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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ ↑(f.pair g) =ᵐ[μ] fun x ↦ (↑f x, ↑g x)

Tactic:
pair_eq_mk

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ ↑(mk (fun x ↦ (↑f x, ↑g x)) ⋯) =ᵐ[μ] fun x ↦ (↑f x, ↑g x)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case cons
δ : Type u_4
X : δ → Type u_5
inst✝¹ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
s : (i : δ) → Set (X i)
a : δ
l : List δ
ih : (Measure.tprod l μ) (Set.tprod l s) = (List.map (fun i ↦ (μ i) (s i)) l).prod
⊢ (Measure.tprod (a :: l) μ) (Set.tprod (a :: l) s) = (List.map (fun i ↦ (μ i) (s i)) (a :: l)).prod

Tactic:
rw [tprod_cons, Set.tprod]
    simp only [foldr_cons, prod_cons, map_cons]
    rw [prod_prod, ih]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Indicator

Goal state before:
pp: α : Type u_1
ε : Type u_2
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s t : Set α
h : s ⊆ t
f : α → ε
a : α
⊢ ‖s.indicator f a‖ₑ ≤ ‖t.indicator f a‖ₑ

Tactic:
simp only [enorm_indicator_eq_indicator_enorm]

Goal state after:
α : Type u_1
ε : Type u_2
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s t : Set α
h : s ⊆ t
f : α → ε
a : α
⊢ s.indicator (fun a ↦ ‖f a‖ₑ) a ≤ t.indicator (fun a ↦ ‖f a‖ₑ) a
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

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✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ 0
hp : p ≠ 0
hp_top : ¬p = ∞
⊢ eLpNorm (s.indicator fun x ↦ c) p μ = ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
exact eLpNorm_indicator_const hs hp hp_top

Goal state after:
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_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≥0∞
h0 : ¬p = 0
h_top : p = ∞
⊢ eLpNorm f ∞ μ ≤ c • eLpNorm g ∞ μ

Tactic:
exact eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul' h

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

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

Tactic:
rw [← mul_comm]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : AEStronglyMeasurable f μ
C : ℝ
hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ eLpNormEssSup f μ < ∞

Tactic:
exact eLpNormEssSup_lt_top_of_ae_bound hfC

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

Goal state before:
pp: s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
⊢ -1 < p * s

Tactic:
rcases lt_or_gt_of_ne hs' with h | h

Goal state after:
case inl
s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
h : s < 0
⊢ -1 < p * s

case inr
s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
h : 0 < s
⊢ -1 < p * s
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p : ℝ≥0∞
hp : 1 ≤ p
⊢ p.HolderConjugate p.conjExponent

Tactic:
have : p ≠ 0 := (zero_lt_one.trans_le hp).ne'

Goal state after:
p : ℝ≥0∞
hp : 1 ≤ p
this : p ≠ 0
⊢ p.HolderConjugate p.conjExponent
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : AEStronglyMeasurable f μ
h : MemLp (fun x ↦ ‖f x‖) p μ
⊢ eLpNorm f p μ < ∞

Tactic:
rw [← eLpNorm_norm]; exact h.2

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: case inr.inr
p : ℝ≥0∞
hp✝ : 1 ≤ p
this : p ≠ 0
hp₁ : 1 < p
hp : p ≠ ∞
⊢ 1 + (p - 1)⁻¹ = (1⁻¹ - p⁻¹)⁻¹

Tactic:
calc
    1 + (p - 1)⁻¹ = (p - 1 + 1) / (p - 1) := by
      rw [ENNReal.add_div, ENNReal.div_self ((tsub_pos_of_lt hp₁).ne') (sub_ne_top hp), one_div]
    _ = (1⁻¹ - p⁻¹)⁻¹ := by
      rw [tsub_add_cancel_of_le, ← inv_eq_iff_eq_inv, div_eq_mul_inv, ENNReal.mul_inv, inv_inv,
        ENNReal.mul_sub, ENNReal.inv_mul_cancel, mul_one] <;> simp [*]

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
⊢ -1 ≤ z / y - 1

Tactic:
rw [le_sub_iff_add_le, neg_add_cancel, div_nonneg_iff]
      exact Or.inl ⟨hz, hy.le⟩

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_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ ∫⁻ (a : α), f a ^ p * (ENNReal.ofReal p)⁻¹ + g a ^ q * (ENNReal.ofReal q)⁻¹ ∂μ = 1

Tactic:
lintegral_add_left'

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_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ ∫⁻ (a : α), f a ^ p * (ENNReal.ofReal p)⁻¹ ∂μ + ∫⁻ (a : α), g a ^ q * (ENNReal.ofReal q)⁻¹ ∂μ = 1

case hf
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_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ AEMeasurable (fun a ↦ f a ^ p * (ENNReal.ofReal p)⁻¹) μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw' : ∑ i ∈ s, w i = 1
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
i : ι
hi : i ∈ s
⊢ w i * z i = w i * x

Tactic:
rcases eq_or_ne (w i) 0 with hwi | hwi

Goal state after:
case inl
ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw' : ∑ i ∈ s, w i = 1
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
i : ι
hi : i ∈ s
hwi : w i = 0
⊢ w i * z i = w i * x

case inr
ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw' : ∑ i ∈ s, w i = 1
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
i : ι
hi : i ∈ s
hwi : w i ≠ 0
⊢ w i * z i = w i * x
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: 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
⊢ (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p

Tactic:
have hp_not_neg : ¬p < 0 := by simp [hp_nonneg]

Goal state after:
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
⊢ (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : α → E
hf : MemLp f p μ
hg : MemLp g p μ
⊢ MemLp (f + -g) p μ

Tactic:
exact hf.add hg.neg

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case h0
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
⊢ ∀ i ∈ s, 0 ≤ (z i ^ w i)⁻¹

pp: case h1
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
x : ι
a✝ : x ∈ s
⊢ (z x ^ w x)⁻¹ ≤ (z x)⁻¹ ^ w x

Tactic:
· exact fun i hi ↦ by positivity [hz i hi]

Goal state after:
case h1
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
x : ι
a✝ : x ∈ s
⊢ (z x ^ w x)⁻¹ ≤ (z x)⁻¹ ^ w x
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case inl
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_nonpos : p ≤ 0
⊢ eLpNorm' f p μ < ∞

pp: case inr
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_pos : 0 < p
⊢ eLpNorm' f p μ < ∞

Tactic:
· rw [le_antisymm hp_nonpos hp_nonneg]
    simp

Goal state after:
case inr
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_pos : 0 < p
⊢ eLpNorm' f p μ < ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
hp_top : ¬p = ∞
hp_pos : 0 < p.toReal
hq_top : ¬q = ∞
hq_eq_zero : q = 0
⊢ False

Tactic:
obtain rfl : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) zero_le

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hq_top : ¬q = ∞
hq_eq_zero : q = 0
hpq : 0 ≤ q
hp0 : 0 ≠ 0
hp_top : ¬0 = ∞
hp_pos : 0 < ENNReal.toReal 0
⊢ False
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 a * g a ^ (p - 1) ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q)

Tactic:
by
  refine (ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf (hg.pow_const _)).trans_eq ?_
  simp [← ENNReal.rpow_mul, hpq.sub_one_mul_conj]

Goal state after:
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
  obtain hf | hf := eq_zero_or_pos (∑ i ∈ s, f i ^ p)
  · exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hf
  obtain hg | hg := eq_zero_or_pos (∑ i ∈ s, g i ^ q)
  · calc
      ∑ i ∈ s, f i * g i = ∑ i ∈ s, g i * f i := by
        congr with i
        rw [mul_comm]
      _ ≤ (∑ i ∈ s, g i ^ q) ^ (1 / q) * (∑ i ∈ s, f i ^ p) ^ (1 / p) :=
        (inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hg)
      _ = (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) := mul_comm _ _
  let f' i := f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
  let g' i := g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
  suffices (∑ i ∈ s, f' i * g' i) ≤ 1 by
    simp_rw [f', g', div_mul_div_comm, ← sum_div] at this
    rwa [div_le_iff₀, one_mul] at this
    positivity
  refine inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq ?_) (le_of_eq ?_)
  · simp_rw [f', div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel₀ hpq.ne_zero, rpow_one,
      div_self hf.ne']
  · simp_rw [g', div_rpow, ← sum_div, ← rpow_mul, one_div, inv_mul_cancel₀ hpq.symm.ne_zero,
      rpow_one, div_self hg.ne']

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p

Tactic:
have hpq : p.HolderConjugate q := .conjExponent hp

Goal state after:
case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : p.HolderConjugate q
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p
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✝² : NormedAddCommGroup E
ι : Type u_4
inst✝¹ : Nonempty ι
inst✝ : SemilatticeSup ι
hp : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
n : ι × ι
⊢ eLpNorm (↑↑(f n.1 - f n.2)) p μ ≠ ∞

Tactic:
exact eLpNorm_ne_top _

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ↥(Lp E 0 μ)
⊢ ‖f‖ = 0

Tactic:
by
  -- Squeezed for performance reasons
  simp_rw [norm_def, eLpNorm_exponent_zero, ENNReal.toReal_zero]

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
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
n : ℕ
hfg : ∀ ε > 0, ∃ N, ∀ n_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ edist (f n_1 x) (g x)} ≤ ε
⊢ ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ edist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n

Tactic:
exact hfg ((2 : ℝ≥0∞)⁻¹ ^ n) (pos_iff_ne_zero.mpr <| pow_ne_zero _ <| by simp)

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_3
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 ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞
this :
  ∑ i ∈ s, ↑(f i).toNNReal * ↑(g i).toNNReal ≤
    (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ * (∑ x ∈ s, ↑(g x).toNNReal ^ q) ^ q⁻¹
i : ι
hi : i ∈ s
⊢ f i * g i = ↑(f i).toNNReal * ↑(g i).toNNReal

Tactic:
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4.e_a.e_a
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 ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞
this :
  ∑ i ∈ s, ↑(f i).toNNReal * ↑(g i).toNNReal ≤
    (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ * (∑ x ∈ s, ↑(g x).toNNReal ^ q) ^ q⁻¹
i : ι
hi : i ∈ s
⊢ f i ^ p = ↑(f i).toNNReal ^ p

Tactic:
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4.e_a.e_a
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 ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞
this :
  ∑ i ∈ s, ↑(f i).toNNReal * ↑(g i).toNNReal ≤
    (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ * (∑ x ∈ s, ↑(g x).toNNReal ^ q) ^ q⁻¹
i : ι
hi : i ∈ s
⊢ g i ^ q = ↑(g i).toNNReal ^ q

Tactic:
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]

Goal state after:
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
hL : MemLp (g ∘ f) p μ
hg : UniformContinuous g
hg' : AntilipschitzWith K' g
g0 : g 0 = 0
⊢ MemLp f p μ

Tactic:
by
  have A : ∀ x, ‖f x‖ ≤ K' * ‖g (f x)‖ := by
    intro x
    -- TODO: add `AntilipschitzWith.le_mul_nnnorm_sub` and `AntilipschitzWith.le_mul_norm`
    rw [← dist_zero_right, ← dist_zero_right, ← g0]
    apply hg'.le_mul_dist
  have B : AEStronglyMeasurable f μ :=
    (hg'.isUniformEmbedding hg).isEmbedding.aestronglyMeasurable_comp_iff.1 hL.1
  exact hL.of_le_mul B (Filter.Eventually.of_forall A)

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✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
f : ↥(Lp E p μ)
⊢ AEEqFun.comp g ⋯ ↑f ∈ Lp F p μ

Tactic:
rw [Lp.mem_Lp_iff_memLp]
    exact (hg.comp_memLp g0 (Lp.memLp f)).ae_eq (AEEqFun.coeFn_comp _ hg.continuous _).symm

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✝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)
⊢ ↑↑(L.compLp (c • f)) =ᵐ[μ] ↑↑(σ c • L.compLp f)

Tactic:
filter_upwards [Lp.coeFn_smul c f, coeFn_compLp L (c • f), Lp.coeFn_smul (σ c) (L.compLp f),
      coeFn_compLp L f] with _ ha1 ha2 ha3 ha4

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✝
ha2 : ↑↑(L.compLp (c • f)) a✝ = L (↑↑(c • f) a✝)
ha3 : ↑↑(σ c • L.compLp f) a✝ = (σ c • ↑↑(L.compLp f)) a✝
ha4 : ↑↑(L.compLp f) a✝ = L (↑↑f a✝)
⊢ ↑↑(L.compLp (c • f)) a✝ = ↑↑(σ c • L.compLp f) a✝
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0∞
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p

Tactic:
have hpq : p.HolderConjugate q := .conjExponent hp

Goal state after:
case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0∞
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : p.HolderConjugate q
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p
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
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ Tendsto (fun n_1 ↦ (f n - f n_1) x) atTop (𝓝 ((f n - f_lim) x))

Tactic:
simp_rw [sub_eq_add_neg]

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
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ Tendsto (fun n_1 ↦ (f n + -f n_1) x) atTop (𝓝 ((f n + -f_lim) x))
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
n : ℕ
⊢ (fun t ↦ (↑x + t)⁻¹ʳ - (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 0] fun t ↦ ‖t‖ ^ n

Tactic:
by
  refine EventuallyEq.trans_isBigO (.fun_sub (inverse_add_nth_order x n) (.refl _ _)) ?_
  simp only [add_sub_cancel_left]
  refine ((isBigO_refl _ _).norm_right.mul (inverse_add_norm x)).trans ?_
  simp only [mul_one, isBigO_norm_left]
  exact ((isBigO_refl _ _).norm_right.const_mul_left _).pow _

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
tfae_6_iff_8 : (∃ C, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
tfae_8_iff_9 : BddAbove (Set.range fun x ↦ ‖f x‖) ↔ ⨆ i, ↑‖f i‖₊ < ⊤
⊢ (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) →
    ∃ p,
      Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑p ∧
        ∀ (k : ι), (normSeminorm 𝕜₂ F).comp ↑(f k) ≤ p

Tactic:
· intro ⟨C, hC⟩
      refine ⟨C.toNNReal • normSeminorm 𝕜 E,
        ((norm_withSeminorms 𝕜 E).continuous_seminorm 0).const_smul C.toNNReal, fun i x ↦ ?_⟩
      exact (hC i x).trans (mul_le_mul_of_nonneg_right (C.le_coe_toNNReal) (norm_nonneg x))

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

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
O : (E →L[𝕜] F) → E →L[𝕜] E := fun f ↦ ↑e.symm ∘SL f
h_O : Continuous O
f' : E →L[𝕜] F
w : (E →L[𝕜] E)ˣ
hw : ↑w = O f'
x : E
⊢ ↑(((unitsEquiv 𝕜 E) w).trans e) x = f' x

Tactic:
simp [O, hw]

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

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

Tactic:
exact nonempty_inter_of_measureReal_lt_add hs h't h's h hu

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f g : α → E
hf : IntegrableAtFilter f l μ
hg : IntegrableAtFilter g l μ
⊢ IntegrableAtFilter (f + -g) l μ

Tactic:
exact hf.add hg.neg

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
G : Type wG
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
z : G
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι 𝕜‖ * ‖z‖ = ‖z‖

Tactic:
norm_mkPiAlgebra,

Goal state after:
𝕜 : Type u
ι : Type v
G : Type wG
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
z : G
⊢ 1 * ‖z‖ = ‖z‖