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

Goal state before:
pp: case h₂
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
p : ℝ≥0∞
f : α → E
g : α → F
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) p μ ≤ eLpNorm f ∞ μ * eLpNorm g p μ

Tactic:
obtain (rfl | rfl | hp) := ENNReal.trichotomy p

Goal state after:
case h₂.inl
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) 0 μ ≤ eLpNorm f ∞ μ * eLpNorm g 0 μ

case h₂.inr.inl
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) ∞ μ ≤ eLpNorm f ∞ μ * eLpNorm g ∞ μ

case h₂.inr.inr
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
p : ℝ≥0∞
f : α → E
g : α → F
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hp : 0 < p.toReal
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) p μ ≤ eLpNorm f ∞ μ * eLpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ

Tactic:
obtain (rfl | rfl | hp) := ENNReal.trichotomy p

Goal state after:
case inl
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : ENNReal.HolderTriple 0 q r
hpqr : 1 / r = 1 / 0 + 1 / q
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f 0 μ * eLpNorm g q μ

case inr.inl
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : ∞.HolderTriple q r
hpqr : 1 / r = 1 / ∞ + 1 / q
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f ∞ μ * eLpNorm g q μ

case inr.inr
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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
i : ι
x✝ : i ∈ s
⊢ 0 ≤ w i * f i ^ p

Tactic:
have := hw i

Goal state after:
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
i : ι
x✝ : i ∈ s
this : 0 ≤ w i
⊢ 0 ≤ w i * f i ^ p
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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
i : ι
x✝ : i ∈ s
⊢ 0 ≤ w i * f i ^ p

Tactic:
by have := hw i; have := hf i; positivity

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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
i : ι
x✝ : i ∈ s
⊢ 0 ≤ w i

Tactic:
have := hw i

Goal state after:
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
i : ι
x✝ : i ∈ s
this : 0 ≤ w i
⊢ 0 ≤ w i
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
f : ↥(Lp E p μ)
hp : 0 < p
hf : eLpNorm (↑↑f) p μ = 0
⊢ f = 0

Tactic:
simp_rw [eq_zero_iff_ae_eq_zero, ← eLpNorm_eq_zero_iff (Lp.aestronglyMeasurable f) hp.ne.symm, hf]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)

Tactic:
convert! Lp_add_le s f g hp using 2 <;> [skip; congr 1; congr 1] <;> apply sum_congr rfl <;>
      intro i hi

Goal state after:
case e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
i : ι
hi : i ∈ s
⊢ (f i + g i) ^ p = |f i + g i| ^ p

case e'_4.e'_5.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
i : ι
hi : i ∈ s
⊢ f i ^ p = |f i| ^ p

case e'_4.e'_6.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
i : ι
hi : i ∈ s
⊢ g i ^ p = |g i| ^ p
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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
H' : (∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ 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)

pp: 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
H' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ 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:
· rcases H' with H' | H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]

Goal state after:
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
H' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ 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)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∑ i ∈ s, w i) ^ (1 - p⁻¹) = ∞ ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = ∞
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

pp: case neg
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : ¬((∑ i ∈ s, w i) ^ (1 - p⁻¹) = ∞ ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = ∞)
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
· rcases H' with H' | H' <;> simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]

Goal state after:
case neg
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : ¬((∑ i ∈ s, w i) ^ (1 - p⁻¹) = ∞ ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = ∞)
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: case neg
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : Filter.comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Tactic:
rw [← h.1] at hc

Goal state after:
case neg
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : Filter.comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: case neg
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : Filter.comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
n : ℤ
hlt : ‖σ₁₂ c ^ n • f x‖ < ↑ε
hle : ‖σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖σ₁₂ c‖ * ‖f x‖
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Tactic:
simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle

Goal state after:
case neg
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : Filter.comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
n : ℤ
hlt : ‖f (c ^ n • x)‖ < ↑ε
hle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖
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_1.2 - x.2) =o[𝓝 x] fun _x ↦ 1

Tactic:
isLittleO_one_iff,

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
⊢ Tendsto (fun x_1 ↦ x_1.2 - x.2) (𝓝 x) (𝓝 0)
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✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι✝) → TopologicalSpace (E i)
inst✝⁵ : (i : ι✝) → AddCommGroup (E i)
inst✝⁴ : (i : ι✝) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGroup F
ι : Type u_5
p : ι → Prop
b : ι → Set F
h : (𝓝 0).HasBasis p b
this✝ : UniformSpace F := IsTopologicalAddGroup.rightUniformSpace F
this : IsUniformAddGroup F
⊢ (𝓝 0).HasBasis (fun Si ↦ IsVonNBounded 𝕜 Si.1 ∧ p Si.2) fun Si ↦ {f | MapsTo (⇑f) Si.1 (b Si.2)}

Tactic:
rw [nhds_induced]

Goal state after:
𝕜 : Type u_1
ι✝ : Type u_2
E : ι✝ → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι✝) → TopologicalSpace (E i)
inst✝⁵ : (i : ι✝) → AddCommGroup (E i)
inst✝⁴ : (i : ι✝) → Module 𝕜 (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGroup F
ι : Type u_5
p : ι → Prop
b : ι → Set F
h : (𝓝 0).HasBasis p b
this✝ : UniformSpace F := IsTopologicalAddGroup.rightUniformSpace F
this : IsUniformAddGroup F
⊢ (Filter.comap toUniformOnFun (𝓝 (toUniformOnFun 0))).HasBasis (fun Si ↦ IsVonNBounded 𝕜 Si.1 ∧ p Si.2) fun Si ↦
    {f | MapsTo (⇑f) Si.1 (b Si.2)}
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: case mp
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' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'
A : MeasurableSet {x | f x ≠ 0}
⊢ AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ

Tactic:
refine ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.stronglyMeasurable.smul g'meas, ?_⟩

Goal state after:
case mp
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' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'
A : MeasurableSet {x | f x ≠ 0}
⊢ (fun x ↦ ↑(f x) • g x) =ᵐ[μ] fun x ↦ ↑(f x) • g' x
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 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
hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0
ε : ι → ℝ
C : ℝ
hε : ∀ (i : ι), 0 < ε i
c : ι → 𝕜
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
by
  by_cases! hm : ∃ i, ‖m i‖ = 0
  · rcases hm with ⟨i, hi⟩
    rw [hf₀ hi, prod_eq_zero (mem_univ i) hi, mul_zero]
  choose δ hδ0 hδm_lt hle_δm _ using fun i => rescale_to_shell_semi_normed (hc i) (hε i) (hm i)
  have hδ0 : 0 < ∏ i, ‖δ i‖ := prod_pos fun i _ => norm_pos_iff.2 (hδ0 i)
  simpa [map_smul_univ, norm_smul, prod_mul_distrib, mul_left_comm C, hδ0] using
    hf (fun i => δ i • m i) hle_δm hδm_lt

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

Goal state before:
pp: case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
l : E →ₗ[𝕜] 𝕜
hl : IsClosed[inst✝²] ↑l.ker
H : ¬finrank 𝕜 ↥l.range = 0
this : finrank 𝕜 ↥l.range = 1
hi : Function.Injective ⇑(l.ker.liftQ l ⋯)
hs : Function.Surjective ⇑(l.ker.liftQ l ⋯)
φ : (E ⧸ l.ker) ≃ₗ[𝕜] 𝕜 := LinearEquiv.ofBijective (l.ker.liftQ l ⋯) ⋯
⊢ Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑l

Tactic:
have hlφ : (l : E → 𝕜) = φ ∘ (LinearMap.ker l).mkQ := by ext; rfl

Goal state after:
case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul 𝕜 E
l : E →ₗ[𝕜] 𝕜
hl : IsClosed[inst✝²] ↑l.ker
H : ¬finrank 𝕜 ↥l.range = 0
this : finrank 𝕜 ↥l.range = 1
hi : Function.Injective ⇑(l.ker.liftQ l ⋯)
hs : Function.Surjective ⇑(l.ker.liftQ l ⋯)
φ : (E ⧸ l.ker) ≃ₗ[𝕜] 𝕜 := LinearEquiv.ofBijective (l.ker.liftQ l ⋯) ⋯
hlφ : ⇑l = ⇑φ ∘ ⇑l.ker.mkQ
⊢ Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑l
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
f : α → β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : AddLeftMono β
hf : Integrable f μ
c : β
c_neg : c < 0
⊢ μ {a | f a ≤ c} < ∞

Tactic:
have : 0 < ‖c‖ := by positivity [c_neg.ne]
  refine lt_of_le_of_lt (measure_mono fun x hx => ?_) (hf.measure_norm_ge_lt_top this)
  have : -c ≤ -f x := by simp; grind
  exact norm_le_norm_of_abs_le_abs <| abs_of_nonpos c_neg.le ▸ this.trans (neg_le_abs _)

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

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ ∫⁻ (a : α), ((fun x ↦ ↑(f x)) * fun a ↦ ‖g a‖ₑ) a ∂μ < ∞ ↔ ∫⁻ (a : α), ‖↑(f a) • g a‖ₑ ∂μ < ∞

pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ AEMeasurable (fun a ↦ ‖g a‖ₑ) (μ.withDensity fun x ↦ ↑(f x))

Tactic:
· simp [enorm_smul]

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ AEMeasurable (fun a ↦ ‖g a‖ₑ) (μ.withDensity fun x ↦ ↑(f x))
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
H : ¬AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable (fun x ↦ ↑(f x) • g x) μ

Tactic:
simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and]

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

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f g : α → E
h : Disjoint (support f) (support g)
hf : StronglyMeasurable f
hg : StronglyMeasurable g
hfg : Integrable (f + g) μ
⊢ Integrable g μ

Tactic:
rw [← indicator_add_eq_right h]

Goal state after:
case refine_2
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f g : α → E
h : Disjoint (support f) (support g)
hf : StronglyMeasurable f
hg : StronglyMeasurable g
hfg : Integrable (f + g) μ
⊢ Integrable ((support g).indicator (f + g)) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case refine_2.inr
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 : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
hz : 0 < ‖z‖₊
⊢ ‖f‖₊ ≤ ‖f.smulRight z‖₊ / ‖z‖₊

Tactic:
opNNNorm_le_iff

Goal state after:
case refine_2.inr
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 : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
hz : 0 < ‖z‖₊
⊢ ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ ‖f.smulRight z‖₊ / ‖z‖₊ * ∏ i, ‖m i‖₊
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
p : Submodule 𝕜 E
hp : IsClosed[inst✝³] ↑p
hq✝ : FiniteDimensional 𝕜 (E ⧸ p)
q : Submodule 𝕜 E
hq : IsCompl p q
⊢ p.ClosedComplemented

Tactic:
exact hq.isTopCompl_of_finiteDimensional_quotient hp |>.closedComplemented

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
A B : Submodule 𝕜 E
B_cofg : FiniteDimensional 𝕜 (E ⧸ B)
hB : IsClosed[inst✝³] ↑B
hAB : Disjoint A B
⊢ A.ClosedComplemented

Tactic:
by
  obtain ⟨C, B_le_C, C_compl_A⟩ := hAB.symm.exists_isCompl
  have C_cofg : FiniteDimensional 𝕜 (E ⧸ C) := CoFG.of_le B_le_C B_cofg
  have hC : IsClosed (C : Set E) := isClosed_mono_of_finiteDimensional_quotient hB B_le_C
  exact C_compl_A.isTopCompl_of_finiteDimensional_quotient hC |>.symm.closedComplemented

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : SeminormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : SeminormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
a : 𝕜
f₀ : E i →L[𝕜] E₁ i
⊢ compContinuousLinearMapL (update f i (a • f₀)) = a • compContinuousLinearMapL (update f i f₀)

Tactic:
by
    ext g x
    change (g fun j ↦ update f i (a • f₀) j <| x j) = a • g fun j ↦ update f i f₀ j (x j)
    convert! g.map_update_smul (fun j ↦ f j (x j)) i a (f₀ (x i)) <;>
      exact apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

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

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
hs : IsLocallyClosed s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ f x * g x) k μ

Tactic:
exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
μ : Measure β
inst✝ : SeparableSpace ↑(Set.range f ∪ {0})
fmeas : Measurable f
hf : Integrable f μ
⊢ Tendsto (fun n ↦ ∫⁻ (x : β), ‖(approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) x - f x‖ₑ ∂μ) atTop (𝓝 0)

Tactic:
by
  apply tendsto_approxOn_L1_enorm fmeas
  · filter_upwards with x using subset_closure (by simp)
  · simpa using hf.2

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : p = ∞
⊢ MemLp (⇑f) p μ

pp: case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ MemLp (⇑f) p μ

Tactic:
· rw [hp_top]; exact memLp_top f μ

Goal state after:
case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ MemLp (⇑f) p μ