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

Goal state before:
pp: z₁ z₂ : ℝ≥0∞
p : ℝ
hp : 1 ≤ p
⊢ (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)

Tactic:
convert!
    rpow_arith_mean_le_arith_mean2_rpow (1 / 2) (1 / 2) (2 * z₁) (2 * z₂) (ENNReal.add_halves 1) hp
      using 1

Goal state after:
case e'_3
z₁ z₂ : ℝ≥0∞
p : ℝ
hp : 1 ≤ p
⊢ (z₁ + z₂) ^ p = (1 / 2 * (2 * z₁) + 1 / 2 * (2 * z₂)) ^ p

case e'_4
z₁ z₂ : ℝ≥0∞
p : ℝ
hp : 1 ≤ p
⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

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

Tactic:
rw [one_div, ← @ENNReal.le_rpow_inv_iff _ _ p⁻¹ (by simp [lt_of_lt_of_le zero_lt_one hp1])]

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

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : (a ^ 1 + b ^ 1) ^ 1 ≤ (a ^ p + b ^ p) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
one_div

Goal state after:
case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : (a ^ 1 + b ^ 1) ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: w₁ w₂ p₁ p₂ : ℝ
hw₁ : 0 < w₁
hw₂ : 0 < w₂
hp₁ : 0 ≤ p₁
hp₂ : 0 ≤ p₂
hw : w₁ + w₂ = 1
⊢ p₁ ^ w₁ * p₂ ^ w₂ = w₁ * p₁ + w₂ * p₂ ↔ p₁ = p₂

Tactic:
have := geom_mean_eq_arith_mean_weighted_iff_of_pos univ ![w₁, w₂] ![p₁, p₂]

Goal state after:
w₁ w₂ p₁ p₂ : ℝ
hw₁ : 0 < w₁
hw₂ : 0 < w₂
hp₁ : 0 ≤ p₁
hp₂ : 0 ≤ p₂
hw : w₁ + w₂ = 1
this :
  (∀ i ∈ univ, 0 < ![w₁, w₂] i) →
    ∑ i, ![w₁, w₂] i = 1 →
      (∀ i ∈ univ, 0 ≤ ![p₁, p₂] i) →
        (∏ i, ![p₁, p₂] i ^ ![w₁, w₂] i = ∑ i, ![w₁, w₂] i * ![p₁, p₂] i ↔
          ∀ j ∈ univ, ∀ k ∈ univ, ![p₁, p₂] j = ![p₁, p₂] k)
⊢ p₁ ^ w₁ * p₂ ^ w₂ = w₁ * p₁ + w₂ * p₂ ↔ p₁ = p₂
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case refine_3.none
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
s : Finset ι
g : α → ℝ≥0∞
f : ι → α → ℝ≥0∞
hg : AEMeasurable g μ
hf : ∀ i ∈ s, AEMeasurable (f i) μ
q : ℝ
p : ι → ℝ
hpq : q + ∑ i ∈ s, p i = 1
hq : 0 ≤ q
hp : ∀ i ∈ s, 0 ≤ p i
hi : none ∈ insertNone s
⊢ 0 ≤ none.elim q p

Tactic:
exact hq

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
⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =
    (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)

Tactic:
have hpp2 : p * p2 = q := by
        symm
        rw [mul_comm, ← div_eq_iff hp0_ne]

Goal state after:
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
hpp2 : p * p2 = q
⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =
    (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case h₂.inr.inr
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
p : ℝ≥0∞
f : α → E
g : α → F
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‖₊
hp : 0 < p.toReal
hp₁ : 0 < p
hp₂ : p < ∞
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) p μ ≤ eLpNorm f ∞ μ * eLpNorm g p μ

Tactic:
simp_rw [eLpNorm_eq_lintegral_rpow_enorm_toReal hp₁.ne' hp₂.ne, eLpNorm_exponent_top,
    eLpNormEssSup, one_div, ENNReal.rpow_inv_le_iff hp, enorm_mul, enorm_norm]

Goal state after:
case h₂.inr.inr
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
p : ℝ≥0∞
f : α → E
g : α → F
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‖₊
hp : 0 < p.toReal
hp₁ : 0 < p
hp₂ : p < ∞
⊢ ∫⁻ (x : α), (‖f x‖ₑ * ‖g x‖ₑ) ^ p.toReal ∂μ ≤
    (essSup (fun x ↦ ‖f x‖ₑ) μ * (∫⁻ (x : α), ‖g x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹) ^ p.toReal
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
i : ι
hi : i ∈ s
⊢ (f i + g i) ^ p = |f i + g i| ^ p

Tactic:
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4.e'_5.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
i : ι
hi : i ∈ s
⊢ f i ^ p = |f i| ^ p

Tactic:
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4.e'_6.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
i : ι
hi : i ∈ s
⊢ g i ^ p = |g i| ^ p

Tactic:
simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]

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
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
n : ℕ
hn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
⊢ (∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i

Tactic:
one_div

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
n : ℕ
hp_pos : 0 < p
hn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
⊢ (∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
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
f : ↥(Lp E p μ)
x : α
hx : ↑↑((c • B g).compLp f) x = (c • B g) (↑↑f x)
hg : ↑↑((B g).compLp f) x = (B g) (↑↑f x)
hsmul : ↑↑(c • (B g).compLp f) x = (c • ↑↑((B g).compLp f)) x
⊢ ↑↑((compLpₗ p μ (B (c • g))) f) x = ↑↑(((RingHom.id 𝕜) c • compLpₗ p μ (B g)) f) x

Tactic:
simp [hx, hsmul, hg]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
f g₁ g₂ : ↥(Lp E p μ)
hg₁₂ : ↑↑g₁ ≤ᵐ[μ] ↑↑g₂
⊢ ↑↑(f + g₁) ≤ᵐ[μ] ↑↑(f + g₂)

Tactic:
filter_upwards [coeFn_add f g₁, coeFn_add f g₂, hg₁₂] with _ h1 h2 h3

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
f g₁ g₂ : ↥(Lp E p μ)
hg₁₂ : ↑↑g₁ ≤ᵐ[μ] ↑↑g₂
a✝ : α
h1 : ↑↑(f + g₁) a✝ = (↑↑f + ↑↑g₁) a✝
h2 : ↑↑(f + g₂) a✝ = (↑↑f + ↑↑g₂) a✝
h3 : ↑↑g₁ a✝ ≤ ↑↑g₂ a✝
⊢ ↑↑(f + g₁) a✝ ≤ ↑↑(f + g₂) a✝
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✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
R : Type u_6
inst✝² : NormedAddCommGroup R
inst✝¹ : StarAddMonoid R
inst✝ : NormedStarGroup R
p : ℝ≥0∞
f : ↥(Lp R p μ)
⊢ star ↑f ∈ Lp R p μ

Tactic:
simpa [Lp.mem_Lp_iff_eLpNorm_lt_top] using Lp.eLpNorm_lt_top f

Goal state after:
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_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

Tactic:
intro n m N hnN hmN
      specialize hx N n m hnN hmN
      rw [_root_.dist_eq_norm,
        ← ENNReal.ofReal_le_iff_le_toReal (ENNReal.ne_top_of_tsum_ne_top hB N),
        ofReal_norm]
      exact hx.le

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁴ : SeminormedRing R
inst✝³ : NormedAlgebra 𝕜 R
inst✝² : Module R E
inst✝¹ : IsBoundedSMul R E
inst✝ : IsScalarTower 𝕜 R E
⊢ ‖lsmul 𝕜 R‖₊ ≤ 1

Tactic:
by
  rw [← NNReal.coe_le_coe]
  simpa using opNorm_lsmul_le

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

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

Tactic:
induction s using Finset.induction_on with
  | empty => simp
  | insert _ _ hx IH =>
    simp only [hx, Finset.mem_insert, iUnion_iUnion_eq_or_left, not_false_eq_true,
      Finset.sum_insert]
    exact (measureReal_union_le _ _).trans (by gcongr)

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

Goal state before:
pp: case pos
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
hp_pos : p ≠ 0
hμs_pos : μ s ≠ 0
hp_top : p = ∞
⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * μ.real s ^ (1 / p.toReal)

pp: case neg
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
hp_pos : p ≠ 0
hμs_pos : μ s ≠ 0
hp_top : ¬p = ∞
⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * μ.real s ^ (1 / p.toReal)

Tactic:
· rw [hp_top, ENNReal.toReal_top, _root_.div_zero, Real.rpow_zero, mul_one]
    exact norm_indicatorConstLp_top hμs_pos

Goal state after:
case neg
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
hp_pos : p ≠ 0
hμs_pos : μ s ≠ 0
hp_top : ¬p = ∞
⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * μ.real s ^ (1 / p.toReal)
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
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ eLpNorm ((s ∆ t).indicator fun x ↦ c) p μ = ‖MemLp.toLp ((s ∆ t).indicator fun x ↦ c) ⋯‖ₑ

Tactic:
Lp.enorm_toLp

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
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ eLpNorm ((s ∆ t).indicator fun x ↦ c) p μ = eLpNorm ((s ∆ t).indicator fun x ↦ c) p μ
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case refine_2
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
⊢ 𝓝 0 ≤ 𝓝 0

Tactic:
· -- Finally, to show `𝓣₀ ≤ 𝓣`, we simply argue that `id = (fun x ↦ x • 1)` is continuous from
    -- `(𝕜, 𝓣₀)` to `(𝕜, 𝓣)` because `(•) : (𝕜, 𝓣₀) × (𝕜, 𝓣) → (𝕜, 𝓣)` is continuous.
    calc
      @nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0 =
          map id (@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0) :=
        map_id.symm
      _ = map (fun x => id x • (1 : 𝕜)) (@nhds 𝕜 hnorm.toUniformSpace.toTopologicalSpace 0) := by
        simp
      _ ≤ @nhds 𝕜 t ((0 : 𝕜) • (1 : 𝕜)) :=
        (@Tendsto.smul_const _ _ _ hnorm.toUniformSpace.toTopologicalSpace t _ _ _ _ _
          tendsto_id (1 : 𝕜))
      _ = @nhds 𝕜 t 0 := by rw [zero_smul]

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
ν : Measure δ
g : δ → ε
f : α → δ
hf : MeasurePreserving f μ ν
hg : AEStronglyMeasurable g ν
⊢ Integrable (g ∘ f) μ ↔ Integrable g ν

Tactic:
rw [← hf.map_eq] at hg ⊢
  exact (integrable_map_measure hg hf.measurable.aemeasurable).symm

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

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

Tactic:
by exact this _ rfl

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
E : Type u_4
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
f : E →ₗ[𝕜] F
s : Submodule 𝕜 E
inst✝ : s.CoFG
h : IsClosed[inst✝⁶] ↑(Submodule.map f s)
t : Submodule 𝕜 E
s_compl_t : IsCompl s t
this : FiniteDimensional 𝕜 ↥t
⊢ IsClosed[inst✝⁶] ↑f.range

Tactic:
← Submodule.map_top,

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
E : Type u_4
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
f : E →ₗ[𝕜] F
s : Submodule 𝕜 E
inst✝ : s.CoFG
h : IsClosed[inst✝⁶] ↑(Submodule.map f s)
t : Submodule 𝕜 E
s_compl_t : IsCompl s t
this : FiniteDimensional 𝕜 ↥t
⊢ IsClosed[inst✝⁶] ↑(Submodule.map f ⊤)
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
ha : ‖f a‖ₑ ≠ ∞
⊢ μ {a} ≠ ∞

Tactic:
measure_singleton

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
ha : ‖f a‖ₑ ≠ ∞
⊢ 0 ≠ ∞
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
hb : ‖f b‖ₑ ≠ ∞
⊢ μ {b} ≠ ∞

Tactic:
measure_singleton

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
hb : ‖f b‖ₑ ≠ ∞
⊢ 0 ≠ ∞
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
inst✝ : NullSingletonClass μ
hb : ‖f b‖ₑ ≠ ∞
⊢ μ {b} ≠ ∞

Tactic:
measure_singleton

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
inst✝ : NullSingletonClass μ
hb : ‖f b‖ₑ ≠ ∞
⊢ 0 ≠ ∞
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case refine_1
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
T : Set ↑s
T_count : T.Countable
hT : s ⊆ ⋃ i ∈ T, u i
w : ↑s
⊢ IsOpen[inst✝³] (u w)

Tactic:
exact u_open _

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
f : Eᵤ
hf : f ∈ F
y : Eᵤ
hy : y ∈ c • U
hx : (fun x ↦ f +ᵥ x) y ∈ U
⊢ (fun x ↦ f +ᵥ x) y ∈ ↑M + c • U

Tactic:
exact ⟨f, Submodule.subset_span hf, y, hy, rfl⟩

Goal state after:
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 y : G
⊢ { toFun := fun m ↦ (f m) (x + y), map_update_add' := ⋯, map_update_smul' := ⋯ } =
    { toFun := fun m ↦ (f m) x, map_update_add' := ⋯, map_update_smul' := ⋯ } +
      { toFun := fun m ↦ (f m) y, map_update_add' := ⋯, map_update_smul' := ⋯ }

Tactic:
ext1; simp

Goal state after:
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')
c : 𝕜
x : G
⊢ { toFun := fun m ↦ (f m) (c • x), map_update_add' := ⋯, map_update_smul' := ⋯ } =
    (RingHom.id 𝕜) c • { toFun := fun m ↦ (f m) x, map_update_add' := ⋯, map_update_smul' := ⋯ }

Tactic:
ext1; simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
⊢ ⇑(toSimpleFunc (negPart f)) =ᵐ[μ] ⇑(toSimpleFunc f).negPart

Tactic:
by
  rw [SimpleFunc.negPart, MeasureTheory.SimpleFunc.negPart]
  filter_upwards [posPart_toSimpleFunc (-f), neg_toSimpleFunc f]
  intro a h₁ h₂
  rw [h₁]
  change max _ _ = max _ _
  rw [h₂]
  simp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
hdp : 0 < d
r' : ℝ := max r 2⁻¹
hr' : r' < 1
hlt : 0 < r'
hdlt : d < d / r'
y₀ : E
hy₀F : y₀ ∈ F
hxy₀ : dist x y₀ < d / r'
⊢ x - y₀ ∉ F

Tactic:
by_contra h

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
hdp : 0 < d
r' : ℝ := max r 2⁻¹
hr' : r' < 1
hlt : 0 < r'
hdlt : d < d / r'
y₀ : E
hy₀F : y₀ ∈ F
hxy₀ : dist x y₀ < d / r'
h : x - y₀ ∈ F
⊢ False