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

Goal state before:
pp: case hbc
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →ₛₗ[σ₁₂] F
C : ℝ
h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖
x : E
⊢ C ≤ max C 0

Tactic:
apply le_max_left

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝¹ g h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝¹ : Monoid M
inst✝ : MulAction M β
f✝ : l.Germ β
f : α → β
⊢ 1 • ↑f = ↑f

Tactic:
simp only [← coe_one, ← coe_smul', one_smul]

Goal state after:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s : Set Ω
hs : s.Finite
hs' : s.Nonempty
⊢ IsProbabilityMeasure (uniformOn s)

Tactic:
apply cond_isProbabilityMeasure_of_finite
  · rwa [Measure.count_ne_zero_iff]
  · exact (Measure.count_apply_lt_top.2 hs).ne

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ n ≤ OuterMeasure.pi m ↔ ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)

Tactic:
OuterMeasure.pi,

Goal state after:
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ n ≤ boundedBy (piPremeasure m) ↔ ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ n ≤ OuterMeasure.pi m ↔ ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)

Tactic:
rw [OuterMeasure.pi, le_boundedBy']; constructor
  · intro h s hs; refine (h _ hs).trans_eq (piPremeasure_pi hs)
  · intro h s hs; refine le_trans (n.mono <| subset_pi_eval_image univ s) (h _ ?_)
    simp [univ_pi_nonempty_iff, hs]

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

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

Tactic:
exact ⟨Classical.choose h, by ext; simp⟩

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

Goal state before:
pp: case h
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
j : ι
a✝ : j ≠ i
⊢ t ∩ (fun b ↦ b i) ⁻¹' s ⊆ t

Tactic:
apply inter_subset_left

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

Goal state before:
pp: α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
⊢ ε * μ {x | ε ≤ ‖f x‖ₑ ^ p.toReal} ≤ eLpNorm f p μ ^ p.toReal

Tactic:
have : 1 / p.toReal * p.toReal = 1 := by
    refine one_div_mul_cancel ?_
    rw [Ne, ENNReal.toReal_eq_zero_iff]
    exact not_or_intro hp_ne_zero hp_ne_top
  rw [← ENNReal.rpow_one (ε * μ { x | ε ≤ ‖f x‖ₑ ^ p.toReal }), ← this, ENNReal.rpow_mul]
  gcongr
  exact pow_mul_meas_ge_le_eLpNorm μ hp_ne_zero hp_ne_top hf ε

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ (∫⁻ (x : α), s.indicator (fun x ↦ ‖c‖ₑ ^ p.toReal) x ∂μ) ^ (1 / p.toReal) = ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
rw [lintegral_indicator_const₀ hs, ENNReal.mul_rpow_of_nonneg, ← ENNReal.rpow_mul,
        mul_one_div_cancel hp_pos.ne', ENNReal.rpow_one]

Goal state after:
case hz
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ 0 ≤ 1 / p.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: case e_f
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f g : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
x : α
⊢ ‖s.piecewise f g x‖ₑ = s.piecewise (fun x ↦ ‖f x‖ₑ) (fun x ↦ ‖g x‖ₑ) x

Tactic:
by_cases hx : x ∈ s <;> simp [hx]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_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 = ∞
hg' : ∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ = 0
⊢ ∀ᵐ (x : α) ∂μ, ‖g x‖ₑ = 0

Tactic:
MeasureTheory.lintegral_eq_zero_iff' (by fun_prop)

Goal state after:
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 = ∞
hg' : (fun a ↦ ‖g a‖ₑ ^ p) =ᵐ[μ] 0
⊢ ∀ᵐ (x : α) ∂μ, ‖g x‖ₑ = 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: case pos
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 = ∞
⊢ eLpNorm' f p μ ≤ c * eLpNorm' g p μ

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 = ∞
⊢ eLpNorm' f p μ ≤ c * eLpNorm' g p μ

Tactic:
· by_cases hg' : eLpNorm' g p μ = 0
    · have : ∀ᵐ (x : α) ∂μ, ‖g x‖ₑ = 0 := by
        simp only [eLpNorm'_eq_lintegral_enorm, one_div, ENNReal.rpow_eq_zero_iff, inv_pos, hp,
          and_true, inv_neg'', hp', and_false, or_false] at hg'
        rw [MeasureTheory.lintegral_eq_zero_iff' (by fun_prop)] at hg'
        exact hg'.mono fun x hx ↦ by simpa [hp, hp'] using hx
      have : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ = 0 := (this.and h).mono fun x ⟨h, h'⟩ ↦ by simp_all
      simpa only [hg', mul_zero, nonpos_iff_eq_zero] using eLpNorm'_eq_zero_of_ae_eq_zero hp this
    · simp_all

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 = ∞
⊢ eLpNorm' f p μ ≤ c * eLpNorm' g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝ : ENorm ε
c : ε
hq_pos : 0 < q
⊢ eLpNorm' (fun x ↦ c) q μ = ‖c‖ₑ * μ Set.univ ^ (1 / q)

Tactic:
rw [eLpNorm'_eq_lintegral_enorm, lintegral_const,
    ENNReal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 ≤ 1 / q)]
  congr
  rw [← ENNReal.rpow_mul]
  suffices hq_cancel : q * (1 / q) = 1 by rw [hq_cancel, ENNReal.rpow_one]
  rw [one_div, mul_inv_cancel₀ (ne_of_lt hq_pos).symm]

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

Goal state before:
pp: 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 := MeasurableEquiv.piUnique X
this : (piPremeasure fun i ↦ (μ i).toOuterMeasure) = ⇑(Measure.map (⇑e.symm) (μ default))
⊢ Measure.map (⇑e) ((OuterMeasure.pi fun i ↦ (μ i).toOuterMeasure).toMeasure ⋯) = μ default

Tactic:
OuterMeasure.pi,

Goal state after:
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 := MeasurableEquiv.piUnique X
this : (piPremeasure fun i ↦ (μ i).toOuterMeasure) = ⇑(Measure.map (⇑e.symm) (μ default))
⊢ Measure.map (⇑e) ((boundedBy (piPremeasure fun i ↦ (μ i).toOuterMeasure)).toMeasure ⋯) = μ default
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
⊢ (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)

Tactic:
have q : 0 < y - x := by rwa [sub_pos]

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
⊢ (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:

Tactic:
_ = exp ((1 + t) / 2 * x + (1 - t) / 2 * (-x)) := by ring_nf
    _ ≤ (1 + t) / 2 * exp x + (1 - t) / 2 * exp (-x) :=
        convexOn_exp.2 (Set.mem_univ _) (Set.mem_univ _) (by linarith) (by linarith) <| by ring
    _ = _ := by rw [cosh_eq, sinh_eq]; ring

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)
⊢ ∫⁻ (a : α), (f * g) a ∂μ = ∫⁻ (a : α), (funMulInvSnorm f p μ * funMulInvSnorm g q μ) a * (npf * nqg) ∂μ

Tactic:
refine lintegral_congr fun a => ?_

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
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)
a : α
⊢ (f * g) a = (funMulInvSnorm f p μ * funMulInvSnorm g q μ) a * (npf * nqg)
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: 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
h2p : ∀ i ∈ s, p i = 0
⊢ (∫⁻ (a : α), f i₀ a ∂μ) ^ p i₀ * ∏ i ∈ s, 1 = (∫⁻ (a : α), f i₀ a ∂μ) ^ p i₀ * ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i

Tactic:
congr 1
            apply prod_congr rfl fun i hi ↦ by rw [h2p i hi, ENNReal.rpow_zero]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

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

Tactic:
rw [← rpow_sum_of_nonneg (hz k hk) (hw · · |>.le), ← sum_mul, hw', rpow_one, one_mul]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

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

Tactic:
have h_rpow : ∀ a : ℝ≥0∞, a ^ q = (a ^ p) ^ (q / p) := fun a => by
    rw [← ENNReal.rpow_mul, mul_div_cancel₀ _ hp_pos.ne']
  have h_rpow_add_rpow_le_add :
    ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p := by
    refine rpow_add_rpow_le_add (a ^ p) (b ^ p) ?_
    rwa [one_le_div hp_pos]
  rw [h_rpow a, h_rpow b, one_div p, ENNReal.le_rpow_inv_iff hp_pos, ← ENNReal.rpow_mul, mul_comm,
    mul_one_div]
  rwa [one_div_div] at h_rpow_add_rpow_le_add

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

Goal state before:
pp: case h₂
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), g a * (f + g) a ^ (p - 1) ∂μ ≤
    (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q)

Tactic:
exact lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hg (hf.add hg)

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

Goal state before:
pp: case inl
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hp0 : 0 ≤ 0
hp1 : 0 ≤ 1
⊢ 1 ≤ 2⁻¹ * 2

Tactic:
ENNReal.inv_mul_cancel two_ne_zero ofNat_ne_top

Goal state after:
case inl
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hp0 : 0 ≤ 0
hp1 : 0 ≤ 1
⊢ 1 ≤ 1
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case refine_1
α : Type u_1
ι : Type u_2
m : MeasurableSpace α
μ : Measure α
l : Filter ι
F : Type u_5
inst✝¹ : PseudoEMetricSpace F
inst✝ : Zero F
f : ι → α → F
g : α → F
hg : TendstoInMeasure μ f l g
s : Set α
ε : ℝ≥0∞
hε : 0 < ε
⊢ (fun x ↦ 0) ≤ fun i ↦ μ {x | ε ≤ edist ((fun i ↦ s.indicator (f i)) i x) (s.indicator g x)}

Tactic:
intro; simp

Goal state after:
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
hf' : 0 ≤ ∑' (i : ι), f i ^ p
⊢ (∑' (i : ι), (f i * g i) ^ r) ^ (1 / r) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)

Tactic:
have hg' : 0 ≤ ∑' i, g i ^ q := tsum_nonneg fun i ↦ rpow_nonneg (hg i) q

Goal state after:
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
hf' : 0 ≤ ∑' (i : ι), f i ^ p
hg' : 0 ≤ ∑' (i : ι), g i ^ q
⊢ (∑' (i : ι), (f i * g i) ^ r) ^ (1 / r) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

Tactic:
by_cases! H : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0

Goal state after:
case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0
H' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = ∞)
⊢ (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞

Tactic:
simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
      ENNReal.sum_eq_top, not_or] using H'

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

Goal state before:
pp: case h
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))

Tactic:
rw [ae_iff]

Goal state after:
case h
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
⊢ StrictMono ns ∧ μ {a | ¬Tendsto (fun i ↦ f (ns i) a) atTop (𝓝 (g a))} = 0
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✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
g : E → F
c✝¹ : ℝ≥0
inst✝ : NormedSpace ℝ E
ν : Measure α
c✝ : ℝ≥0∞
hc : c✝ ≠ ∞
h : μ ≤ c✝ • ν
c : ℝ
f : ↥(Lp E p ν)
⊢ ↑↑(MemLp.toLp ↑↑(c • f) ⋯) =ᵐ[μ] ↑↑((RingHom.id ℝ) c • MemLp.toLp ↑↑f ⋯)

Tactic:
grw [MemLp.coeFn_toLp, Lp.coeFn_smul, MemLp.coeFn_toLp]

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✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
g : E → F
c✝¹ : ℝ≥0
inst✝ : NormedSpace ℝ E
ν : Measure α
c✝ : ℝ≥0∞
hc : c✝ ≠ ∞
h : μ ≤ c✝ • ν
c : ℝ
f : ↥(Lp E p ν)
⊢ ↑↑(c • f) =ᵐ[μ] (RingHom.id ℝ) c • ↑↑f
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
f : α ⊕ β ↪ γ
⊢ Disjoint (Set.range ⇑(inl.trans f, inr.trans f).1) (Set.range ⇑(inl.trans f, inr.trans f).2)

Tactic:
by
      rw [Set.disjoint_left]
      rintro _ ⟨a, h⟩ ⟨b, rfl⟩
      simp only at h
      have : Sum.inl a = Sum.inr b := f.injective h
      simp only [reduceCtorEq] at this

Goal state after:
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Unique α
x✝ : α ↪ β
⊢ (fun x ↦ { toFun := fun x_1 ↦ x, inj' := ⋯ }) ((fun f ↦ f default) x✝) = x✝

Tactic:
by
    ext x
    simp_rw [Function.Embedding.coeFn_mk]
    congr 1
    exact Subsingleton.elim _ x

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
hzero : Tendsto (fun x_1 ↦ -↑x⁻¹ * x_1) (𝓝 0) (𝓝 0)
⊢ (fun x_1 ↦ (1 - -↑x⁻¹ * x_1)⁻¹ʳ * ↑x⁻¹) =O[𝓝 0] fun _t ↦ 1 * 1

Tactic:
exact (inverse_one_sub_norm.comp_tendsto hzero).mul (isBigO_const_const _ one_ne_zero _)

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

Goal state before:
pp: case hg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ (fun x_1 ↦ ‖x.1‖ * ‖x_1.2 - x.2‖) =o[𝓝 x] fun _x ↦ 1

Tactic:
apply IsLittleO.const_mul_left

Goal state after:
case hg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ (fun x_1 ↦ ‖x_1.2 - x.2‖) =o[𝓝 x] fun _x ↦ 1
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: case hbc
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E →L[𝕜] F →L[𝕜] G
x : E
y : F
⊢ ‖f‖ ≤ max ‖f‖ 1

Tactic:
apply le_max_left

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
x : α
hx : ↑(f x) • g x = g' x
h'x : ↑(f x) ≠ 0
⊢ ↑(f x) ≠ 0

Tactic:
simp only [Ne, ENNReal.coe_eq_zero] at h'x

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
x : α
hx : ↑(f x) • g x = g' x
h'x : ¬f x = 0
⊢ ↑(f x) ≠ 0
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : μ u ≠ ∞
⊢ μ u < μ s + μ t

Tactic:
apply (ENNReal.toReal_lt_toReal hu _).1

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : μ u ≠ ∞
⊢ (μ u).toReal < (μ s + μ t).toReal

α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : μ u ≠ ∞
⊢ μ s + μ t ≠ ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
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:
· exact norm_indicatorConstLp hp_pos hp_top

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

Goal state before:
pp: case inl
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf : Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
h✝ : IsEmpty ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
refine ⟨‖f 0‖ + 1, add_pos_of_nonneg_of_pos (norm_nonneg _) zero_lt_one, fun m => ?_⟩
    obtain rfl : m = 0 := funext (IsEmpty.elim ‹_›)
    simp [univ_eq_empty, zero_le_one]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
⊢ IntegrableOn f (range e) μ ↔ Integrable (f ∘ e) (Measure.comap e μ)

Tactic:
he.integrableOn_iff_comap .rfl,

Goal state after:
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
⊢ IntegrableOn (f ∘ e) (e ⁻¹' range e) (Measure.comap e μ) ↔ Integrable (f ∘ e) (Measure.comap e μ)