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

Goal state before:
pp: α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
s : Finset ι
g : α → ℝ≥0∞
f : ι → α → ℝ≥0∞
hg : AEMeasurable g μ
hf : ∀ i ∈ s, AEMeasurable (f i) μ
q : ℝ
p : ι → ℝ
hpq : q + ∑ i ∈ s, p i = 1
hq : 0 ≤ q
hp : ∀ i ∈ s, 0 ≤ p i
⊢ ∫⁻ (t : α), ∏ j ∈ insertNone s, (j.elim (g t) fun j ↦ f j t) ^ j.elim q p ∂μ ≤
    ∏ j ∈ insertNone s, (∫⁻ (t : α), j.elim (g t) fun j ↦ f j t ∂μ) ^ j.elim q p

Tactic:
refine ENNReal.lintegral_prod_norm_pow_le _ ?_ ?_ ?_

Goal state after:
case refine_1
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
s : Finset ι
g : α → ℝ≥0∞
f : ι → α → ℝ≥0∞
hg : AEMeasurable g μ
hf : ∀ i ∈ s, AEMeasurable (f i) μ
q : ℝ
p : ι → ℝ
hpq : q + ∑ i ∈ s, p i = 1
hq : 0 ≤ q
hp : ∀ i ∈ s, 0 ≤ p i
⊢ ∀ i ∈ insertNone s, AEMeasurable (fun t ↦ i.elim (g t) fun j ↦ f j t) μ

case refine_2
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
s : Finset ι
g : α → ℝ≥0∞
f : ι → α → ℝ≥0∞
hg : AEMeasurable g μ
hf : ∀ i ∈ s, AEMeasurable (f i) μ
q : ℝ
p : ι → ℝ
hpq : q + ∑ i ∈ s, p i = 1
hq : 0 ≤ q
hp : ∀ i ∈ s, 0 ≤ p i
⊢ ∑ i ∈ insertNone s, i.elim q p = 1

case refine_3
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
s : Finset ι
g : α → ℝ≥0∞
f : ι → α → ℝ≥0∞
hg : AEMeasurable g μ
hf : ∀ i ∈ s, AEMeasurable (f i) μ
q : ℝ
p : ι → ℝ
hpq : q + ∑ i ∈ s, p i = 1
hq : 0 ≤ q
hp : ∀ i ∈ s, 0 ≤ p i
⊢ ∀ i ∈ insertNone s, 0 ≤ i.elim q p
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
p : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
g : α → F
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
⊢ ↑c * eLpNorm g ∞ μ * eLpNorm f p μ = ↑c * eLpNorm f p μ * eLpNorm g ∞ μ

Tactic:
by
      simp only [mul_assoc]; rw [mul_comm (eLpNorm _ _ _)]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr.calc_2
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
⊢ (∑ i ∈ s, (w i ^ (1 - p⁻¹)) ^ (1 - p⁻¹)⁻¹) ^ (1 / (1 - p⁻¹)⁻¹) * (∑ i ∈ s, (w i ^ p⁻¹ * f i) ^ p) ^ (1 / p) =
    (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
have hp₀ : p ≠ 0 := by positivity

Goal state after:
case inr.calc_2
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
hp₀ : p ≠ 0
⊢ (∑ i ∈ s, (w i ^ (1 - p⁻¹)) ^ (1 - p⁻¹)⁻¹) ^ (1 / (1 - p⁻¹)⁻¹) * (∑ i ∈ s, (w i ^ p⁻¹ * f i) ^ p) ^ (1 / p) =
    (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case refine_1
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)
hB : B = (∑' (i : ι), g i ^ q) ^ (1 / q)
⊢ ∑' (i : ι), f i * g i ≤ A * B

pp: case refine_2
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)
hB : B = (∑' (i : ι), g i ^ q) ^ (1 / q)
⊢ HasSum (fun i ↦ f i * g i) (∑' (i : ι), f i * g i)

Tactic:
· simpa [hA, hB] using H₂

Goal state after:
case refine_2
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)
hB : B = (∑' (i : ι), g i ^ q) ^ (1 / q)
⊢ HasSum (fun i ↦ f i * g i) (∑' (i : ι), f i * g i)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case refine_1
ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ p) (B ^ p)
hp' : p ≠ 0
H₁ : Summable fun i ↦ (f i + g i) ^ p
H₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
hB : B = (∑' (i : ι), g i ^ p) ^ (1 / p)
⊢ (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ A + B

pp: case refine_2
ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ p) (B ^ p)
hp' : p ≠ 0
H₁ : Summable fun i ↦ (f i + g i) ^ p
H₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
hB : B = (∑' (i : ι), g i ^ p) ^ (1 / p)
⊢ HasSum (fun i ↦ (f i + g i) ^ p) (((∑' (i : ι), (f i + g i) ^ p) ^ (1 / p)) ^ p)

Tactic:
· simpa [hA, hB] using H₂

Goal state after:
case refine_2
ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ p) (B ^ p)
hp' : p ≠ 0
H₁ : Summable fun i ↦ (f i + g i) ^ p
H₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
hB : B = (∑' (i : ι), g i ^ p) ^ (1 / p)
⊢ HasSum (fun i ↦ (f i + g i) ^ p) (((∑' (i : ι), (f i + g i) ^ p) ^ (1 / p)) ^ p)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
K : Type u_8
inst✝ : RCLike K
f : α → K
⊢ MemLp (fun x ↦ RCLike.re (f x)) p μ ∧ MemLp (fun x ↦ RCLike.im (f x)) p μ ↔ MemLp f p μ

Tactic:
refine ⟨?_, fun hf => ⟨hf.re, hf.im⟩⟩
  rintro ⟨hre, him⟩
  convert! MeasureTheory.MemLp.add (ε := K) hre.ofReal (him.ofReal.const_mul RCLike.I)
  ext1 x
  rw [Pi.add_apply, mul_comm, RCLike.re_add_im]

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

Goal state before:
pp: 𝕜✝ : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜✝
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜✝ E
R✝ : Type u_3
inst✝⁷ : NonUnitalSeminormedRing R✝
inst✝⁶ : NormedSpace 𝕜✝ R✝
inst✝⁵ : IsScalarTower 𝕜✝ R✝ R✝
inst✝⁴ : SMulCommClass 𝕜✝ R✝ R✝
𝕜 : Type u_4
R : Type u_5
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedRing R
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : NormOneClass R
x : R
⊢ ‖x‖ ≤ ‖(mul 𝕜 R) x‖

Tactic:
by
      convert! ratio_le_opNorm ((mul 𝕜 R) x) (1 : R)
      simp [norm_one]

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

Goal state before:
pp: case mpr
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 : DecidableEq ι
inst✝ : TopologicalSpace F
f : ((i : ι) → E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F
⊢ f ∈
      {f |
        Continuous[Pi.topologicalSpace, inst✝] ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
          (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
                (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
            ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
              (toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
                c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)} →
    f ∈ range toUniformOnFun

Tactic:
rintro ⟨hcont, hadd, hsmul⟩
    exact ⟨⟨⟨f, by intro; convert! hadd, by intro; convert! hsmul⟩, hcont⟩, rfl⟩

Goal state after:
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:
by
  refine (LinearMap.snd 𝕜 E F).isLinear.with_bound 1 fun x => ?_
  rw [one_mul]
  exact le_max_right _ _

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✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
⊢ Tendsto (fun x ↦ ↑(‖c‖₊ * x ^ (1 / p.toReal))) (𝓝 0) (𝓝 ↑0)

Tactic:
ENNReal.tendsto_coe

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
⊢ Tendsto (fun x ↦ ‖c‖₊ * x ^ (1 / p.toReal)) (𝓝 0) (𝓝 0)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: case e'_5
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
hp₁ : Fact (1 ≤ p)
β : Type u_3
l : Filter β
t : β → Set α
ht : ∀ (b : β), MeasurableSet (t b)
hμt : ∀ (b : β), μ (t b) ≠ ∞
hp : p ≠ ∞
h : Tendsto (fun b ↦ μ (t b ∆ s)) l (𝓝 0)
hp₀ : p ≠ 0
⊢ 0 = ‖c‖ * ENNReal.toReal 0 ^ (1 / p.toReal)

Tactic:
simp [ENNReal.toReal_eq_zero_iff, hp, hp₀]

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 ∈ balancedCore 𝕜 {ξ₀}ᶜ
⊢ False

Tactic:
exact notMem_compl_iff.mpr (mem_singleton ξ₀) ((balancedCore_subset _) this)

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
ν : Measure δ
g : δ → ε
f : α → δ
hf : MeasurePreserving f μ ν
hg : AEStronglyMeasurable g (Measure.map f μ)
⊢ Integrable (g ∘ f) μ ↔ Integrable g (Measure.map f μ)

Tactic:
exact (integrable_map_measure hg hf.measurable.aemeasurable).symm

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : IsOrderedAddMonoid β
f g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ Integrable (f ⊓ g) μ

Tactic:
by
  rw [← memLp_one_iff_integrable] at hf hg ⊢
  exact hf.inf hg

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ AEMeasurable f Measure.count

Tactic:
refine (measurable_zero.measurable_of_countable_ne ?_).aemeasurable

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ {x | 0 x ≠ f x}.Countable
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝¹ : Fintype ι
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
⊢ ∀ (n : ℕ), Fintype.card ι = n → Continuous[inst✝⁵, Pi.topologicalSpace] ⇑ξ.equivFun

Tactic:
intro n hn

Goal state after:
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝¹ : Fintype ι
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
n : ℕ
hn : Fintype.card ι = n
⊢ Continuous[inst✝⁵, Pi.topologicalSpace] ⇑ξ.equivFun
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

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

Tactic:
rw [sub_eq_add_neg]
  exact hf.add hg.neg

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Continuous[inst✝³, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
a : α
⊢ IntegrableAtFilter f (𝓝 a) μ

Tactic:
rw [← nhdsWithin_univ]
  exact hf.continuousOn.integrableAt_nhdsWithin MeasurableSet.univ (mem_univ a)

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

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

Tactic:
exact hab.le

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

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

Tactic:
exact hab.le

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

Goal state before:
pp: case h_e
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
C : ℝ
h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
g : Eₗ →SL[σ₁₂] F
H : ↑g ∘ₛₗ e = f
⊢ IsUniformInducing ⇑e.range.subtypeL

Tactic:
exact isUniformEmbedding_subtype_val.isUniformInducing

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

Goal state before:
pp: α : Type u_1
F : Type u_3
𝕜 : Type u_6
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : SMulZeroClass 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : DistribSMul 𝕜 F
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α →ₛ E
hf : Integrable (⇑f) μ
b : E
x✝ : b ∈ f.range
⊢ (T (⇑f ⁻¹' {b})) (c • b) = c • (T (⇑f ⁻¹' {b})) b

Tactic:
by rw [h_smul]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ ∫ (a : α), ↑↑f a ∂μ

Tactic:
simp only [integral_eq_setToFun]
  exact continuous_setToFun (dominatedFinMeasAdditive_weightedSMul μ)

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h✝ : c ≠ 0
h : Tendsto (u - fun x ↦ c) l (𝓝 0)
this : Tendsto (fun x ↦ (u - fun x ↦ c) x - -c) l (𝓝 (0 - -c))
⊢ Tendsto u l (𝓝 c)

Tactic:
simp only [Pi.sub_apply, sub_neg_eq_add, sub_add_cancel, zero_add] at this

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h✝ : c ≠ 0
h : Tendsto (u - fun x ↦ c) l (𝓝 0)
this : Tendsto (fun x ↦ u x) l (𝓝 c)
⊢ Tendsto u l (𝓝 c)
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case succ
α : Type u_1
β : Type u_3
inst✝ : NormedField β
t u : α → β
l : Filter α
h : t ~[l] u
n✝ : ℕ
ih : t ^ n✝ ~[l] u ^ n✝
⊢ t ^ (n✝ + 1) ~[l] u ^ (n✝ + 1)

Tactic:
simpa [pow_succ] using ih.mul h

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄

Tactic:
simp only [edist_nndist]
  norm_cast
  apply dist_vsub_vsub_le

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ ∫ (x : α), f x (p • a + q • b) ∂μ ≤ ∫ (x : α), p • f x a + q • f x b ∂μ

Tactic:
refine integral_mono_ae ?lhs ?rhs ?ae_le
                  case lhs =>
                    refine hf_int _ ?_
                    rw [convex_iff_add_mem] at hs
                    exact hs ha hb hp hq hpq
                  case rhs => fun_prop (disch := aesop)
                  case ae_le =>
                    filter_upwards [hf_conv] with x hx
                    exact hx.2 ha hb hp hq hpq

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
⊢ ‖f z‖ ≤ c / r * ‖z‖

Tactic:
by_cases z_zero : z = 0

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : z = 0
⊢ ‖f z‖ ≤ c / r * ‖z‖

case neg
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
⊢ ‖f z‖ ≤ c / r * ‖z‖
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G'' : Type u_7
G' : Type u_8
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : PartialOrder G'
inst✝⁵ : IsOrderedAddMonoid G'
inst✝⁴ : NormedSpace ℝ G'
inst✝³ : NormedAddCommGroup G''
inst✝² : PartialOrder G''
inst✝¹ : IsOrderedAddMonoid G''
inst✝ : NormedSpace ℝ G''
T : Set α → G'' →L[ℝ] G'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G''), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(α →₁ₛ[μ] G'')
hfg : f ≤ g
⊢ setToL1S T f ≤ setToL1S T g

Tactic:
by
  rw [← sub_nonneg] at hfg ⊢
  rw [← setToL1S_sub h_zero h_add]
  exact setToL1S_nonneg h_zero h_add hT_nonneg hfg

Goal state after:
Module: Mathlib.Topology.Instances.Matrix

Goal state before:
pp: X : Type u_1
n : Type u_5
R : Type u_8
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace R
inst✝⁴ : Fintype n
inst✝³ : Mul R
inst✝² : AddCommMonoid R
inst✝¹ : ContinuousAdd R
inst✝ : ContinuousMul R
A B : X → n → R
hA : Continuous A
hB : Continuous B
⊢ Continuous fun x ↦ A x ⬝ᵥ B x

Tactic:
dsimp only [dotProduct]

Goal state after:
X : Type u_1
n : Type u_5
R : Type u_8
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace R
inst✝⁴ : Fintype n
inst✝³ : Mul R
inst✝² : AddCommMonoid R
inst✝¹ : ContinuousAdd R
inst✝ : ContinuousMul R
A B : X → n → R
hA : Continuous A
hB : Continuous B
⊢ Continuous fun x ↦ ∑ i, A x i * B x i
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : IsOrderedAddMonoid β
⊢ Injective fun f ↦ f.toFun

Tactic:
by simp [Injective]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖star f‖ ≤ ‖f‖

Tactic:
by
    rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
      norm_star, BoundedContinuousFunction.norm_mkOfCompact]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
f : ↥(Lp G 1 μ)
⊢ ↑↑(Integrable.toL1 ↑↑f ⋯) =ᵐ[0] ↑↑(0 f)

Tactic:
simp only [ae_zero, EventuallyEq, eventually_bot]

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε
n : ℕ
⊢ n ≥ N → thickenedIndicatorAux (δseq n) E x = 0

Tactic:
n_large

Goal state after:
case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε
n : ℕ
n_large : n ≥ N
⊢ thickenedIndicatorAux (δseq n) E x = 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
c : ℝ≥0
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f

Tactic:
rw! [

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
c : ℝ≥0
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
⊢ ∫ (x : X) in univ, f x ∂μ = ∫ (x : X), f x ∂μ

Tactic:
by rw [Measure.restrict_univ]

Goal state after: