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

Goal state before:
pp: α : Type u_1
γ : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace γ
inst✝¹ : CommMonoid γ
inst✝ : ContinuousMul γ
ι : Type u_5
s : Finset ι
f : ι → α →ₘ[μ] γ
⊢ ↑(∏ i ∈ s, f i) =ᵐ[μ] fun x ↦ ∏ i ∈ s, ↑(f i) x

Tactic:
grw [coeFn_finsetProd]

Goal state after:
α : Type u_1
γ : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace γ
inst✝¹ : CommMonoid γ
inst✝ : ContinuousMul γ
ι : Type u_5
s : Finset ι
f : ι → α →ₘ[μ] γ
⊢ ∏ i ∈ s, ↑(f i) =ᵐ[μ] fun x ↦ ∏ i ∈ s, ↑(f i) x
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
⊢ eLpNormEssSup (s.indicator fun x ↦ c) μ ≤ ‖c‖ₑ

Tactic:
by
  obtain rfl | hμ0 := eq_or_ne μ 0
  · simp
  · exact (eLpNormEssSup_indicator_le s fun _ => c).trans (eLpNormEssSup_const c hμ0).le

Goal state after:
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≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ
hp : 0 < p
⊢ eLpNorm' f p μ ≤ c • eLpNorm' g p μ

Tactic:
eLpNorm'_eq_lintegral_enorm

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≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ
hp : 0 < p
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ c • (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)
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
⊢ eLpNorm' f p μ ≤ c • eLpNorm' g p μ

Tactic:
eLpNorm'_eq_lintegral_enorm

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) ^ (1 / p) ≤ c • (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : ¬c = ∞
this : c ^ p ≠ ∞
⊢ eLpNorm' f p μ ≤ c * eLpNorm' g p μ

Tactic:
eLpNorm'_eq_lintegral_enorm

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : ¬c = ∞
this : c ^ p ≠ ∞
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ c * (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : IsFiniteMeasure μ
c : F
hc_ne_zero : c ≠ 0
hq_ne_zero : q ≠ 0
⊢ (‖c‖ₑ ^ q) ^ (1 / q) * μ Set.univ ^ (1 / q) = ‖c‖ₑ * μ Set.univ ^ (1 / q)

Tactic:
congr
    rw [← ENNReal.rpow_mul]
    suffices hp_cancel : q * (1 / q) = 1 by rw [hp_cancel, ENNReal.rpow_one]
    rw [one_div, mul_inv_cancel₀ hq_ne_zero]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case e_6
ι : Type u_1
inst✝¹ : Fintype ι
X : ι → Type u_4
inst✝ : Unique ι
m : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
e : ((i : ι) → X i) ≃ᵐ X default := ⋯
s : Set ((i : ι) → X i)
⊢ eval default '' s = ⇑e.symm ⁻¹' s

Tactic:
exact e.toEquiv.image_eq_preimage_symm s

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: α₁ : Type u_4
β₁ : Type u_5
α₂ : Type u_6
β₂ : Type u_7
inst✝⁴ : Fintype α₁
inst✝³ : Fintype α₂
inst✝² : MeasurableSpace β₁
inst✝¹ : MeasurableSpace β₂
μ : α₁ → Measure β₁
ν : α₂ → Measure β₂
inst✝ : ∀ (i : α₂), SigmaFinite (ν i)
eα : α₁ ≃ α₂
eβ : β₁ ≃ᵐ β₂
hm : ∀ (i : α₁), MeasurePreserving (⇑eβ) (μ i) (ν (eα i))
⊢ MeasurePreserving (⇑(MeasurableEquiv.arrowCongr' eα eβ)) (Measure.pi fun i ↦ μ i) (Measure.pi fun i ↦ ν i)

Tactic:
by
  convert!
    (measurePreserving_piCongrLeft (fun i : α₂ ↦ ν i) eα).comp
      (measurePreserving_pi μ (fun i : α₁ ↦ ν (eα i)) hm)
  simp only [MeasurableEquiv.arrowCongr', Equiv.arrowCongr', Equiv.arrowCongr, EquivLike.coe_coe,
    comp_def, MeasurableEquiv.coe_mk, Equiv.coe_fn_mk, MeasurableEquiv.piCongrLeft,
    Equiv.piCongrLeft, Equiv.symm_symm, Equiv.piCongrLeft', eq_rec_constant, Equiv.coe_fn_symm_mk]

Goal state after:
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
s : Set α
f : α → ε
hsf : Function.support f ⊆ s
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm' f p.toReal (μ.restrict s) = eLpNorm' f p.toReal μ

Tactic:
eLpNorm'_eq_lintegral_enorm

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
s : Set α
f : α → ε
hsf : Function.support f ⊆ s
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ (∫⁻ (a : α) in s, ‖f a‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) = (∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal)
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 < y - x
⊢ 1 + -(p * ((y - x) / y)) < (1 + (x / y - 1)) ^ p

Tactic:
← mul_neg,

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 < y - x
⊢ 1 + p * -((y - x) / y) < (1 + (x / y - 1)) ^ p
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
hp0 : p.toReal = 0
a✝ : α
⊢ 1 < ∞

Tactic:
exact ENNReal.one_lt_top

Goal state after:
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
hf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
hg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0
npf : ℝ≥0∞ := (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)
nqg : ℝ≥0∞ := (∫⁻ (c : α), g c ^ q ∂μ) ^ (1 / q)
⊢ npf * nqg ≠ ∞

Tactic:
simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case neg
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
hp_pos : 0 < p
h_zero : ¬a + b = 0
h_nonzero : ¬(a = 0 ∧ b = 0)
h_add : a / (a + b) + b / (a + b) = 1
h : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p ≤ 1
hab_0 : (a + b) ^ p ≠ 0
h_mul : (a + b) ^ p * (a ^ p * ((a + b) ^ p)⁻¹ + b ^ p * ((a + b) ^ p)⁻¹) ≤ (a + b) ^ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Tactic:
mul_add,

Goal state after:
case neg
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
hp_pos : 0 < p
h_zero : ¬a + b = 0
h_nonzero : ¬(a = 0 ∧ b = 0)
h_add : a / (a + b) + b / (a + b) = 1
h : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p ≤ 1
hab_0 : (a + b) ^ p ≠ 0
h_mul : (a + b) ^ p * (a ^ p * ((a + b) ^ p)⁻¹) + (a + b) ^ p * (b ^ p * ((a + b) ^ p)⁻¹) ≤ (a + b) ^ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case refine_2
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞
⊢ (∑ i ∈ s, w i * z i) ^ p ≠ ∞ →
    ∑ i ∈ s, w i * z i ^ p ≠ ∞ → ((∑ i ∈ s, w i * z i) ^ p).toNNReal ≤ (∑ i ∈ s, w i * z i ^ p).toNNReal

Tactic:
intro h_top_rpow_sum _
    -- show hypotheses needed to put the `.toNNReal` inside the sums.
    have h_top : ∀ a : ι, a ∈ s → w a * z a ≠ ⊤ :=
      haveI h_top_sum : ∑ i ∈ s, w i * z i ≠ ⊤ := by
        intro h
        rw [h, top_rpow_of_pos hp_pos] at h_top_rpow_sum
        exact h_top_rpow_sum rfl
      fun a ha => (lt_top_of_sum_ne_top h_top_sum ha).ne
    have h_top_rpow : ∀ a : ι, a ∈ s → w a * z a ^ p ≠ ⊤ := by
      intro i hi
      specialize h_top i hi
      rwa [Ne, ← h_top_iff_rpow_top i hi]
    -- put the `.toNNReal` inside the sums.
    simp_rw [toNNReal_sum h_top_rpow, toNNReal_rpow, toNNReal_sum h_top, toNNReal_mul,
      toNNReal_rpow]
    -- use corresponding nnreal result
    refine
      NNReal.rpow_arith_mean_le_arith_mean_rpow s (fun i => (w i).toNNReal)
        (fun i => (z i).toNNReal) ?_ hp
    -- verify the hypothesis `∑ i ∈ s, (w i).toNNReal = 1`, using `∑ i ∈ s, w i = 1` .
    have h_sum_nnreal : ∑ i ∈ s, w i = ↑(∑ i ∈ s, (w i).toNNReal) := by
      push_cast
      congr! with i hi
      refine (coe_toNNReal (lt_top_of_sum_ne_top ?_ hi).ne).symm
      exact hw'.symm ▸ ENNReal.one_ne_top
    rwa [← coe_inj, ← h_sum_nnreal]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case neg
p : ℝ≥0∞
h : p ∉ Set.Ioo 0 1
⊢ 1 < ∞

Tactic:
exact ENNReal.one_lt_top

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

Goal state before:
pp: case neg
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
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
eLpNorm'_eq_lintegral_enorm

Goal state after:
case neg
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
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case neg
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
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
eLpNorm'_eq_lintegral_enorm

Goal state after:
case neg
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
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case neg
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
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
eLpNorm'_eq_lintegral_enorm

Goal state after:
case neg
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
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)
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 ↔ w₁ ≠ 0 ∧ w₂ ≠ 0 ∧ p₁ ≠ p₂

Tactic:
have := geom_mean_lt_arith_mean_weighted_iff_of_nonneg 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, ![w₁, w₂] j ≠ 0 ∧ ![w₁, w₂] k ≠ 0 ∧ ![p₁, p₂] j ≠ ![p₁, p₂] k)
⊢ p₁ ^ w₁ * p₂ ^ w₂ < w₁ * p₁ + w₂ * p₂ ↔ w₁ ≠ 0 ∧ w₂ ≠ 0 ∧ p₁ ≠ p₂
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε' : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
p q : ℝ≥0∞
f : α → ε'
s : Set α
hfq : MemLp ((toMeasurable μ s).indicator f) q μ
hf : ∀ x ∉ s, f x = 0
hs : μ s ≠ ∞
hpq : p ≤ q
this : (toMeasurable μ s).indicator f = f
⊢ MemLp ((toMeasurable μ s).indicator f) p μ

Tactic:
memLp_indicator_iff_restrict (measurableSet_toMeasurable μ s)

Goal state after:
α : Type u_1
ε' : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
p q : ℝ≥0∞
f : α → ε'
s : Set α
hfq : MemLp f q (μ.restrict (toMeasurable μ s))
hf : ∀ x ∉ s, f x = 0
hs : μ s ≠ ∞
hpq : p ≤ q
this : (toMeasurable μ s).indicator f = f
⊢ MemLp f p (μ.restrict (toMeasurable μ s))
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
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

Tactic:
have h :
    (∫⁻ a : α, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a : α, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a : α, (f + g) a ^ p ∂μ) ^ (1 / q) :=
    lintegral_rpow_add_le_add_eLpNorm_mul_lintegral_rpow_add hpq hf hg

Goal state after:
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
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  ∫⁻ (a : α), (f + g) a ^ p ∂μ ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / q)
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case refine_2
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : EDist E
inst✝ : IsFiniteMeasure μ
f : ι → α → E
l : Filter ι
g : α → E
hfin : ∀ (ε : ℝ≥0∞) (i : ι), μ {x | ε ≤ edist (f i x) (g x)} ≠ ∞
h : ∀ (ε : ℝ≥0∞), 0 < ε → Tendsto (fun i ↦ (μ {x | ε ≤ edist (f i x) (g x)}).toNNReal) l (𝓝 0)
ε : ℝ≥0∞
hε : 0 < ε
⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)

Tactic:
rw [← ENNReal.tendsto_toNNReal_iff ENNReal.zero_ne_top (hfin ε)]

Goal state after:
case refine_2
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : EDist E
inst✝ : IsFiniteMeasure μ
f : ι → α → E
l : Filter ι
g : α → E
hfin : ∀ (ε : ℝ≥0∞) (i : ι), μ {x | ε ≤ edist (f i x) (g x)} ≠ ∞
h : ∀ (ε : ℝ≥0∞), 0 < ε → Tendsto (fun i ↦ (μ {x | ε ≤ edist (f i x) (g x)}).toNNReal) l (𝓝 0)
ε : ℝ≥0∞
hε : 0 < ε
⊢ Tendsto (ENNReal.toNNReal ∘ fun x ↦ μ {x_1 | ε ≤ edist (f x x_1) (g x_1)}) l (𝓝 (ENNReal.toNNReal 0))
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ
p q r : ℝ
hpqr : p.HolderTriple q r
hf : ∀ (i : ι), 0 ≤ f i
hg : ∀ (i : ι), 0 ≤ g i
hf_sum : Summable fun i ↦ f i ^ p
hg_sum : Summable fun i ↦ g i ^ q
⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧
    ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)

Tactic:
lift f to ι → ℝ≥0 using hf
  lift g to ι → ℝ≥0 using hg
  -- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
  beta_reduce at *
  norm_cast at *
  exact NNReal.summable_and_Lr_rpow_le_Lp_mul_Lq_tsum hpqr hf_sum hg_sum

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

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
f f' : ↥(Lp E p μ)
a : α
⊢ ↑↑(hg.compLp g0 f) a = (g ∘ ↑↑f) a →
    ↑↑(hg.compLp g0 f') a = (g ∘ ↑↑f') a →
      ↑↑(hg.compLp g0 f - hg.compLp g0 f') a = (↑↑(hg.compLp g0 f) - ↑↑(hg.compLp g0 f')) a →
        ↑↑(f - f') a = (↑↑f - ↑↑f') a → ‖↑↑(hg.compLp g0 f - hg.compLp g0 f') a‖ ≤ ↑c * ‖↑↑(f - f') a‖

Tactic:
ha1

Goal state after:
α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
f f' : ↥(Lp E p μ)
a : α
ha1 : ↑↑(hg.compLp g0 f) a = (g ∘ ↑↑f) a
⊢ ↑↑(hg.compLp g0 f') a = (g ∘ ↑↑f') a →
    ↑↑(hg.compLp g0 f - hg.compLp g0 f') a = (↑↑(hg.compLp g0 f) - ↑↑(hg.compLp g0 f')) a →
      ↑↑(f - f') a = (↑↑f - ↑↑f') a → ‖↑↑(hg.compLp g0 f - hg.compLp g0 f') a‖ ≤ ↑c * ‖↑↑(f - f') 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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g✝ : E → F
c : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
f g : ↥(Lp E p μ)
a : α
⊢ ↑↑(f + g) a = (↑↑f + ↑↑g) a →
    ↑↑(L.compLp (f + g)) a = L (↑↑(f + g) a) →
      ↑↑(L.compLp f) a = L (↑↑f a) →
        ↑↑(L.compLp g) a = L (↑↑g a) →
          ↑↑(L.compLp f + L.compLp g) a = (↑↑(L.compLp f) + ↑↑(L.compLp g)) a →
            ↑↑(L.compLp (f + g)) a = ↑↑(L.compLp f + L.compLp g) a

Tactic:
ha1

Goal state after:
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g✝ : E → F
c : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
f g : ↥(Lp E p μ)
a : α
ha1 : ↑↑(f + g) a = (↑↑f + ↑↑g) a
⊢ ↑↑(L.compLp (f + g)) a = L (↑↑(f + g) a) →
    ↑↑(L.compLp f) a = L (↑↑f a) →
      ↑↑(L.compLp g) a = L (↑↑g a) →
        ↑↑(L.compLp f + L.compLp g) a = (↑↑(L.compLp f) + ↑↑(L.compLp g)) a →
          ↑↑(L.compLp (f + g)) a = ↑↑(L.compLp f + L.compLp g) 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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g : E → F
c✝ : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
c : 𝕜
f : ↥(Lp E p μ)
a✝ : α
⊢ ↑↑(c • f) a✝ = (c • ↑↑f) a✝ →
    ↑↑(L.compLp (c • f)) a✝ = L (↑↑(c • f) a✝) →
      ↑↑(σ c • L.compLp f) a✝ = (σ c • ↑↑(L.compLp f)) a✝ →
        ↑↑(L.compLp f) a✝ = L (↑↑f a✝) → ↑↑(L.compLp (c • f)) a✝ = ↑↑(σ c • L.compLp f) a✝

Tactic:
ha1

Goal state after:
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g : E → F
c✝ : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
c : 𝕜
f : ↥(Lp E p μ)
a✝ : α
ha1 : ↑↑(c • f) a✝ = (c • ↑↑f) a✝
⊢ ↑↑(L.compLp (c • f)) a✝ = L (↑↑(c • f) a✝) →
    ↑↑(σ c • L.compLp f) a✝ = (σ c • ↑↑(L.compLp f)) a✝ →
      ↑↑(L.compLp f) a✝ = L (↑↑f a✝) → ↑↑(L.compLp (c • f)) a✝ = ↑↑(σ c • L.compLp f) a✝
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: case inr.inl
α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
b₁ : β
a₂ : α
f_eq : g b₁ = f a₂
⊢ False

Tactic:
exact disj.le_bot ⟨⟨a₂, rfl⟩, ⟨b₁, f_eq⟩⟩

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
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
ε : ℝ≥0∞
hε : ε > 0
N : ℕ
h_B : B N ≤ ε
n : ℕ
hn : n ≥ N
h_sub : eLpNorm (f n - f_lim) p μ ≤ liminf (fun m_1 ↦ eLpNorm (f n - f m_1) p μ) atTop
⊢ eLpNorm (f n - f_lim) p μ ≤ ε

Tactic:
refine h_sub.trans ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
ε : ℝ≥0∞
hε : ε > 0
N : ℕ
h_B : B N ≤ ε
n : ℕ
hn : n ≥ N
h_sub : eLpNorm (f n - f_lim) p μ ≤ liminf (fun m_1 ↦ eLpNorm (f n - f m_1) p μ) atTop
⊢ liminf (fun m_1 ↦ eLpNorm (f n - f m_1) p μ) atTop ≤ ε
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜₁ : Type u_2
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_8
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedAddCommGroup G
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedSpace 𝕜₁ E
inst✝⁸ : NontriviallyNormedField 𝕜₂
inst✝⁷ : NormedSpace 𝕜₂ F
inst✝⁶ : NontriviallyNormedField 𝕜₃
inst✝⁵ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₃₂ : 𝕜₃ →+* 𝕜₂
inst✝⁴ : RingHomInvPair σ₂₃ σ₃₂
inst✝³ : RingHomInvPair σ₃₂ σ₂₃
σ₁₃ : 𝕜₁ →+* 𝕜₃
inst✝² : RingHomIsometric σ₁₃
inst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝ : RingHomIsometric σ₁₂
e : F ≃ₛₗᵢ[σ₂₃] G
f : E →SL[σ₁₂] F
r : ℝ≥0
⊢ ‖↑↑e ∘SL f‖₊ ≤ r ↔ ‖f‖₊ ≤ r

Tactic:
by simp [opNNNorm_le_iff]

Goal state after: