mathlib-initiative/mathlib-tactics huggingface dataset:
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✝ : NormedAddCommGroup F
f : α → F
⊢ eLpNorm' (fun a ↦ ‖f a‖) q μ = eLpNorm' f q μ

Tactic:
simp [eLpNorm'_eq_lintegral_enorm]

Goal state after:
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
f : α → ε
⊢ eLpNorm' (fun a ↦ ‖f a‖ₑ) q μ = eLpNorm' f q μ

Tactic:
simp [eLpNorm'_eq_lintegral_enorm]

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

Goal state before:
pp: case inl
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 ≤ -1
hs' : -1 ≠ 0
⊢ 1 + -p < 0

Tactic:
add_neg_lt_iff_lt_add,

Goal state after:
case inl
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 ≤ -1
hs' : -1 ≠ 0
⊢ 1 < 0 + p
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: case left
p q r : ℝ≥0
hr : r ≠ 0
h : (↑p).HolderTriple ↑q ↑r
⊢ p ≠ 0

Tactic:
rintro rfl
        apply hr
        exact_mod_cast (coe_zero ▸ h).unique _ _ r 0

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

Goal state before:
pp: case right
p q r : ℝ≥0
hr : r ≠ 0
h : (↑p).HolderTriple ↑q ↑r
⊢ q ≠ 0

Tactic:
rintro rfl
        apply hr
        exact_mod_cast (coe_zero ▸ h).unique _ _ r 0

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

Goal state before:
pp: case inr
s : ℝ
hs : -1 ≤ s
p : ℝ
hp1✝ : 0 ≤ p
hp2 : p ≤ 1
hp1 : 0 < p
⊢ (1 + s) ^ p ≤ 1 + p * s

Tactic:
rcases eq_or_lt_of_le hp2 with (rfl | hp2)

Goal state after:
case inr.inl
s : ℝ
hs : -1 ≤ s
hp1✝ : 0 ≤ 1
hp2 : 1 ≤ 1
hp1 : 0 < 1
⊢ (1 + s) ^ 1 ≤ 1 + 1 * s

case inr.inr
s : ℝ
hs : -1 ≤ s
p : ℝ
hp1✝ : 0 ≤ p
hp2✝ : p ≤ 1
hp1 : 0 < p
hp2 : p < 1
⊢ (1 + s) ^ p ≤ 1 + p * s
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

Tactic:
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

Goal state after:
case inr.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ ∑ i ∈ s, f' i * g' i ≤ 1
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ p
pos : 0 < p
H₁ :
  ∀ (s : Finset ι), ∑ i ∈ s, (f i + g i) ^ p ≤ ((∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)) ^ p
bdd : BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i + g i) ^ p)
H₂ : Summable fun i ↦ (f i + g i) ^ p
⊢ (∑' (i : ι), (f i + g i) ^ p) ^ p⁻¹ ≤ (∑' (i : ι), f i ^ p) ^ p⁻¹ + (∑' (i : ι), g i ^ p) ^ p⁻¹

Tactic:
NNReal.rpow_inv_le_iff pos,

Goal state after:
ι : Type u
f g : ι → ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ p
pos : 0 < p
H₁ :
  ∀ (s : Finset ι), ∑ i ∈ s, (f i + g i) ^ p ≤ ((∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)) ^ p
bdd : BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i + g i) ^ p)
H₂ : Summable fun i ↦ (f i + g i) ^ p
⊢ ∑' (i : ι), (f i + g i) ^ p ≤ ((∑' (i : ι), f i ^ p) ^ p⁻¹ + (∑' (i : ι), g i ^ p) ^ p⁻¹) ^ p
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
p : ℝ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
a : α
ha : Tendsto (fun n ↦ f n a) atTop (𝓝 (f_lim a))
⊢ (fun a ↦ ‖f_lim a‖ₑ ^ p) a = (fun a ↦ liminf (fun m ↦ ‖f m a‖ₑ ^ p) atTop) a

Tactic:
dsimp only

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
p : ℝ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
a : α
ha : Tendsto (fun n ↦ f n a) atTop (𝓝 (f_lim a))
⊢ ‖f_lim a‖ₑ ^ p = liminf (fun m ↦ ‖f m a‖ₑ ^ p) atTop
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ (fun x ↦ ‖f_lim x‖ₑ) x = (fun x ↦ liminf (fun m ↦ ‖f m x‖ₑ) atTop) x

Tactic:
dsimp only

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ ‖f_lim x‖ₑ = liminf (fun m ↦ ‖f m x‖ₑ) atTop
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case hx
ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
⊢ 0 ≤ ∑ i ∈ s, w i * f i ^ p

pp: case hx
ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
⊢ 0 ≤ ∑ i ∈ s, w i

Tactic:
· exact sum_nonneg fun i _ ↦ by have := hw i; have := hf i; positivity

Goal state after:
case hx
ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ
hw : ∀ (i : ι), 0 ≤ w i
hf : ∀ (i : ι), 0 ≤ f i
⊢ 0 ≤ ∑ i ∈ s, w i
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
c : ℝ≥0
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊
this : (eLpNorm (↑↑f) p μ).toNNReal ≤ (↑c * eLpNorm (↑↑g) p μ).toNNReal
⊢ (eLpNorm (↑↑f) p μ).toNNReal ≤ c * (eLpNorm (↑↑g) p μ).toNNReal

pp: case ha
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
c : ℝ≥0
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊
this : eLpNorm (↑↑f) p μ ≤ c • eLpNorm (↑↑g) p μ
⊢ eLpNorm (↑↑f) p μ ≠ ∞

pp: case hb
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
c : ℝ≥0
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊
this : eLpNorm (↑↑f) p μ ≤ c • eLpNorm (↑↑g) p μ
⊢ c • eLpNorm (↑↑g) p μ ≠ ∞

Tactic:
ENNReal.toNNReal_mul,

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
c : ℝ≥0
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊
this : (eLpNorm (↑↑f) p μ).toNNReal ≤ (↑c).toNNReal * (eLpNorm (↑↑g) p μ).toNNReal
⊢ (eLpNorm (↑↑f) p μ).toNNReal ≤ c * (eLpNorm (↑↑g) p μ).toNNReal

case ha
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
c : ℝ≥0
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊
this : eLpNorm (↑↑f) p μ ≤ c • eLpNorm (↑↑g) p μ
⊢ eLpNorm (↑↑f) p μ ≠ ∞

case hb
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
c : ℝ≥0
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊
this : eLpNorm (↑↑f) p μ ≤ c • eLpNorm (↑↑g) p μ
⊢ c • eLpNorm (↑↑g) p μ ≠ ∞
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ (i : ι), 0 ≤ f i
hg : ∀ (i : ι), 0 ≤ g i
A B : ℝ
hA : 0 ≤ A
hB : 0 ≤ B
hfA : HasSum (fun i ↦ f i ^ p) (A ^ p)
hgB : HasSum (fun i ↦ g i ^ p) (B ^ p)
⊢ ∃ C, 0 ≤ C ∧ C ≤ A + B ∧ HasSum (fun i ↦ (f i + g i) ^ p) (C ^ p)

Tactic:
lift f to ι → ℝ≥0 using hf
  lift g to ι → ℝ≥0 using hg
  lift A to ℝ≥0 using hA
  lift B to ℝ≥0 using hB
  -- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
  beta_reduce at hfA hgB
  norm_cast at hfA hgB
  obtain ⟨C, hC₁, hC₂⟩ := NNReal.Lp_add_le_hasSum hp hfA hgB
  use C
  -- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
  beta_reduce
  norm_cast
  exact ⟨zero_le, hC₁, hC₂⟩

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
⊢ ∫⁻ (a : α), liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) atTop ^ p ∂μ ≤
    liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), B i) atTop ^ p

Tactic:
rw [← h_tsum_1]

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 : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
⊢ ∫⁻ (a : α), liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) atTop ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h₁
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
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ 2 ^ n * 2 ^ 1 = ∞ → (2 ^ k)⁻¹ ≠ 0

pp: case h₂
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
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ (2 ^ k)⁻¹ = ∞ → 2 ^ n * 2 ^ 1 ≠ 0

pp: case h₁
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
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ (2 ^ k)⁻¹ * 2 ^ 1 = ∞ → 2 ^ n ≠ 0

pp: case h₂
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
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ 2 ^ n = ∞ → (2 ^ k)⁻¹ * 2 ^ 1 ≠ 0

Tactic:
all_goals simp

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

Goal state before:
pp: case pos.refine_1
α : Type u_1
m✝ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x ↦ ‖(f n - f m) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
n m : ℕ
⊢ ∀ N ≤ n, N ≤ m → dist (f n x) (f m x) ≤ (B N).toReal

Tactic:
N

Goal state after:
case pos.refine_1
α : Type u_1
m✝ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x ↦ ‖(f n - f m) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
n m N : ℕ
⊢ N ≤ n → N ≤ m → dist (f n x) (f m x) ≤ (B N).toReal
Module: Mathlib.MeasureTheory.Function.LpOrder

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : Fact (1 ≤ p)
f g : ↥(Lp E p μ)
hfg : |f| ≤ |g|
⊢ ‖f‖ ≤ ‖g‖

Tactic:
by
      rw [← coeFn_le] at hfg
      simp_rw [Lp.norm_def, ENNReal.toReal_le_toReal (Lp.eLpNorm_ne_top f) (Lp.eLpNorm_ne_top g)]
      refine eLpNorm_mono_ae ?_
      filter_upwards [hfg, Lp.coeFn_abs f, Lp.coeFn_abs g] with x hx hxf hxg
      rw [hxf, hxg] at hx
      exact HasSolidNorm.solid hx

Goal state after:
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
⊢ (fun x ↦
        match x with
        | ⟨(f, g), disj⟩ =>
          {
            toFun := fun x ↦
              match x with
              | Sum.inl a => f a
              | Sum.inr b => g b,
            inj' := ⋯ })
      ((fun f ↦ ⟨(inl.trans f, inr.trans f), ⋯⟩) f) =
    f

Tactic:
dsimp only

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α ⊕ β ↪ γ
⊢ {
      toFun := fun x ↦
        match x with
        | Sum.inl a => (inl.trans f) a
        | Sum.inr b => (inr.trans f) b,
      inj' := ⋯ } =
    f
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
h_B : ∃ N, B N ≤ ε
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - f_lim) p μ ≤ ε

Tactic:
obtain ⟨N, h_B⟩ := h_B

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 ≥ N, eLpNorm (f n - f_lim) p μ ≤ ε
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✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
hp : 1 ≤ p
f : ℕ → α → E
hf : ∀ (n : ℕ), MemLp (f n) p μ
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
⊢ ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)

Tactic:
by
  obtain ⟨f_lim, h_f_lim_meas, h_lim⟩ :
      ∃ f_lim : α → E, StronglyMeasurable f_lim ∧
        ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) :=
    exists_stronglyMeasurable_limit_of_tendsto_ae (fun n => (hf n).1)
      (ae_tendsto_of_cauchy_eLpNorm (fun n => (hf n).1) hp hB h_cau)
  have h_tendsto' : atTop.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0) :=
    cauchy_tendsto_of_tendsto (fun m => (hf m).1) f_lim hB h_cau h_lim
  have h_ℒp_lim : MemLp f_lim p μ :=
    memLp_of_cauchy_tendsto hp hf f_lim h_f_lim_meas.aestronglyMeasurable h_tendsto'
  exact ⟨f_lim, h_ℒp_lim, h_tendsto'⟩

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F✝ : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : (i : ι) → TopologicalSpace (E i)
inst✝¹² : (i : ι) → AddCommGroup (E i)
inst✝¹¹ : (i : ι) → Module 𝕜 (E i)
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜 F✝
inst✝⁸ : UniformSpace F✝
inst✝⁷ : IsUniformAddGroup F✝
inst✝⁶ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
F : Type u_4
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : UniformSpace F
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
hF : T2Space F
H :
  ∀ {m : (i : ι) → E i},
    Continuous[topologicalSpace ((i : ι) → E i) F {s | IsVonNBounded 𝕜 s}, inst✝³.toTopologicalSpace] fun f ↦
      (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E F)

Tactic:
completeSpace_iff_isComplete_range isUniformInducing_toUniformOnFun,

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F✝ : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : (i : ι) → TopologicalSpace (E i)
inst✝¹² : (i : ι) → AddCommGroup (E i)
inst✝¹¹ : (i : ι) → Module 𝕜 (E i)
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜 F✝
inst✝⁸ : UniformSpace F✝
inst✝⁷ : IsUniformAddGroup F✝
inst✝⁶ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
F : Type u_4
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : UniformSpace F
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
hF : T2Space F
H :
  ∀ {m : (i : ι) → E i},
    Continuous[topologicalSpace ((i : ι) → E i) F {s | IsVonNBounded 𝕜 s}, inst✝³.toTopologicalSpace] fun f ↦
      (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsComplete (range toUniformOnFun)
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : SeminormedRing A
inst✝⁵ : Algebra 𝕜 A
E : Type u_3
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module A E
inst✝¹ : IsBoundedSMul A E
inst✝ : IsScalarTower 𝕜 A E
x : A
y : E
⊢ ‖(x, y).1 • (x, y).2‖ ≤ ‖x‖ * ‖y‖

Tactic:
norm_smul_le

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : SeminormedRing A
inst✝⁵ : Algebra 𝕜 A
E : Type u_3
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module A E
inst✝¹ : IsBoundedSMul A E
inst✝ : IsScalarTower 𝕜 A E
x : A
y : E
⊢ ‖(x, y).1‖ * ‖(x, y).2‖ ≤ ‖x‖ * ‖y‖
Module: Mathlib.Topology.Algebra.Module.Simple

Goal state before:
pp: R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
l : M →ₗ[R] N
⊢ IsClosed ↑l.ker ∨ Dense ↑l.ker

Tactic:
rcases l.surjective_or_eq_zero with (hl | rfl)

Goal state after:
case inl
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
⊢ IsClosed ↑l.ker ∨ Dense ↑l.ker

case inr
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
⊢ IsClosed ↑(ker 0) ∨ Dense ↑(ker 0)
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
⊢ IsStrictMap f ↔ IsHomeomorph ⇑(quotientKerEquivRange f)

Tactic:
simp only [IsStrictMap, isHomeomorph_iff_isQuotientMap_injective, Equiv.injective, and_true]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
⊢ IsQuotientMap (rangeFactorization f) ↔ IsQuotientMap ⇑(quotientKerEquivRange f)
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : MeasurableSingletonClass α
inst✝ : SigmaFinite μ
s : Finset α
⊢ ∑ b ∈ s, μ.real {b} = μ.real ↑s

Tactic:
by
  simp [measureReal_def, ← ENNReal.toReal_sum (fun _ _ ↦ ne_of_lt measure_singleton_lt_top)]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : IsFiniteMeasure μ
c : E
inst✝ : NeZero μ
hp_zero : p ≠ 0
this : μ ≠ 0
⊢ ‖c‖ * (μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)

Tactic:
← ENNReal.toReal_rpow

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : IsFiniteMeasure μ
c : E
inst✝ : NeZero μ
hp_zero : p ≠ 0
this : μ ≠ 0
⊢ ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal) = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ∞
⊢ ‖c‖ * (μ Set.univ ^ (1 / p.toReal)).toReal = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)

Tactic:
← ENNReal.toReal_rpow

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
c : E
hp_zero : p ≠ 0
hp_top : p ≠ ∞
⊢ ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal) = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
s : Set α
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
h_compLp :
  ↑↑((ContinuousLinearMap.toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1)) =ᵐ[μ] fun a ↦
    (ContinuousLinearMap.toSpanSingleton ℝ x) (↑↑(indicatorConstLp 2 hs hμs 1) a)
y : α
hy : ↑↑(indicatorConstLp 2 hs hμs 1) y = s.indicator (fun x ↦ 1) y
⊢ s.indicator (fun x_1 ↦ x) y =
    (fun a ↦ (ContinuousLinearMap.toSpanSingleton ℝ x) (↑↑(indicatorConstLp 2 hs hμs 1) a)) y

Tactic:
dsimp only

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
s : Set α
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
h_compLp :
  ↑↑((ContinuousLinearMap.toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1)) =ᵐ[μ] fun a ↦
    (ContinuousLinearMap.toSpanSingleton ℝ x) (↑↑(indicatorConstLp 2 hs hμs 1) a)
y : α
hy : ↑↑(indicatorConstLp 2 hs hμs 1) y = s.indicator (fun x ↦ 1) y
⊢ s.indicator (fun x_1 ↦ x) y = (ContinuousLinearMap.toSpanSingleton ℝ x) (↑↑(indicatorConstLp 2 hs hμs 1) y)