ℝ≥0∞
bound : ∀ᶠ (i : ι) in u, eLpNorm (f i) p μ ≤ C
h_tendsto : TendstoInMeasure μ f u g
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
⊢ eLpNorm g p μ ≤ C

Tactic:
obtain ⟨l, hl⟩ := h_tendsto.exists_seq_tendsto_ae'
  exact Lp.eLpNorm_le_of_ae_tendsto (hl.1.eventually bound) (fun n => hf (l n)) hl.2

Goal state after:
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
c : 𝕜
⊢ ‖(toSpanSingleton 𝕜 E x) c‖ = ‖x‖ * ‖c‖

Tactic:
rw [mul_comm]
  exact norm_smul _ _

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Mul

Goal state before:
pp: 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
⊢ ‖lsmul 𝕜 R‖₊ = 1

Tactic:
by
  rw [← NNReal.coe_inj]
  simp

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Mul

Goal state before:
pp: 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
⊢ ‖(lsmul 𝕜 R) a‖₊ = ‖a‖₊

Tactic:
by
  rw [← NNReal.coe_inj]
  simp

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✝⁸ : NormedField 𝕜
inst✝⁷ : (i : ι) → TopologicalSpace (E i)
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : DecidableEq ι
inst✝¹ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
hcont : Continuous[Pi.topologicalSpace, inst✝¹] ((toFun {s | IsVonNBounded 𝕜 s}) f)
hadd :
  ∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
      (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)
hsmul :
  ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
    (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) = c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)
inst✝ : DecidableEq ι
⊢ ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i), f (update m i (c • x)) = c • f (update m i x)

Tactic:
convert! hsmul

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
x : E × F
⊢ ‖(LinearMap.fst 𝕜 E F) x‖ ≤ ‖x‖

Tactic:
exact le_max_left _ _

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsFiniteMeasure μ
hs : NullMeasurableSet s μ
⊢ μ.real s + μ.real sᶜ = μ.real univ

Tactic:
by
  rw [← measureReal_union₀' hs aedisjoint_compl_right, union_compl_self]

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsFiniteMeasure μ
h₁ : NullMeasurableSet s μ
⊢ μ.real sᶜ = μ.real univ - μ.real s

Tactic:
by
  linarith [measureReal_add_measureReal_compl₀ h₁]

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

Goal state before:
pp: α : Type u_1
ι : Type u_3
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (μ univ).toReal < ∑ i ∈ s, (μ (t i)).toReal
⊢ ∑ i ∈ s, μ (t i) ≠ ∞

Tactic:
exact (ENNReal.sum_lt_top.mpr (fun i hi ↦ measure_lt_top ..)).ne

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

Goal state before:
pp: case inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
η : ℝ≥0
hη_pos : 0 < η
hη_le : ‖c‖ₑ * ↑η ^ (1 / p.toReal) ≤ ε
⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun x ↦ c) p μ ≤ ε

Tactic:
refine ⟨η, hη_pos, fun s hs => ?_⟩

Goal state after:
case inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
η : ℝ≥0
hη_pos : 0 < η
hη_le : ‖c‖ₑ * ↑η ^ (1 / p.toReal) ≤ ε
s : Set α
hs : μ s ≤ ↑η
⊢ eLpNorm (s.indicator fun x ↦ c) p μ ≤ ε
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
c : E
⊢ indicatorConstLp p ⋯ ⋯ c = 0

Tactic:
simp only [indicatorConstLp, Set.indicator_empty', MemLp.toLp_zero]

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : IsFiniteMeasure μ
f : α → ε
p : ℝ≥0∞
hf : MemLp f p μ
⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ

Tactic:
by_cases h_zero : p = 0

Goal state after:
case pos
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : IsFiniteMeasure μ
f : α → ε
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : p = 0
⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ

case neg
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : IsFiniteMeasure μ
f : α → ε
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : ¬p = 0
⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ
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✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
p : ℝ≥0∞
hf : MemLp f p μ
⊢ Integrable (fun x ↦ ‖f x‖ ^ p.toReal) μ

Tactic:
by_cases h_zero : p = 0

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : p = 0
⊢ Integrable (fun x ↦ ‖f x‖ ^ p.toReal) μ

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : ¬p = 0
⊢ Integrable (fun x ↦ ‖f x‖ ^ p.toReal) μ
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case pos
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
⊢ ξ ∈ Metric.closedBall 0 ε

Tactic:
hξ0

Goal state after:
case pos
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
⊢ 0 ∈ Metric.closedBall 0 ε
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✝³ : NormedAddCommGroup β
𝕜 : Type u_8
inst✝² : NormedRing 𝕜
inst✝¹ : MulActionWithZero 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
f : α → 𝕜
hf : Integrable f μ
g : α → β
g_aestronglyMeasurable : AEStronglyMeasurable g μ
ess_sup_g : essSup (fun x ↦ ‖g x‖ₑ) μ ≠ ∞
⊢ Integrable (fun x ↦ f x • g x) μ

Tactic:
rw [← memLp_one_iff_integrable] at *
  refine ⟨hf.1.smul g_aestronglyMeasurable, ?_⟩
  have hg' : eLpNorm g ∞ μ ≠ ∞ := by rwa [eLpNorm_exponent_top]
  calc
    eLpNorm (fun x : α => f x • g x) 1 μ ≤ _ := by
      simpa using! MeasureTheory.eLpNorm_smul_le_mul_eLpNorm g_aestronglyMeasurable hf.1
        (p := 1) (q := ∞)
    _ < ∞ := ENNReal.mul_lt_top hf.2 hg'.lt_top

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ AEMeasurable f Measure.count

Tactic:
refine (measurable_zero.measurable_of_countable_ne ?_).aemeasurable
    simpa only [Ne, Pi.zero_apply, eq_comm, Function.support] using hs'

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ AEMeasurable f Measure.count

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂Measure.count, f x ∈ t

Tactic:
· refine (measurable_zero.measurable_of_countable_ne ?_).aemeasurable
    simpa only [Ne, Pi.zero_apply, eq_comm, Function.support] using hs'

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂Measure.count, f x ∈ t
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 : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hx : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
⊢ ‖f x • ↑↑u x‖ₑ = ((fun x ↦ ↑(f x)) * fun a ↦ ‖↑↑u a‖ₑ) x

Tactic:
simp [NNReal.smul_def, enorm_smul]

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
H : Type u_9
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup H
𝕜 : Type u_10
𝕜' : Type u_11
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜'
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : NormedSpace 𝕜 H
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : RingHomInvPair σ σ'
inst✝ : RingHomInvPair σ' σ
φ : α → H
L : H ≃SL[σ] E
h : Integrable (fun a ↦ L (φ a)) μ
⊢ Integrable φ μ

Tactic:
by simpa using ContinuousLinearMap.integrable_comp (L.symm : E →SL[σ'] H) h

Goal state after: