mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MeanInequalities

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

pp: ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
⊢ 0 ≤ x

Tactic:
hw',

Goal state after:
ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
⊢ x ^ 1 = x

ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
⊢ 0 ≤ x
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁷ : CommRing 𝕜
inst✝¹⁶ : LinearOrder 𝕜
inst✝¹⁵ : IsStrictOrderedRing 𝕜
inst✝¹⁴ : CommRing E
inst✝¹³ : LinearOrder E
inst✝¹² : IsStrictOrderedRing E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : LinearOrder F
inst✝⁹ : IsOrderedAddMonoid F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜 G
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module 𝕜 F
inst✝⁴ : Module E F
inst✝³ : IsScalarTower 𝕜 E F
inst✝² : SMulCommClass 𝕜 E F
inst✝¹ : IsOrderedModule 𝕜 F
inst✝ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : MonovaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (a * 1) • f x • g x + (b * 1) • f y • g y = a • f x • g x + b • f y • g y

Tactic:
mul_one

Goal state after:
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 ∈ s, w i) * x = x

Tactic:
hw',

Goal state after:
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
⊢ 1 * x = x
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁸ : CommRing 𝕜
inst✝¹⁷ : LinearOrder 𝕜
inst✝¹⁶ : IsStrictOrderedRing 𝕜
inst✝¹⁵ : CommRing E
inst✝¹⁴ : LinearOrder E
inst✝¹³ : IsStrictOrderedRing E
inst✝¹² : AddCommGroup F
inst✝¹¹ : LinearOrder F
inst✝¹⁰ : IsOrderedAddMonoid F
inst✝⁹ : AddCommGroup G
inst✝⁸ : Module 𝕜 G
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : Module E F
inst✝⁴ : IsScalarTower 𝕜 E F
inst✝³ : SMulCommClass 𝕜 E F
inst✝² : IsOrderedModule 𝕜 F
inst✝¹ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
inst✝ : IsOrderedModule 𝕜 E
hf : ConcaveOn 𝕜 s f
hg : ConcaveOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : AntivaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x • g x + b • f y • g y = (a * 1) • f x • g x + (b * 1) • f y • g y

Tactic:
mul_one

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

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

Tactic:
hw',

Goal state after:
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∀ j ∈ s, ∀ k ∈ s, z j = z k
k : ι
hk : k ∈ s
⊢ z k ^ 1 = 1 * z k
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case pos
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : p = q
⊢ eLpNorm' f q μ ≤ eLpNorm' f q μ * 1

Tactic:
mul_one

Goal state after:
case pos
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : p = q
⊢ eLpNorm' f q μ ≤ eLpNorm' f q 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✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
hf : AEStronglyMeasurable f μ
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
by
  have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq
  by_cases hpq_eq : p = q
  · rw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_one]
  have hpq : p < q := lt_of_le_of_ne hpq hpq_eq
  let g := fun _ : α => (1 : ℝ≥0∞)
  have h_rw : (∫⁻ a, ‖f a‖ₑ ^ p ∂μ) = ∫⁻ a, (‖f a‖ₑ * g a) ^ p ∂μ :=
    lintegral_congr fun a => by simp [g]
  repeat' rw [eLpNorm'_eq_lintegral_enorm]
  rw [h_rw]
  let r := p * q / (q - p)
  have hpqr : 1 / p = 1 / q + 1 / r := by simp [field]
  calc
    (∫⁻ 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.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε : Type u_2
m : MeasurableSpace α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
μ : Measure α
inst✝ : IsProbabilityMeasure μ
hf : AEStronglyMeasurable f μ
h_le_μ : eLpNorm' f p μ ≤ eLpNorm' f q μ * 1
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ

Tactic:
mul_one

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
μ : Measure α
inst✝ : IsProbabilityMeasure μ
hf : AEStronglyMeasurable f μ
h_le_μ : eLpNorm' f p μ ≤ eLpNorm' f q μ
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: a b : ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
⊢ a * b ≤ a ^ p / p.toNNReal + b ^ q / q.toNNReal

Tactic:
simpa [hpq.nonneg, hpq.symm.nonneg] using young_inequality a b hpq.toNNReal

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

Goal state before:
pp: case h₁
α : Type u_2
inst✝ : MeasurableSpace α
p q r : ℝ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp0_ne : p ≠ 0
hp0 : 0 ≤ p
hq0_lt : 0 < q
hq0_ne : q ≠ 0
h_one_div_r : 1 / r = 1 / p - 1 / q
p2 : ℝ := ⋯
q2 : ℝ := ⋯
hp2q2 : p2.HolderConjugate q2
⊢ ∫⁻ (a : α), f a ^ p * g a ^ p ∂μ ≤
    (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2)

Tactic:
simp_rw [ENNReal.rpow_mul]

Goal state after:
case h₁
α : Type u_2
inst✝ : MeasurableSpace α
p q r : ℝ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp0_ne : p ≠ 0
hp0 : 0 ≤ p
hq0_lt : 0 < q
hq0_ne : q ≠ 0
h_one_div_r : 1 / r = 1 / p - 1 / q
p2 : ℝ := q / p
q2 : ℝ := p2.conjExponent
hp2q2 : p2.HolderConjugate q2
⊢ ∫⁻ (a : α), f a ^ p * g a ^ p ∂μ ≤
    (∫⁻ (a : α), (f a ^ p) ^ p2 ∂μ) ^ (1 / p2) * (∫⁻ (a : α), (g a ^ p) ^ q2 ∂μ) ^ (1 / q2)
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
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) * 1 ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Tactic:
mul_one

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
hg : AEMeasurable g μ
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
Module: Mathlib.Analysis.MeanInequalities

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

Tactic:
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
      have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
        refine fun y => mul_div_cancel_left_of_imp fun h => ?_
        simp [h, hpq.ne_zero]
      simp only [Set.mem_ofPred_eq, div_rpow, ← sum_div, ← rpow_mul,
        div_mul_cancel₀ _ hpq.symm.ne_zero, rpow_one, div_le_iff₀ hf, one_mul, hpq.mul_eq_add, ←
        rpow_sub' A, add_sub_cancel_right, le_refl, true_and, ← mul_div_assoc, B]
      rw [div_eq_iff, ← rpow_add hf.ne', one_div, one_div, hpq.inv_add_inv_eq_one, rpow_one]
      simpa [hpq.symm.ne_zero] using hf.ne'

Goal state after:
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✝ : PseudoMetricSpace E
f : ι → α → E
l : Filter ι
g : α → E
h : ∀ (ε : ℝ), 0 < ε → Tendsto (fun i ↦ μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
i : ι
a : α
⊢ ε ≤ edist (f i a) (g a) ↔ ε.toReal ≤ dist (f i a) (g a)

Tactic:
rw [edist_dist, ENNReal.le_ofReal_iff_toReal_le hε_top (by positivity)]

Goal state after:
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
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : ↥(Lp E p μ)
⊢ Tendsto (fun b ↦ dist (f b) f_lim) fi (𝓝 0) ↔ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)

Tactic:
simp_rw [dist_def]

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : ↥(Lp E p μ)
⊢ Tendsto (fun b ↦ (eLpNorm (↑↑(f b) - ↑↑f_lim) p μ).toReal) fi (𝓝 0) ↔
    Tendsto (fun n ↦ eLpNorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)
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
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖
⊢ ‖f‖ ≤ ‖g‖

Tactic:
rw [norm_def, norm_def]

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
f : ↥(Lp E p μ)
g : ↥(Lp F p μ)
h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖
⊢ (eLpNorm (↑↑f) p μ).toReal ≤ (eLpNorm (↑↑g) p μ).toReal
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f (p * 1) μ ≠ ∞

Tactic:
mul_one

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f p μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f (p * 1) μ ≠ ∞

Tactic:
mul_one

Goal state after:
case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f p μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case e'_7
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ≥0∞
f : α → ε
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ
⊢ p = p * 1

Tactic:
mul_one

Goal state after:
case e'_7
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ≥0∞
f : α → ε
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ
⊢ p = p
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case e'_7
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
q : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ
⊢ p = p * 1

Tactic:
mul_one

Goal state after:
case e'_7
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
q : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ
⊢ p = p
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✝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‖
⊢ ‖c ^ n‖⁻¹ * 1 ≤ ↑ε⁻¹ * ‖c‖ * ‖f x‖

Tactic:
mul_one

Goal state after:
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‖
⊢ ‖c ^ n‖⁻¹ ≤ ↑ε⁻¹ * ‖c‖ * ‖f x‖
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
⊢ IsBoundedLinearMap 𝕜 fun x ↦ x.2

Tactic:
refine (LinearMap.snd 𝕜 E F).isLinear.with_bound 1 fun x => ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
x : E × F
⊢ ‖(LinearMap.snd 𝕜 E F) x‖ ≤ 1 * ‖x‖
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
g_emb : IsEmbedding g
Φ : Quotient (ker (g ∘ f)) ≃ₜ Quotient (ker f) := Homeomorph.Quotient.congrRight ⋯
key : g ∘ kerLift f ∘ ⇑Φ = kerLift (g ∘ f)
⊢ IsStrictMap f ↔ IsStrictMap (g ∘ f)

Tactic:
simp_rw [isStrictMap_iff_isEmbedding_kerLift, ← g_emb.of_comp_iff, ← key]

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
g_emb : IsEmbedding g
Φ : Quotient (ker (g ∘ f)) ≃ₜ Quotient (ker f) := Homeomorph.Quotient.congrRight ⋯
key : g ∘ kerLift f ∘ ⇑Φ = kerLift (g ∘ f)
⊢ IsEmbedding (g ∘ kerLift f) ↔ IsEmbedding (g ∘ kerLift f ∘ ⇑Φ)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: case mp.ht
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' : ∀ᵐ (x : α) ∂μ.withDensity fun x ↦ ↑(f x), g x = g' x
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x

Tactic:
ae_withDensity_iff hf.coe_nnreal_ennreal

Goal state after:
case mp.ht
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' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
⊢ ∀ᵐ (x : α) ∂μ.withDensity fun x ↦ ↑(f x), g x = (↑(f x))⁻¹ • g' x

Tactic:
ae_withDensity_iff hf.coe_nnreal_ennreal

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hm : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ.real (s \ t) = μ.real (s ∪ t) - μ.real t

Tactic:
rw [union_comm, ← measureReal_add_sdiff hm h₂ h₁]
  ring

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hst : s ⊆ t
h_nulldiff : μ.real (t \ s) = 0
h : μ (t \ s) ≠ ∞
⊢ μ.real s = μ.real t

Tactic:
rw [measureReal_eq_zero_iff h] at h_nulldiff
  simp [measureReal_def, measure_eq_measure_of_null_sdiff hst h_nulldiff]

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
s : Set α
μ₁ μ₂ : Measure α
h₁ : μ₁ s ≠ ∞
h₂ : μ₂ s ≠ ∞
⊢ (μ₁ + μ₂).real s = μ₁.real s + μ₂.real s

Tactic:
by
  simp only [measureReal_def, add_apply, ENNReal.toReal_add h₁ h₂]

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
hp_pos : p ≠ 0
hμs_pos : μ s ≠ 0
hp_top : p = ∞
⊢ ‖indicatorConstLp ∞ hs hμs c‖ = ‖c‖ * 1

Tactic:
mul_one

Goal state after:
case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
hp_pos : p ≠ 0
hμs_pos : μ s ≠ 0
hp_top : p = ∞
⊢ ‖indicatorConstLp ∞ hs hμs c‖ = ‖c‖
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
inst✝ : IsFiniteMeasure μ
f : α → ε
p : ℕ
hf : MemLp f (↑p) μ
⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p) μ

Tactic:
by simpa using hf.integrable_enorm_rpow'

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝¹ : {ξ₀}ᶜ ∈ 𝓝 0
this✝ : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
this : ξ₀ * 1 ∈ balancedCore 𝕜 {ξ₀}ᶜ
⊢ False

Tactic:
mul_one

Goal state after:
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝¹ : {ξ₀}ᶜ ∈ 𝓝 0
this✝ : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
this : ξ₀ ∈ balancedCore 𝕜 {ξ₀}ᶜ
⊢ False
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
this : f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x
⊢ Integrable f (μ.restrict {x}) ↔ ‖f x‖ₑ = 0 ∨ μ {x} < ∞

Tactic:
integrable_congr this,

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
this : f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x
⊢ Integrable (fun x_1 ↦ f x) (μ.restrict {x}) ↔ ‖f x‖ₑ = 0 ∨ μ {x} < ∞
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

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

Tactic:
IntegrableAtFilter,

Goal state after:
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
⊢ (∃ s ∈ map e l, IntegrableOn f s (Measure.map e μ)) ↔ ∃ s ∈ l, IntegrableOn (f ∘ e) s μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: 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
hflt : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → E
⊢ Integrable (fun x ↦ ↑(f x).toNNReal • g x) μ ↔ Integrable (fun x ↦ (f x).toReal • g x) μ

Tactic:
ENNReal.toReal

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

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

Tactic:
IntegrableAtFilter,

Goal state after:
α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
l : Filter α
inst✝ : MeasurableSpace β
e : α → β
he : MeasurableEmbedding e
f : β → ε
μ : Measure β
⊢ (∃ s ∈ map e l, IntegrableOn f s μ) ↔ ∃ s ∈ map e l, IntegrableOn f s (Measure.map e (Measure.comap e μ))
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
ha : ‖f a‖ₑ ≠ ∞
⊢ μ {a} ≠ ∞

Tactic:
rw [measure_singleton]; exact ENNReal.zero_ne_top

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
hb : ‖f b‖ₑ ≠ ∞
⊢ μ {b} ≠ ∞

Tactic:
rw [measure_singleton]; exact ENNReal.zero_ne_top

Goal state after: