mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case pos
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
hst : s ∩ t = ∅
⊢ (uniformOn s) (t ∩ u) = (uniformOn (s ∩ t)) u * (uniformOn s) t

pp: case neg
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
hst : ¬s ∩ t = ∅
⊢ (uniformOn s) (t ∩ u) = (uniformOn (s ∩ t)) u * (uniformOn s) t

Tactic:
· rw [hst, uniformOn_empty_meas, Measure.coe_zero, Pi.zero_apply, zero_mul,
      uniformOn_eq_zero_iff hs, ← Set.inter_assoc, hst, Set.empty_inter]

Goal state after:
case neg
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
hst : ¬s ∩ t = ∅
⊢ (uniformOn s) (t ∩ u) = (uniformOn (s ∩ t)) u * (uniformOn s) t
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : ConditionallyCompleteLattice β
f : α → β
inst✝ : Nonempty α
hμ : ∀ (a : α), μ {a} ≠ 0
hf : BddAbove (range f)
⊢ essSup f μ = ⨆ a, f a

Tactic:
by rw [essSup, ae_eq_top.2 hμ, limsup_top_eq_ciSup hf]

Goal state after:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
| (Measure.count s)⁻¹ * Measure.count (s ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count s) +
    (Measure.count t)⁻¹ * Measure.count (t ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count t)

Tactic:
mul_comm (Measure.count (s ∪ t))⁻¹,

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
| (Measure.count s)⁻¹ * Measure.count (s ∩ u) * (Measure.count s * (Measure.count (s ∪ t))⁻¹) +
    (Measure.count t)⁻¹ * Measure.count (t ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count t)
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
| (Measure.count s)⁻¹ * Measure.count (s ∩ u) * (Measure.count s * (Measure.count (s ∪ t))⁻¹) +
    (Measure.count t)⁻¹ * Measure.count (t ∩ u) * ((Measure.count (s ∪ t))⁻¹ * Measure.count t)

Tactic:
mul_comm (Measure.count (s ∪ t))⁻¹,

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
| (Measure.count s)⁻¹ * Measure.count (s ∩ u) * (Measure.count s * (Measure.count (s ∪ t))⁻¹) +
    (Measure.count t)⁻¹ * Measure.count (t ∩ u) * (Measure.count t * (Measure.count (s ∪ t))⁻¹)
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteLattice β
hμ : ∀ (a : α), μ {a} ≠ 0
f : α → β
⊢ liminf f ⊤ = ⨅ i, f i

Tactic:
liminf_top_eq_iInf

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteLattice β
hμ : ∀ (a : α), μ {a} ≠ 0
f : α → β
⊢ ⨅ i, f i = ⨅ i, f i
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}) ^ (1 / p.toReal) ≤ eLpNorm f p μ

Tactic:
rw [eLpNorm_eq_lintegral_rpow_enorm_toReal hp_ne_zero hp_ne_top]
  gcongr
  exact mul_meas_ge_le_lintegral₀ (hf.enorm.pow_const _) ε

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

Goal state before:
pp: case neg
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∞
hε : ε ≠ 0
hmeas_top : ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0
h : ¬ε = ∞
⊢ μ {x | ε ≤ ‖f x‖ₑ} ≤ ε⁻¹ ^ p.toReal * eLpNorm f p μ ^ p.toReal

Tactic:
have hεpow : ε ^ p.toReal ≠ 0 := (ENNReal.rpow_pos (pos_iff_ne_zero.2 hε) h).ne.symm
    have hεpow' : ε ^ p.toReal ≠ ∞ := by finiteness
    rw [ENNReal.inv_rpow, ← ENNReal.mul_le_mul_iff_right hεpow hεpow', ← mul_assoc,
      ENNReal.mul_inv_cancel hεpow hεpow', one_mul]
    exact mul_meas_ge_le_pow_eLpNorm' μ hp_ne_zero hp_ne_top hf ε

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

Goal state before:
pp: case e_a
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
⊢ q * q⁻¹ = 1

Tactic:
mul_inv_cancel₀ hq_ne_zero

Goal state after:
case e_a
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
⊢ 1 = 1
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
hyz' : 0 < z / y
h : z / y = 1
⊢ z - y ≤ 0

Tactic:
div_eq_one_iff_eq hy.ne'

Goal state after:
x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
hyz' : 0 < z / y
h : z = y
⊢ z - y ≤ 0
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
hxy' : 0 < x / y
h : x / y = 1
⊢ y - x ≤ 0

Tactic:
div_eq_one_iff_eq hy.ne'

Goal state after:
x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
hxy' : 0 < x / y
h : x = y
⊢ y - x ≤ 0
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case inr.inr.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) < log (1 + s) * p

Tactic:
rw [← div_lt_iff₀ hp', ← div_lt_div_right_of_neg hs']

Goal state after:
case inr.inr.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) / s < log (1 + p * s) / p / s
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
⊢ eLpNorm' f p.toReal (c • μ) = c ^ (1 / p).toReal • eLpNorm' f p.toReal μ

Tactic:
rw [eLpNorm'_smul_measure ENNReal.toReal_nonneg]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
⊢ c ^ (1 / p.toReal) * eLpNorm' f p.toReal μ = c ^ (1 / p).toReal • eLpNorm' f p.toReal μ
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ
h : p.HolderConjugate q
⊢ p / q = p - 1

Tactic:
field_simp [h.symm.ne_zero]

Goal state after:
p q : ℝ
h : p.HolderConjugate q
⊢ p = q * (p - 1)
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0
h : p.HolderConjugate q
⊢ p / q = p - 1

Tactic:
field_simp [h.symm.ne_zero]

Goal state after:
p q : ℝ≥0
h : p.HolderConjugate q
⊢ p = q * (p - 1)
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y : 𝕜
hx : x ∈ s
hxy : x < y
hxy' : f x < f y
⊢ StrictMonoOn f (s ∩ Set.Ici y)

Tactic:
by
  intro u hu v hv huv
  have step1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z := by
    intro z hz
    refine hf.lt_right_of_left_lt hx hz.1 ?_ hxy'
    rw [openSegment_eq_Ioo (hxy.trans hz.2)]
    exact ⟨hxy, hz.2⟩
  rcases eq_or_lt_of_le hu.2 with (rfl | hu2)
  · exact step1 ⟨hv.1, huv⟩
  · refine hf.lt_right_of_left_lt ?_ hv.1 ?_ (step1 ⟨hu.1, hu2⟩)
    · apply hf.1.segment_subset hx hu.1
      rw [segment_eq_Icc (hxy.le.trans hu.2)]
      exact ⟨hxy.le, hu.2⟩
    · rw [openSegment_eq_Ioo (hu2.trans huv)]
      exact ⟨hu2, huv⟩

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 * g) a ∂μ ≤ 1

Tactic:
calc
    (∫⁻ a : α, (f * g) a ∂μ) ≤
        ∫⁻ a : α, f a ^ p / ENNReal.ofReal p + g a ^ q / ENNReal.ofReal q ∂μ :=
      lintegral_mono fun a => young_inequality (f a) (g a) hpq
    _ = 1 := by
      simp only [div_eq_mul_inv]
      rw [lintegral_add_left']
      · rw [lintegral_mul_const'' _ (hf.pow_const p), lintegral_mul_const', hf_norm, hg_norm,
          one_mul, one_mul, hpq.inv_add_inv_ennreal]
        simp [hpq.symm.pos]
      · exact (hf.pow_const _).mul_const _

Goal state after:
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 = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
h_top : ∀ a ∈ s, w a * z a ≠ ∞
h_top_rpow : ∀ a ∈ s, w a * z a ^ p ≠ ∞
⊢ (∑ x ∈ s, (w x).toNNReal * (z x).toNNReal) ^ p ≤ ∑ x ∈ s, (w x).toNNReal * (z x ^ p).toNNReal

Tactic:
toNNReal_rpow

Goal state after:
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 = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
h_top : ∀ a ∈ s, w a * z a ≠ ∞
h_top_rpow : ∀ a ∈ s, w a * z a ^ p ≠ ∞
⊢ (∑ x ∈ s, (w x).toNNReal * (z x).toNNReal) ^ p ≤ ∑ x ∈ s, (w x).toNNReal * (z x).toNNReal ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case pos
p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
hp_pos : 0 < p
h_top : a + b = ∞
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

pp: case neg
p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
hp_pos : 0 < p
h_top : ¬a + b = ∞
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Tactic:
· rw [← @ENNReal.rpow_eq_top_iff_of_pos (a + b) p hp_pos] at h_top
    rw [h_top]
    exact le_top

Goal state after:
case neg
p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
hp_pos : 0 < p
h_top : ¬a + b = ∞
⊢ a ^ p + b ^ p ≤ (a + b) ^ p
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case insert.inr
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
⊢ ∫⁻ (a : α), ∏ i ∈ insert i₀ s, f i a ^ p i ∂μ ≤ ∏ i ∈ insert i₀ s, (∫⁻ (a : α), f i a ∂μ) ^ p i

Tactic:
have hpi₀ : 0 ≤ 1 - p i₀ := by
        simp_rw [sub_nonneg, ← hp, single_le_sum h2p (mem_insert_self ..)]

Goal state after:
case insert.inr
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
hpi₀ : 0 ≤ 1 - p i₀
⊢ ∫⁻ (a : α), ∏ i ∈ insert i₀ s, f i a ^ p i ∂μ ≤ ∏ i ∈ insert i₀ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:

Tactic:
(∫⁻ a : α, (‖f a‖ₑ * g a) ^ p ∂μ) ^ (1 / p) ≤
        (∫⁻ a : α, ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * (∫⁻ a : α, g a ^ r ∂μ) ^ (1 / r) :=
      ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr μ hf.enorm aemeasurable_const
    _ = (∫⁻ a : α, ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q) := by
      rw [hpqr]; simp [r, g]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ q) (B ^ q)
H₁ : Summable fun i ↦ f i * g i
H₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
⊢ B = (∑' (i : ι), g i ^ q) ^ (1 / q)

Tactic:
rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]

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

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
h1 : ¬p = 1
hp1_lt : 1 < p
hpq : p.HolderConjugate p.conjExponent
h0 : ∫⁻ (a : α), (f + g) a ^ p ∂μ = 0
⊢ 0 ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Tactic:
@ENNReal.zero_rpow_of_pos (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
h1 : ¬p = 1
hp1_lt : 1 < p
hpq : p.HolderConjugate p.conjExponent
h0 : ∫⁻ (a : α), (f + g) a ^ p ∂μ = 0
⊢ 0 ≤ (∫⁻ (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: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SeminormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
l : Filter ι
f : ι → α → E
g : α → E
⊢ TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ≥0∞), 0 < ε → ε ≠ ∞ → Tendsto (fun i ↦ μ.real {x | ε ≤ ‖f i x - g x‖ₑ}) l (𝓝 0)

Tactic:
tendstoInMeasure_iff_enorm

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : SeminormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
l : Filter ι
f : ι → α → E
g : α → E
⊢ (∀ (ε : ℝ≥0∞), 0 < ε → ε ≠ ∞ → Tendsto (fun i ↦ μ {x | ε ≤ ‖f i x - g x‖ₑ}) l (𝓝 0)) ↔
    ∀ (ε : ℝ≥0∞), 0 < ε → ε ≠ ∞ → Tendsto (fun i ↦ μ.real {x | ε ≤ ‖f i x - g x‖ₑ}) l (𝓝 0)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
l : Filter ι
f f' : ι → α → E
g g' : α → E
h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i
h_right : g =ᵐ[μ] g'
h_tendsto : TendstoInMeasure μ f l g
ε : ℝ≥0∞
hε : 0 < ε
this : (fun i ↦ μ {x | ε ≤ edist (f' i x) (g' x)}) =ᶠ[l] fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}
⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist (f' i x) (g' x)}) l (𝓝 0)

Tactic:
rw [tendsto_congr' this]

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
l : Filter ι
f f' : ι → α → E
g g' : α → E
h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i
h_right : g =ᵐ[μ] g'
h_tendsto : TendstoInMeasure μ f l g
ε : ℝ≥0∞
hε : 0 < ε
this : (fun i ↦ μ {x | ε ≤ edist (f' i x) (g' x)}) =ᶠ[l] fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}
⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
l : Filter ι
f f' : ι → α → E
g g' : α → E
h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i
h_right : g =ᵐ[μ] g'
h_tendsto : TendstoInMeasure μ f l g
ε : ℝ≥0∞
hε : 0 < ε
⊢ (fun i ↦ μ {x | ε ≤ edist (f' i x) (g' x)}) =ᶠ[l] fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}

Tactic:
filter_upwards [h_left] with i h_ae_eq

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
l : Filter ι
f f' : ι → α → E
g g' : α → E
h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i
h_right : g =ᵐ[μ] g'
h_tendsto : TendstoInMeasure μ f l g
ε : ℝ≥0∞
hε : 0 < ε
i : ι
h_ae_eq : f i =ᵐ[μ] f' i
⊢ μ {x | ε ≤ edist (f' i x) (g' x)} = μ {x | ε ≤ edist (f i x) (g x)}
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
α : Type ?u.18
E : Type u_6
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
⊢ 0 ∈ {f | eLpNorm (↑f) p μ < ∞}

Tactic:
simp [eLpNorm_congr_ae AEEqFun.coeFn_zero, eLpNorm_zero]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
⊢ (eLpNorm (↑0) p μ).toNNReal = 0

Tactic:
simp [eLpNorm_congr_ae AEEqFun.coeFn_zero, eLpNorm_zero]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
p : ℝ
hp : 1 ≤ p
f g : ι → ℝ≥0
A B : ℝ≥0
hfA : HasSum (fun a ↦ f a ^ p) (A ^ p)
hgB : HasSum (fun a ↦ g a ^ p) (B ^ p)
C : ℝ≥0
hC₁ : C ≤ A + B
hC₂ : HasSum (fun i ↦ (f i + g i) ^ p) (C ^ p)
⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A + ↑B ∧ HasSum (fun i ↦ ((fun i ↦ ↑(f i)) i + (fun i ↦ ↑(g i)) i) ^ p) (C ^ p)

Tactic:
use C

Goal state after:
case h
ι : Type u
p : ℝ
hp : 1 ≤ p
f g : ι → ℝ≥0
A B : ℝ≥0
hfA : HasSum (fun a ↦ f a ^ p) (A ^ p)
hgB : HasSum (fun a ↦ g a ^ p) (B ^ p)
C : ℝ≥0
hC₁ : C ≤ A + B
hC₂ : HasSum (fun i ↦ (f i + g i) ^ p) (C ^ p)
⊢ 0 ≤ ↑C ∧ ↑C ≤ ↑A + ↑B ∧ HasSum (fun i ↦ ((fun i ↦ ↑(f i)) i + (fun i ↦ ↑(g i)) i) ^ p) (↑C ^ p)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < B N
n : ℕ
f_norm_diff : ℕ → α → ℝ := fun i x ↦ ‖f (i + 1) x - f i x‖
⊢ eLpNorm' (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i

Tactic:
have hgf_norm_diff :
    ∀ n,
      (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) =
        ∑ i ∈ Finset.range (n + 1), f_norm_diff i :=
    fun n => funext fun x => by simp [f_norm_diff]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < B N
n : ℕ
f_norm_diff : ℕ → α → ℝ := fun i x ↦ ‖f (i + 1) x - f i x‖
hgf_norm_diff :
  ∀ (n : ℕ), (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) = ∑ i ∈ Finset.range (n + 1), f_norm_diff i
⊢ eLpNorm' (fun x ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' (i : ℕ), B i
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
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
x : α
hx : x ∈ sᶜ
ε : ℝ≥0∞
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, edist (f (ns n) x) (g x) < ε

Tactic:
hs,

Goal state after:
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
x : α
hx : x ∈ (limsup S atTop)ᶜ
ε : ℝ≥0∞
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, edist (f (ns n) x) (g x) < ε
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →SL[σ₁₂] F
inst✝ : RingHomIsometric σ₁₂
hn : ‖f‖ = 0
x : E
⊢ ‖f‖ * ‖x‖ = 0

Tactic:
hn,

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →SL[σ₁₂] F
inst✝ : RingHomIsometric σ₁₂
hn : ‖f‖ = 0
x : E
⊢ 0 * ‖x‖ = 0
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : (i : ι) → TopologicalSpace (E i)
inst✝⁸ : (i : ι) → AddCommGroup (E i)
inst✝⁷ : (i : ι) → Module 𝕜 (E i)
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_4} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
hF : ¬T2Space F
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E (SeparationQuotient F))

Tactic:
exact this inferInstance

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

Goal state before:
pp: 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
⊢ IsOpen (range toContinuousLinearMap)

Tactic:
by
  rw [isOpen_iff_mem_nhds, forall_mem_range]
  refine fun e => IsOpen.mem_nhds ?_ (mem_range_self _)
  let O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (e.symm : F →L[𝕜] E).comp f
  have h_O : Continuous O := (isBoundedBilinearMap_comp (𝕜 := 𝕜) (F := F) (G := E)).continuous_right
  convert! show IsOpen (O ⁻¹' {x | IsUnit x}) from Units.isOpen.preimage h_O using 1
  ext f'
  constructor
  · rintro ⟨e', rfl⟩
    exact ⟨(e'.trans e.symm).toUnit, rfl⟩
  · rintro ⟨w, hw⟩
    use (unitsEquiv 𝕜 E w).trans e
    ext x
    simp [O, hw]

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

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : Submodule R M
f : M →L[R] ↥p
hf : ∀ (x : ↥p), f ↑x = x
⊢ IsTopCompl p (↑f).ker

Tactic:
by
  simpa using p.subtypeL.isTopCompl_range_ker_of_leftInverse f hf

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
α' : Type u_8
inst✝ : MeasurableSpace α'
f : α → α'
g : α' → ε
hg : AEStronglyMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
⊢ Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ

Tactic:
simp_rw [← memLp_one_iff_integrable]
  exact memLp_map_measure_iff hg hf

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
f : α → δ
hf : MeasurableEmbedding f
g : δ → ε
⊢ Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ

Tactic:
simp_rw [← memLp_one_iff_integrable]
  exact hf.memLp_map_measure_iff

Goal state after: