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

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n ↦ approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
h_toLp : ∀ (n : ℕ), ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᵐ[μ] ⇑(x n)
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)

Tactic:
have hx_nonneg_Lp : ∀ n, 0 ≤ toLp (x n) (hx_memLp n) := by
    intro n
    rw [← Lp.simpleFunc.coeFn_le, Lp.simpleFunc.toLp_eq_toLp]
    filter_upwards [Lp.simpleFunc.coeFn_zero p μ G, h_toLp n] with a ha0 ha_toLp
    rw [ha0, ha_toLp]
    exact hx_nonneg n a

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n ↦ approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
h_toLp : ∀ (n : ℕ), ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᵐ[μ] ⇑(x n)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case refine_1
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : ↥(Lp E p μ) → Prop
indicatorConst :
  ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α → E⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    Disjoint (support f) (support g) →
      motive (MemLp.toLp f hf) → motive (MemLp.toLp g hg) → motive (MemLp.toLp f hf + MemLp.toLp g hg)
isClosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive f}
f : ↥(Lp E p μ)
⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), motive ↑(simpleFunc.indicatorConst p hs ⋯ c)

Tactic:
exact fun c s => indicatorConst c

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T T' : Set α → E →L[ℝ] F'
c : ℝ
h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T' s = c • T s
f : α →ₛ E
hf : Integrable (⇑f) μ
this : ∀ x ∈ {x ∈ f.range | x ≠ 0}, T' (⇑f ⁻¹' {x}) = c • T (⇑f ⁻¹' {x})
x : E
hx : x ∈ {x ∈ f.range | x ≠ 0}
⊢ (c • T (⇑f ⁻¹' {x})) x = c • (T (⇑f ⁻¹' {x})) x

Tactic:
_root_.smul_apply

Goal state after:
α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T T' : Set α → E →L[ℝ] F'
c : ℝ
h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T' s = c • T s
f : α →ₛ E
hf : Integrable (⇑f) μ
this : ∀ x ∈ {x ∈ f.range | x ≠ 0}, T' (⇑f ⁻¹' {x}) = c • T (⇑f ⁻¹' {x})
x : E
hx : x ∈ {x ∈ f.range | x ≠ 0}
⊢ c • (T (⇑f ⁻¹' {x})) x = c • (T (⇑f ⁻¹' {x})) x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
L : Type u_6
inst✝ : RCLike L
r : L
f : α → L
⊢ ∫ (a : α), f a / r ∂μ = (∫ (a : α), f a ∂μ) / r

Tactic:
simpa only [← div_eq_mul_inv] using integral_mul_const r⁻¹ f

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
s : Set E
inst✝ : SeparableSpace ↑s
hfi : Integrable f μ
hfm : Measurable f
hs : ∀ᵐ (x : α) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
y₀ : E
h₀ : y₀ ∈ s
h₀i : Integrable (fun x ↦ y₀) μ
hfi' : ∀ (n : ℕ), Integrable (⇑(approxOn f hfm s y₀ h₀ n)) μ
⊢ Tendsto (fun n ↦ SimpleFunc.integral μ (approxOn f hfm s y₀ h₀ n)) atTop (𝓝 (∫ (x : α), f x ∂μ))

Tactic:
simp only [SimpleFunc.integral_eq_integral _ (hfi' _), integral, L1.integral]

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
s : Set E
inst✝ : SeparableSpace ↑s
hfi : Integrable f μ
hfm : Measurable f
hs : ∀ᵐ (x : α) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
y₀ : E
h₀ : y₀ ∈ s
h₀i : Integrable (fun x ↦ y₀) μ
hfi' : ∀ (n : ℕ), Integrable (⇑(approxOn f hfm s y₀ h₀ n)) μ
⊢ Tendsto
    (fun n ↦
      if x : CompleteSpace E then
        if hf : Integrable (fun x ↦ (approxOn f hfm s y₀ h₀ n) x) μ then
          L1.integralCLM (Integrable.toL1 (fun x ↦ (approxOn f hfm s y₀ h₀ n) x) hf)
        else 0
      else 0)
    atTop
    (𝓝
      (if x : CompleteSpace E then
        if hf : Integrable (fun x ↦ f x) μ then L1.integralCLM (Integrable.toL1 (fun x ↦ f x) hf) else 0
      else 0))
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
⊢ Tendsto (u / v) l (𝓝 1) → u ~[l] v

Tactic:
· exact isEquivalent_of_tendsto_one

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
v v' : V
p p' : P
⊢ dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'

Tactic:
simpa using dist_triangle (v +ᵥ p) (v' +ᵥ p) (v' +ᵥ p')

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

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝¹ : Countable ι
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : ∀ (i : ι), Integrable f (μ i)
h : Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂μ i
i : ι
x✝ : X
⊢ 0 x✝ ≤ (fun a ↦ ‖f a‖) x✝

Tactic:
by positivity

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: case mp
R : Type u_1
inst✝² : PseudoMetricSpace R
inst✝¹ : Monoid R
inst✝ : LipschitzMul R
s t : Set R
s_bdd : Bornology.IsBounded s
t_bdd : Bornology.IsBounded t
bdd : Bornology.IsBounded (s ×ˢ t)
C : ℝ≥0
mul_lip : LipschitzWith C fun p ↦ p.1 * p.2
p a : R
a_in_s : a ∈ s
b : R
b_in_t : b ∈ t
eq_p : (fun x1 x2 ↦ x1 * x2) a b = p
⊢ ∃ a b, (a ∈ s ∧ b ∈ t) ∧ a * b = p

Tactic:
exact ⟨a, b, ⟨a_in_s, b_in_t⟩, eq_p⟩

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_6
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ∞
hf : MemLp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖f a‖ ^ p.toReal) x

Tactic:
by positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv

Goal state after:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s
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 : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
h : f =ᵐ[μ] g
⊢ setToFun μ T hT f = setToFun μ T hT g

Tactic:
by
  by_cases hF : CompleteSpace F; swap
  · simp [setToFun, hF]
  by_cases hfi : Integrable f μ
  · have hgi : Integrable g μ := hfi.congr h
    rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 h]
  · have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
    rw [setToFun_undef hT hfi, setToFun_undef hT hgi]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
h'f : MemLp (⇑f) 1 μ
g : ↥(α →₁ₛ[μ] E) := f.toLp h'f
A : ⇑f =ᵐ[μ] ↑↑↑g
⊢ setToFun μ T hT ⇑f = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x

Tactic:
rw [setToFun_congr_ae hT A, L1.setToFun_eq_setToL1 hT, L1.setToL1_eq_setToL1SCLM]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
h'f : MemLp (⇑f) 1 μ
g : ↥(α →₁ₛ[μ] E) := f.toLp h'f
A : ⇑f =ᵐ[μ] ↑↑↑g
⊢ (L1.SimpleFunc.setToL1SCLM α E μ hT) g = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖g a‖ ^ q) x

Tactic:
by positivity

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖f a‖ ^ p) x

Tactic:
by positivity

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖f a‖ * ‖g a‖) x

Tactic:
by positivity

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
𝕜 : Type u_2
inst✝⁶ : PseudoMetricSpace 𝕜
inst✝⁵ : TopologicalSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : Zero 𝕜
inst✝² : Zero β
inst✝¹ : SMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
c₁ c₂ : 𝕜
f : α →ᵇ β
⊢ 0 ≤ dist c₁ c₂ * dist f 0

Tactic:
by positivity

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝¹ : TopologicalSpace α
R : Type u_1
inst✝ : NonUnitalSeminormedRing R
f g : α →ᵇ R
⊢ 0 ≤ ‖f‖ * ‖g‖

Tactic:
by positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
⊢ ProperSpace E

Tactic:
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
⊢ ProperSpace E
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : α →ᵇ R
h : f * g = 0
hfg : ∀ (x : α), f x = 0 ∨ g x = 0
hfg' : ∀ (x : α), ‖(f + g) x‖ = max ‖f x‖ ‖g x‖
key : ∀ (c : ℝ), 0 ≤ c → (‖f + g‖ ≤ c ↔ max ‖f‖ ‖g‖ ≤ c)
⊢ ‖f + g‖ ≤ max ‖f‖ ‖g‖

Tactic:
rw [key]; positivity

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

Goal state before:
pp: α : Type u_1
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : SeminormedAddCommGroup E
f : C(α, E)
inst✝ : Fintype α
⊢ 0 ≤ ‖⇑f‖

Tactic:
by positivity

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

Goal state before:
pp: α : Type u_1
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : SeminormedAddCommGroup E
f : C(α, E)
inst✝ : Fintype α
⊢ 0 ≤ ‖f‖

Tactic:
by positivity

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 : CompactSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : C(α, β)
⊢ ‖f‖ * ‖f‖ ≤ ‖star f * f‖

Tactic:
rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
      ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompactSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : C(α, β)
⊢ ∀ (x : α), ‖f x‖ ≤ √‖star f * f‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β →ₛ F
hf_int : Integrable (⇑f) μ
⊢ ∫ (x : β), f x ∂μ = ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) • x

Tactic:
by
  simp_rw [← f.coe_toLargerSpace_eq hm]
  rw [SimpleFunc.integral_eq_sum _ hf_int]
  congr 1

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

Goal state before:
pp: case pos
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
borelize G

Goal state after:
case pos
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : ¬infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ infEDist y E * (↑δ.toNNReal)⁻¹ ≤ edist x y * (↑δ.toNNReal)⁻¹ + infEDist x E * (↑δ.toNNReal)⁻¹

Tactic:
← add_mul,

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : ¬infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ infEDist y E * (↑δ.toNNReal)⁻¹ ≤ (edist x y + infEDist x E) * (↑δ.toNNReal)⁻¹
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case cons
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
f : α → E
ι : Type u_7
s✝ : Finset ι
μ : ι → Measure α
T : ι → Set α → E →L[ℝ] F
C : ι → ℝ
hTs : ∀ (i : ι), DominatedFinMeasAdditive (μ i) (T i) (C i)
i : ι
s : Finset ι
his : i ∉ s
hs' : s.Nonempty
ih : (∀ i ∈ s, Integrable f (μ i)) → setToFun (∑ i ∈ s, μ i) (∑ i ∈ s, T i) ⋯ f = ∑ i ∈ s, setToFun (μ i) (T i) ⋯ f
hf : ∀ i_1 ∈ Finset.cons i s his, Integrable f (μ i_1)
⊢ setToFun (∑ i ∈ Finset.cons i s his, μ i) (∑ i ∈ Finset.cons i s his, T i) ⋯ f =
    ∑ i ∈ Finset.cons i s his, setToFun (μ i) (T i) ⋯ f

Tactic:
| @cons i s his hs' ih =>

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

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
hposbdd : ∫ (ω : α), max (f ω) 0 ∂μ ≤ μ.real univ • ↑r
⊢ ∫ (a : α), |f a| ∂μ ≤ (2 * μ univ * ↑r).toReal

Tactic:
←
    max_zero_add_max_neg_zero_eq_abs_self,

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
hposbdd : ∫ (ω : α), max (f ω) 0 ∂μ ≤ μ.real univ • ↑r
⊢ ∫ (a : α), max (f a) 0 + max (-f a) 0 ∂μ ≤ (2 * μ univ * ↑r).toReal
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
val✝ : OrderTop α
h : Tendsto f (pure ⊤) (𝓝 b)
this : rightLim f ⊤ = f ⊤
⊢ Tendsto (rightLim f) (pure ⊤) (𝓝 b)

Tactic:
tendsto_nhds_unique h (tendsto_pure_nhds f ⊤),

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
val✝ : OrderTop α
h : Tendsto f (pure ⊤) (𝓝 b)
this : rightLim f ⊤ = f ⊤
⊢ Tendsto (rightLim f) (pure ⊤) (𝓝 (f ⊤))
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in S, f x ∂μ))

Tactic:
refine Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun ε ε0 => ?_

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ
ε0 : 0 < ε
⊢ ∀ᶠ (x : ι) in atTop, ∫ (x : X) in s x, f x ∂μ ∈ Metric.closedBall (∫ (x : X) in S, f x ∂μ) ε
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case convert_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
this :
  Tendsto (fun x ↦ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop
    (𝓝 (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))
i : ι
hi : i₀ ≤ i
⊢ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s i, f x ∂μ = ∫ (x : X) in s i, f x ∂μ

Tactic:
setIntegral_sdiff (hsm i) hi₀ (h_anti hi),

Goal state after:
case convert_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
this :
  Tendsto (fun x ↦ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop
    (𝓝 (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))
i : ι
hi : i₀ ≤ i
⊢ ∫ (x : X) in s i₀, f x ∂μ - (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i, f x ∂μ) = ∫ (x : X) in s i, f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
⊢ IntegrableOn f (⋃ n, s i₀ \ s n) μ

Tactic:
· rw [← sdiff_iInter]
    exact hi₀.mono_set sdiff_subset

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : Countable G
inst✝ : SMulInvariantMeasure G α μ
hμ : μ ≠ 0
h : IsFundamentalDomain G s μ
⊢ μ s ≠ 0

Tactic:
have hc := measure_univ_pos.mpr hμ

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : Countable G
inst✝ : SMulInvariantMeasure G α μ
hμ : μ ≠ 0
h : IsFundamentalDomain G s μ
hc : 0 < μ univ
⊢ μ s ≠ 0
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
⊢ MeasurableSet (Ioi c)

Tactic:
by
  refine OuterMeasure.ofFunction_caratheodory fun t => ?_
  have : Nonempty R := ⟨c⟩
  simp only [length_eq]
  refine le_iInf fun a => le_iInf fun b => le_iInf fun h => ?_
  simp only [← length_eq]
  rw [← length_sdiff_botSet, inter_sdiff_right_comm, ← length_sdiff_botSet (s := t \ Ioi c),
    sdiff_sdiff_comm]
  grw [h]
  rcases le_total a c with hac | hac <;> rcases le_total b c with hbc | hbc
  · simp only [Ioc_inter_Ioi, f.length_Ioc, hac, hbc, le_refl, Ioc_eq_empty,
      max_eq_right, min_eq_left, Ioc_sdiff_Ioi, f.length_empty, zero_add, not_lt]
  · simp only [hac, hbc, Ioc_inter_Ioi, Ioc_sdiff_Ioi, f.length_Ioc, min_eq_right,
      ← ENNReal.ofReal_add, f.mono hac, f.mono hbc, sub_nonneg,
      sub_add_sub_cancel, le_refl,
      max_eq_right]
  · simp only [hbc, le_refl, Ioc_eq_empty, Ioc_inter_Ioi, min_eq_left, Ioc_sdiff_Ioi,
      f.length_empty, zero_add, or_true, le_sup_iff, f.length_Ioc, not_lt]
  · simp only [hac, hbc, Ioc_inter_Ioi, Ioc_sdiff_Ioi, f.length_Ioc, min_eq_right,
      le_refl, Ioc_eq_empty, add_zero, max_eq_left, f.length_empty, not_lt]

Goal state after: